Banach spaces with few operators

Bernard Maurey

Laboratoire d’Analyse et Mathématiques Appliquées, UMR 8050

Université de Marne la Vallée

Boulevard Descartes, Cité Descartes, Champs sur Marne 77454 Marne la Vallée Cedex 2, FRANCE

Preprint submitted to Elsevier Preprint 18 July 2003

1 Introduction

Several natural questions about the linear structure of infinite-dimensional normed spaces, that were asked since the early days of the theory, remained without answer for many years. Here are two examples (in this article, Banach space means infinite-dimensional Banach space, real or complex):

A. Is it true that every Banach space is isomorphic to its (closed) hyperplanes?

B. If a Banach space X is isomorphic to every infinite-dimensional subspace of itself, does it follow that X is isomorphic to ℓ_{2}?

These two questions come from Banach’s book [Ba]. Question B was called the homogeneous Banach space problem in modern times. Question A is attributed to Banach; actually, Banach’s book contains a weaker question, formulated in terms of linear dimension, which amounts to asking whether every Banach space embeds isomorphically in its hyperplanes; this simply asks whether X is isomorphic to some proper subspace of itself. Let us formulate two other general questions, less ancient than the first two. Recall that a sequence (e_{n})_{n≥0} of non zero vectors is unconditional if there exists a constant C such that

∥∑_{i=1}^{m} ±a_{i}e_{i}∥ ≤ C ∥∑_{i=1}^{m} a_{i}e_{i}∥

for every m ≥ 0, all scalars (a_{i})_{i=1}^{m} and every choice of signs ±1; the best constant C is called unconditionality constant of (e_{n})_{n≥0}. An unconditional sequence is a basis for its closed linear span; unconditional bases were earlier called absolute bases, see [BP].

C. Does every Banach space contain an infinite unconditional basic sequence?

D. Is it possible to decompose every Banach space as a topological sum of two infinite-dimensional subspaces?

Question D was formulated around 1970 by Lindenstrauss [L2]. Question C appears in Bessaga-Pełczyński [BP] in 1958, but was considered several years before, since it asks for a natural improvement of the classical result from Banach’s book, according to which every Banach space contains a subspace with basis.

All these questions have been answered during the last decade, most of them in the negative direction that seems to indicate that there is no hope for a structure theory of general Banach spaces. There is one notable example though of a positive answer, the homogeneous space problem; interestingly enough, one of the “negative” objects discovered during the period 1990–95 plays a little rôle in the positive solution of Question B: at some point in the proof, one has to exclude the possibility that the homogeneous space could be hereditarily indecomposable. Despite my rather pessimistic comments above, the results and examples obtained since 1990 represent a significant progress of our understanding of infinite-dimensional Banach spaces. The solutions of the different problems have various points of contact, and introduce new notions that underlie several of the constructions and proofs.

Let us agree that for the rest of this paper, the word subspace will always indicate an infinite-dimensional vector subspace of a Banach space (but not necessarily a closed subspace). Beside the preceding problems, one of the main questions that remained a mystery for years is the following: does every Banach space contain a subspace isomorphic to some ℓ_{p} (for 1 ≤ p < +∞) or to c_{0}? This question was settled negatively in 1974 by the famous example of Tsirelson [Ts], who constructed a reflexive Banach space that does not contain any ℓ_{p}. Tsirelson’s solution has had an enormous influence on most of what is discussed here. After being disappointed by Tsirelson’s example, the structureseekers had to look for more modest questions, for example the following.

E. Does every Banach space contain a reflexive subspace, or else a subspace isomorphic to either c_{0} or ℓ_{1}?

This question conforms to the common experience that non reflexivity is often related to the presence of c_{0} or ℓ_{1}; indeed, a theorem of James [J1] says that a Banach space X with unconditional basis is reflexive if and only if it does not contain isomorphs of c_{0} or ℓ_{1}. There is therefore a loose connection between Questions E and C; any counterexample to E has to be a counterexample to C as well.

As we have said before, it is obvious when we look back that the first giant step in the direction of all the results mentioned in this article was done around 1974 by Tsirelson, (in his only paper about Banach spaces, as he likes to point out!). As far as I know, Tsirelson’s space was the first example of a space where the norm is defined by an inductive procedure that “forces” some specific property to hold, but somehow, nothing more than the desired property. The same year, Krivine proved the finite-dimensional counterpart, that goes in the opposite (positive) direction and says roughly that every Banach space contains ℓ_{p}^{n}’s of arbitrary large finite dimension n. Almost 20 years passed before Tsirelson’s breakthrough was extended to a solution of the above mentioned problems; during these years, it was still hoped by many that techniques using Krivine’s ideas could lead, for example, to a positive solution of question C.

A difficulty common to these questions is that one has to analyze whether or not some particular phenomenon will occur in every subspace of a given Banach space X; this rather vague question can be put to precise terms as follows: what do we know about subsets A of the unit sphere of X that meet every infinite-dimensional vector subspace? Here, I want to call such a set A a (linearly) inevitable set (these sets were called in [GM1] by the inexpressive term asymptotic; I feel that one should reform this poor terminology). Can we say that two inevitable sets A_{1}, A_{2} are in a sense so big that they must intersect, or at least almost intersect, meaning that dist(A_{1}, A_{2}) = 0? An equivalent question asks whether every enlargement A_{ε} of an inevitable set A must contain an infinite-dimensional vector subspace. This type of problem reminds Ramsey theory, but nobody could exploit this analogy before Gowers, in his dichotomy theorem (see below). It was realized by V. Milman (see [M1,M2]), a few years before Tsirelson’s example, that if we can prove that any two inevitable sets A_{1} and A_{2} in the unit sphere of a given space X almost intersect, then X must contain some ℓ_{p}, p ∈ [1, ∞), or c_{0}. In [M1], Milman defines a notion of spectrum for a uniformly continuous function on the unit sphere, and shows that when the spectrum of every such function is non-empty, then X contains some ℓ_{p} or c_{0}; if inevitable sets almost intersect, then this spectrum is non empty. This question about intersecting inevitable sets is not totally absurd, since the answer is positive in one case: the result of Gowers’ paper [G1] implies that any two inevitable subsets of the unit sphere of c_{0} almost intersect.

It follows indirectly that Tsirelson gave the first example of two inevitable sets A_{1} and A_{2} that are separated by some δ > 0, that is ∥a_{1} − a_{2}∥ ≥ δ for all a_{1} ∈ A_{1}, a_{2} ∈ A_{2}. Tsirelson’s space thus gives us a first clue, but the real start was the modification of Tsirelson’s space constructed by Schlumprecht [Sc], which provides us with an infinite sequence (A_{n})_{n≥0} of such separated inevitable sets, giving a lot of possibilities for coding things in the sphere. This will be explained in section 4.

For Gowers and the author of this article, this was the decisive information for constructing a space with no infinite unconditional sequence; during the summer of 1991, after hearing Schlumprecht at the Banach space conference in Jerusalem, we both constructed an example of a space with no unconditional sequence; the two examples X(G) and X(M) were very similar. Gowers’ example was the first to be presented to a few specialists. The construction will be indicated in section 7; it is rather intricate, but the fundamental principle, that seems to me very clear, is presented in a separated section (section 4) containing the following partial result: if we have a sequence of well separated inevitable sets in a Banach space X, then for any given C ≥ 1 we can renorm X in such a way that any infinite basic sequence in X has unconditionality constant larger than this C in the new norm. Obviously we are on the way to a negative answer to question C. The reader who wants to get an idea about what is going on with the failure of unconditionality must read section 4. The technical problem for solving question C itself is to mix the simple idea from section 4 with an inductive definition of a norm a la Tsirelson-Schlumprecht. This will be done in section 7.

Schlumprecht’s example is an ad hoc example of a space containing a sequence of separated inevitable subsets (A_{n}), but this strange situation did not seem likely to happen in the most regular of all spaces, namely ℓ_{2}. It was therefore a big surprise when Odell and Schlumprecht showed that one can move the sets (A_{n}) from Schlumprecht’s space S to ℓ_{2} by a non linear procedure, and get a sequence (B_{n}) of inevitable subsets in the unit sphere of ℓ_{2} that are somewhat orthogonal [OS]. On the other hand, it is still unknown whether Tsirelson’s space T contains a sequence of well separated inevitable subsets, or whether T satisfies the opposite property of having bounded distortion. This subject of distortion will not be discussed here. The reader may consult [BLi, Chapter 13] for a complete description, that includes the results of [OS].

The space X(G) (or X(M)) was basically intended to be a counterexample to the unconditional sequence problem C, but it quickly appeared to have a very radical property: when seeing Gowers’ preprint about X(G), W.B. Johnson observed that the space had the additional property that for every pair (Y, Z) of subspaces, we have that inf ∥y − z∥ is zero, when y and z run in the unit spheres of Y and Z. This means that Y and Z can never form a topological sum; in other words, every subspace of X(G) is indecomposable. The paper [GM1] therefore introduced the first example X_{gm} of a hereditarily indecomposable space (in short: HI), thus solving negatively question D. As it happens sometimes, question D was solved by proving much more than asked. Obviously, a HI space cannot contain an infinite unconditional sequence, since the span of such a sequence is clearly decomposable (in odd and even, for example), and X_{gm} of course solves negatively question C. Finding examples of indecomposable spaces that are not HI took some more time (the shift space X_{s} mentioned later in this introduction is such an example; other examples are given by non-HI duals of HI spaces, as given by Ferenczi or Argyros-Felouzis, see below); at this point of the story, it seems easier to get a HI space than a genuine indecomposable space (one that is not HI)!

The HI spaces are very rigid, in the following sense: every bounded linear operator T on a complex HI space X has the form λId + S, where λ ∈ ℂ and S is a strictly singular operator. Let us recall that a famous open question is the existence of a Banach space X such that every operator on X would have the form λId + K, with K compact. The existence of HI spaces is far from solving this “λId + K problem”, but it does give comparable spectral consequences, because it is known that strictly singular operators allow to extend the classical Riesz spectral theory of compact operators (see [LT, 2.c] for example). It follows that operators on a complex HI space X have a discrete spectrum, consisting of a converging sequences of eigenvalues, together with the limit (that ought to be λ, if T = λId + S); if T is Fredholm on a HI space X, it has always index 0 (this result also holds in the real case). This implies easily that a HI space is not isomorphic to any proper subspace, thus solving negatively question A (it solves exactly the question from Banach’s book, but the question had become more popular under the weaker form that asks whether hyperplanes are isomorphic to the whole space; as for question D, this weaker form was solved by proving much more). These relatively easy spectral consequences of the HI property are presented in section 5; the interested reader can jump directly to that section.

Gowers has constructed several further examples; the most striking is a space X_{g} not containing any reflexive subspace, and containing no c_{0} or ℓ_{1} [G3]; this is a counterexample to question E. The construction is partly similar to that of [GM1], but in a much more difficult context which requires new ideas; I like to think that this example X_{g} is a sort of generalized James-tree space (see [J3,LS]), where every vector in the unit ball of X^{∗} is a potential node for the tree. To some extent, constructing this space X_{g} is like building X_{gm} on this abstract tree.

It was very tempting to relate the HI property of a Banach space X to the fact that X does not contain any infinite unconditional sequence. This was done by T. Gowers in his beautiful “dichotomy Theorem” [G5], see also [G6].

Let X be an arbitrary infinite-dimensional Banach space. Either X contains an infinite unconditional sequence, or X contains a HI subspace.

For proving this theorem, Gowers has found a very satisfactory way to extend Ramsey theory to a linear setting. This result also explains why it is not so surprising to get a HI space when one simply looks for a space with no infinite unconditional sequence. It also gives a profound reason for introducing HI spaces. Some questions about general Banach spaces can then be divided into two cases: the “usual” case, where unconditional bases exist, and the “exotic” case, where we may find a HI space in the middle of our road. This dichotomy was the missing piece for the solution of question B, for which Komorowski and Tomczak had proved the following result: a Banach space X with unconditional basis, non trivial type and not containing ℓ_{2}, contains a subspace not isomorphic to X (see [KT,KTb]). They deduce the following partial solution to question B: if a homogeneous Banach space X contains an infinite unconditional sequence, then X is isomorphic to ℓ_{2}. Now, by the dichotomy theorem, a homogeneous space X must contain an unconditional sequence: otherwise, it contains a HI subspace, hence it is HI itself by homogeneity, but clearly a HI space is not homogeneous! Combining [KT] and [G5] solves question B: every homogeneous space X is isomorphic to ℓ_{2}.

As we have said, every operator on a complex HI space X has the form λId+S, where S is strictly singular. Ferenczi [F2] proved a more general result (which was previously checked by hand in [GM1] for the specific example X_{gm}): every operator from a subspace Y ⊂ X to X has the form λi + S, where i is the inclusion map and S is strictly singular. Actually, this property of a complex Banach space X is clearly equivalent to the fact that X is HI. One could be fully satisfied to have examples of spaces that have as few operators as possible. However, there are other questions which assume some structure for the space and then ask whether further structure follows. Roughly speaking, the results of [GM2] state that given an algebra of maps satisfying certain conditions, one can replace the multiple of the inclusion map in the statement above by the restriction to Y of some element of the algebra. These examples illustrate the following principle: you will find inside the space constructed in this way, nothing more than what you decide to put at the start; we obtain in this way a space X_{s} with an isometric right shift S but no infinite unconditional sequence; every operator T ∈ L(X_{s}) is a perturbation of an absolutely summable series of iterates of the shift S and its adjoint. In this space all complemented subspaces are trivial (finite-dimensional or finitecodimensional), which makes this space a bizarre example of a prime space (a Banach space isomorphic to every infinite-dimensional complemented subspace; here the shift and its iterates provide isomorphisms between X_{s} and the finite-codimensional subspaces). Recall that the “normal” known prime spaces are c_{0}, ℓ_{p}, for p ∈ [1, +∞) (see [Pe]) and also ℓ_{∞} (see [L1]). These results from [GM2] are described in section 9. This paper [GM2] also gives a space isomorphic to its subspaces of codimension two but not to its hyperplanes and a space isomorphic to its cube but not to its square (Gowers had previously given in [G4] the first example of this cube-not-square phenomenon).

The shift space X_{s} provides a good illustration for Gowers’ dichotomy theorem. Indeed, this space X_{s} does not contain any infinite unconditional sequence; but X_{s} is not HI, because it has a non trivial operator, the shift S; one can also see directly that for every λ ∈ ℂ with modulus one, we may find a subspace Y_{λ} of X on which the shift S is almost equal to λId (we generate Y_{λ} from a sequence of almost eigenvectors of S, corresponding to the spectral value λ); when μ ≠ λ, the two subspaces Y_{λ} and Y_{μ} can be chosen to form a topological direct sum, and this explains why X is not HI. By Gowers’ dichotomy theorem, every subspace of X_{s} must contain a further subspace which is HI. One can check that the subspaces Y_{λ} are examples of such HI subspaces.

Let us mention further results in the HI direction. Kalton [Ka] has constructed an example of a quasi-Banach space X with the very strange property that there is a vector x ≠ 0 such that every closed infinite-dimensional subspace of X contains x. It follows that this quasi-Banach space does not contain any infinite basic sequence. This example is related to an example of Gowers [G2] of a space with unconditional basis not isomorphic to its hyperplanes; Kalton’s construction uses the technique of twisted sums together with the properties of the space in [G2]. Argyros and Deliyanni [AD] constructed HI spaces without using Schlumprecht’s space, by a technique called mixed Tsirelson’s norms; their example is also an asymptotically-ℓ_{1} space. V. Ferenczi [F1] constructed a uniformly convex HI space, by adding to the tools from [GM1] the tool of complex interpolation for families of Banach spaces developed by Coifman, Cwikel, Rochberg, Sagher and Weiss. P. Habala [Ha] constructed a space such that no infinite-dimensional subspace has the Gordon-Lewis property (this property is a weak form of unconditional structure for a Banach space), and Ferenczi-Habala unite in [FH].

Ferenczi showed that the dual of the example X_{gm} in [GM1] is also HI and that every quotient of this space is still HI [F4]. This type of question is not yet clarified in general. What is clear is that the dual of a reflexive indecomposable space is indecomposable; therefore, if every quotient of a reflexive space is HI, then every subspace of a quotient is indecomposable and this property clearly passes to the dual. However Ferenczi [F4] gave an example of a HI space such that the dual is not HI. This phenomenon was widely extended by Argyros and Felouzis [AF], who showed that for every p > 1, the space ℓ_{p} (or c_{0} when p = +∞) is isomorphic to a quotient of some HI space X_{p} (the corresponding result is obviously false for ℓ_{1}). The dual space X_{p}^{∗}, which contains ℓ_{q}, is clearly not a HI space. This sheds more light on the non stability of the HI property under duality. Argyros and Felouzis obtain this quotient result as a consequence of a factorization theorem through HI spaces: every operator which is thin in some sense factors through a HI space. This is, in a way, a very perverse result, as one is normally trying to factor operators through nice spaces, like what Davis, Figiel, Johnson and Pełczyński did in [DFJP] for weakly compact operators! A sketch for the results of [AF] is presented in section 8. A variant of the interpolation method of [DFJP] will be used for the general factorization result. Obtaining ℓ_{p} as quotient of a HI space requires a clever construction which is also sketched in section 8.

1.1 Other spaces with few operators

As we have said, complex HI spaces have few operators. But HI spaces are not the first examples of spaces that have, in some sense, few operators, and there is a series of works on this theme. The space constructed in [GM1] could be regarded as the infinite-dimensional analogue of the random finitedimensional spaces introduced by Gluskin [Gl1,Gl2]. It was shown by Szarek, that operators on Gluskin’s spaces all approximate, in a certain precise sense, multiples of the identity. Although the proofs in [GM1] and [Gl1] are very different, there are some points of contact, such as the idea of constructing a unit ball with just a few “spikes”. Since these spikes must, under a wellbounded operator, map to other spikes, if they can be chosen in a very nonsymmetrical way, a well-bounded operator is forced to approximate a multiple of the identity. Gluskin achieved the lack of symmetry by choosing the spikes randomly, whereas in [GM1] they were constructed directly (in the dual space) using some infinite (and not too difficult) combinatorics.

Gluskin’s spaces were “glued” together to produce several infinite-dimensional examples of interest by, amongst others, Bourgain [Bo], Szarek [Sz1,Sz2], Mankiewicz [Ma] and Read [Re]. Some of these gluing methods were not at all straightforward. Several of the properties of these spaces are shared by the spaces constructed in [GM1] and [GM2]. For instance, Bourgain’s example is a complex Banach space X such that X and its opposite space \overline{X} are not isomorphic complex spaces. Szarek’s example is a real reflexive Banach space X that does not admit a complex structure (because X does not have any operator T such that T^{2} = −Id). The complex HI space from [GM1] has Bourgain’s property, while its real version satisfies Szarek’s conclusion (but these two facts are not stated in [GM1]).

A related direction is the search for examples in Banach algebras. One particular question is the existence of non-zero homomorphisms from the algebra of bounded linear operators L(X) to ℂ, or equivalently of closed ideals I such that L(X)/I is ℂ, or more generally a commutative Banach algebra. The result on the “shift space” in [GM2] can be compared to those of Mankiewicz [Ma]: we have in [GM2] another example of a complex Banach space such that there exists a (unital) algebra homomorphism from L(X) into a commutative Banach algebra. It follows (as is recalled in [Ma]) that X is not isomorphic to any power Y^{n} of a Banach space Y, for any n ≥ 2 (see also Figiel [Fi] for an early related example of a reflexive space not isomorphic to its square). Indeed, if φ is a non zero multiplicative functional from L(X) to ℂ, and if X = Y^{n}, there is a natural homomorphism i from M_{n} to L(X). But then φ∘i would be a non zero multiplicative functional on M_{n}, which is not possible, as soon as n ≥ 2.

Let us mention two results that have little in common with the present paper, apart from their statements. Kalton and Roberts [KR] have given results for quasi-Banach spaces. They constructed subspaces of L_{p}, 0 < p < 1, where the only continuous linear operators are the multiples of the identity. Shelah [Sh] has constructed examples of non-separable Banach spaces for which every bounded linear operator has the form λId + S, where S has separable range. The proof used some heavy machinery from logic (diamond axiom, or V = L) which was avoided later in [ShS]. In the fall of 1999, Argyros and Tolias announced that they can obtain the same conclusion as Shelah, from a space X_{a} that is a relative of the HI family (see [AT]). This new example lies somewhere between the spaces from [AD] and [G3]. As in the case of the James tree space, the space X_{a} is a space of sequences indexed by a tree. The dual X_{a}^{∗} is non separable, but every operator on X_{a}^{∗} has the form λId + S, where S has separable range.

At the end of this introduction, I must confess that part of this paper has been realized by the well-known cut-and-paste technique, applied to the two papers [GM1] and [GM2]; as a result, some portions of the present paper may sound curiously too English to the reader: they were stolen from T. Gowers’ writing of our papers.

2 Ancestors

In order that the reader sees where our methods come from, we have to say a few words about the ancestor of this story, namely the space T constructed by Tsirelson [Ts] (see also [FJ,CS]). Let us first fix some notation about sequence spaces. Let c_{00} denote the space of finitely supported scalar sequences. Given two subsets E, F ⊂ ℕ, we say that E < F if max E < min F. Let (e_{n})_{n=0}^{∞} be the standard basis of c_{00}. Given a vector x = ∑_{n=0}^{+∞} x_{n}e_{n} its support, denoted supp(x), is the set of n such that x_{n} ≠ 0. If x, y ∈ c_{00}, we write x < y when supp(x) < supp(y). We also write n < x when n ∈ ℕ and n < min supp(x). If x_{1} < … < x_{n}, then we say that the vectors x_{1}, …, x_{n} are successive.

Let C > 1 be fixed and let B_{T}^{∗} be the smallest convex subset of B(c_{0}) ∩ c_{00} that contains ±e_{i} for each i ≥ 0 and such that

x_{1}^{∗} + ⋯ + x_{n}^{∗} ∈ C B_{T}^{∗}

whenever x_{1}^{∗}, …, x_{n}^{∗} are successive in B_{T}^{∗} and n < x_{1}^{∗}. The Tsirelson norm is then defined on c_{00} by

∥x∥_{T} = sup{|x^{∗}(x)| : x^{∗} ∈ B_{T}^{∗} }

and T is the completion of c_{00} under this norm. This point of view agrees with Tsirelson’s original presentation. A dual formulation, given by Figiel and Johnson [FJ], introduces the Tsirelson norm as the smallest norm on c_{00} satisfying ∥e_{i}∥ = 1 for every i ≥ 0 and

∥x_{1} + ⋯ + x_{n}∥ ≥ \frac{1}{C} ∑_{i=1}^{n} ∥x_{i}∥

for every n ≥ 2 and all n < x_{1} < … < x_{n}. The original choice of C was C = 2; very interesting effects can be achieved by varying the constant C and mixing the norms obtained in this way (see [AD] as one example). Recall briefly why T is a counterexample to the ℓ_{p}-containment problem: since ∥x_{1}+⋯+x_{n}∥ ≥ \frac{1}{2} n whenever n norm one vectors satisfy n < x_{1} < … < x_{n}, it is quite clear that among ℓ_{p} spaces, only ℓ_{1} can embed into T; Tsirelson excluded this possibility by showing that T is reflexive; Figiel and Johnson gave a quantitative proof, which is closer to the spirit of this paper; they showed that T does not contain a (9/8)-isomorph of ℓ_{1} (see also [LT, 2.e.1]). But James [J2] proved (by the well-known James’ blocking technique) that if X contains an isomorph of ℓ_{1}, then X contains a (1+ε)-isomorph of ℓ_{1} for every ε > 0. Therefore T does not contain ℓ_{1}, and as a consequence, T does not contain any subspace isomorphic to an ℓ_{p} space or to c_{0}.

A second example, extremely important for us, is the space S constructed by Schlumprecht [Sc], which is a very useful variation of the construction of T. The constant factor C in Tsirelson’s construction

x_{1}^{∗} + ⋯ + x_{n}^{∗} ∈ C B_{T}^{∗}

is replaced by a variable value f(n) depending upon the number n of vectors in the sum; this function f should tend to infinity, but slowlier than any power n^{α}, α > 0. Schlumprecht chooses f(n) = log_{2}(n + 1). Let B_{S}^{∗} be the smallest convex subset of B(c_{0}) ∩ c_{00} containing ±e_{i} for each i ≥ 0 and such that

x_{1}^{∗} + ⋯ + x_{n}^{∗} ∈ f(n) B_{S}^{∗}

whenever x_{1}^{∗}, …, x_{n}^{∗} are successive in B_{S}^{∗} and n ≥ 2. The norm is then defined on c_{00} by

∥x∥_{S} = sup{|x^{∗}(x)| : x^{∗} ∈ B_{S}^{∗}}

and S is the completion of c_{00} for this norm. It is useful to observe that B_{S}^{∗} is obtained as the union of an increasing family of convex sets (B_{n})_{n≥0}, where B_{0} is the intersection of c_{00} with the unit ball of ℓ_{1}, and B_{n+1} is obtained from B_{n} by adding all vectors x of the form x = f(m)^{−1}(x_{1} + ⋯ + x_{m}) with m ≥ 2 arbitrary and x_{1}, …, x_{m} successive elements of B_{n}, and letting B_{n+1} be the convex hull of this extended set. This remark helps to show several properties of the space, for example the fact that the unit vector basis is 1-unconditional in S, by checking this inductively for ∥ . ∥_{n} = sup{|x^{∗}(x)| : x^{∗} ∈ B_{n}}.

An alternative description of S defines the norm as the solution of some implicit equation. We could say that the Schlumprecht norm on c_{00} is the smallest norm ∥ . ∥ on c_{00} such that the unit vector basis is normalized and

∥x_{1} + ⋯ + x_{n}∥ ≥ (∑_{i=1}^{n} ∥x_{i}∥)/f(n)

for every integer n ≥ 2 and every sequence of n successive vectors in S. In other words, the Schlumprecht norm is the solution to the implicit equation

∥x∥_{S} = max(∥x∥_{c_{0}}, sup{ f(n)^{−1}(∑_{i=1}^{n} ∥x_{i}∥_{S}) :

n ≥ 2, x = ∑_{i=1}^{n} x_{i}, x_{1} < … < x_{n}}).

3 Inevitable behaviours

In order to support some intuition for the notion of inevitable set (formally defined in the next section), we recall some easy facts, together with a few words about Schlumprecht’s space S. We begin by a well known blocking procedure for constructing ℓ_{1}^{n}, originating in James [J2].

Lemma 1 Let n ≥ 2 be an integer and 0 < ε < 1; suppose that N is an integer that can be written as N = n^{k} for some k ≥ 1, and let (x_{i})_{i=1}^{N} be norm one vectors in a normed space X, such that

∥∑_{i=1}^{N} ±x_{i}∥ ≥ (1 − ε/n)^{k}N = (n − ε)^{k}

for all signs ±1. There exists a sequence of n blocks y_{1}, …, y_{n} from (x_{i})_{i=1}^{N} that is (1 − ε)^{−1}-equivalent to the unit vector basis of ℓ_{1}^{n}.

Let us briefly sketch the proof: we consider successive generations of blockings, the first one being the obvious blocks x_{i} of length one, i = 1, …, n^{k}, and for ℓ = 0, …, k − 1 the next generation (numbered ℓ + 1) consists of n^{k−ℓ−1} new vectors z = ∑_{j=1}^{n} ε_{j}z_{j}, that are blocks of n consecutive elements z_{1}, …, z_{n} from the preceding generation, with some signs ε_{j} = ±1 chosen such that ∥ ∑_{j=1}^{n} ε_{j}z_{j}∥ is minimal. The last generation is just one single block using all n^{k} vectors. If on our way from ℓ = 0 to ℓ = k we never encounter an inequality of the form

∥∑_{j=1}^{n} ε_{j}z_{j}∥ ≥ (n − ε) max_{1≤j≤n} ∥z_{j}∥,

we get at the end a choice of signs (ε_{i})_{i=1}^{n^{k}} such that ∥ ∑_{i=1}^{N} ε_{i}x_{i}∥ < (n − ε)^{k}, which contradicts the hypothesis. We must therefore get at some stage a family of n blocks (z_{j}) that give, after we rescale them into (y_{j})

∥y_{j}∥ ≤ 1, j = 1, …, n; ∥∑_{j=1}^{n} ±y_{j}∥ ≥ n − ε

for all signs. It is easy to conclude, using simple convexity arguments, that this sequence (y_{j}) is well equivalent to the unit vector basis of ℓ_{1}^{n}.

Let us give an application of this lemma. Suppose that X is a Banach space with basis, and that f is a non-decreasing function on [1, +∞) that tends to +∞ at infinity, but slowlier than any power function t^{α} (α > 0, for example f(t) = log t). Suppose that for every N ≥ 2, all sequences of successive norm one vectors x_{1}, …, x_{N} in X satisfy ∥ ∑_{i=1}^{N} ±x_{i}∥ ≥ N/f(N). This is of course true for ℓ_{1}, or for an Orlicz sequence space ℓ_{M} with M(t) ∼ t/ log(1/t) as t → 0; it is also true, and more interesting in our context, for the Tsirelson space T or Schlumprecht’s space S.

If ε > 0 and n ≥ 2 are given, we can choose k large enough so that f(n^{k}) ≤ (1−ε/n)^{−k}. If N = n^{k}, we are in a position to apply Lemma 1. It follows easily that every subspace Y of X contains almost isometric copies of ℓ_{1}^{n}, spanned by small perturbations of successive vectors. We can describe a scheme L_{1} for getting subsets of the unit sphere of X which intersect every subspace, namely we can describe a decreasing sequence of sets L_{1}^{n} that intersect every subspace: let L_{1}^{n} consist of all norm one vectors x ∈ X such that there exist successive vectors y_{1} < … < y_{n} that are (1 + 2^{−n})-equivalent to the unit vector basis of ℓ_{1}^{n}, and

∥x − \frac{1}{n} ∑_{i=1}^{n} y_{i}∥ < 2^{−n}.

Then L_{1}^{n} intersects every subspace of X. Of course the intersection of the classes (L_{1}^{n})_{n≥1} is in general empty; what we call L_{1} is not a class of vectors, but a symbolic notation that is meant to represent the “idea” of any L_{1}^{n} with large n. In the present case, we say that the scheme L_{1} is inevitable in X.

The next natural thing to do, if we want to know whether X contains, not only ℓ_{1}^{n}s, but also ℓ_{1}, is to look for sums x + y, where x is in L_{1}^{m} and y in some L_{1}^{n}, with n much larger than m and much larger than the “size” of x; if (x + y)/∥x + y∥ is again in some L_{1}^{k}, with k large, then we are in the right direction for building ℓ_{1}. Let us write symbolically this set of vectors x + y, with x ∈ L_{1}^{m}, y ∈ L_{1}^{n}, as L_{1}^{m} ∗ L_{1}^{n}, and let the notation L_{1} ∗ L_{1} represent the scheme lim_{m} lim_{n} L_{1}^{m}∗L_{1}^{n}. This scheme L_{1}∗L_{1} is related to the limit behaviour of the norm of sums x+y, where a first large m is given, with a vector x ∈ L_{1}^{m}, and where n is then chosen much larger than m and than the “size” of x, with a vector y ∈ L_{1}^{n}; the notion of size of a vector x as to be precised: it could be for example the ℓ_{1}-norm of x, or the product of the ℓ_{∞}-norm of x by the smallest N such that supp(x) ≤ N. When the scheme L_{1} is inevitable, then obviously L_{1} ∗ L_{1}, L_{1} ∗ L_{1} ∗ L_{1}, and so on, are also inevitable (extending here the use of the word inevitable from subsets of the unit sphere to bounded sets A ⊂ X on which the norm function is bounded below by some κ > 0). When we look for ℓ_{1}, our first step is to check whether the schemes 2 L_{1} and L_{1} ∗ L_{1} have something in common. Indeed, when x and y are blocks of a given ℓ_{1} basis, with x ∈ L_{1}^{m}, y ∈ L_{1}^{n}, x < y and m < n, then \frac{1}{2}(x + y) belongs to L_{1}^{k(m)}, with k(m) → +∞.

This is precisely this first step that goes wrong with T. When we look for the scheme S_{2} = L_{1} ∗ L_{1}, we get a new inevitable scheme, distinct from 2 L_{1}.

Indeed, the Figiel-Johnson argument for proving that T does not contain ℓ_{1} shows that the inevitable set L_{1}^{m} ∗ L_{1}^{n} is well separated from the inevitable sets 2 L_{1}^{k}, when n is very large and k > 20, say. We may go further, and look for S_{3} = L_{1} ∗ L_{1} ∗ L_{1}, and so on; in T, we do not seem to get much more by having this infinity of possibilities; what we do get is that, in some sense, we cannot get more from T than what was input at the start: if we have x_{j} ∈ L_{1}^{n_{j}}, j = 1, …, k, with n_{j+1} very large with respect to the “size” of x_{1}, …, x_{j}, then (taking the defining constant C equal to 1/2)

\frac{k}{2} ≤ ∥∑_{j=1}^{k} x_{j}∥ ≤ (1 + ε) \frac{k}{2}.

Let us turn now to Schlumprecht’s space S. Since f(n) = log_{2}(n + 1) grows slowlier than any power of n, we get ℓ_{1}^{n}s everywhere, hence the scheme L_{1} is inevitable in S, and so are the next schemes S_{k}, k ≥ 2. But now we get something new and very interesting.

Lemma 2 Schlumprecht’s lemma. If x_{1}, …, x_{k} is a successive sequence in S, such that x_{j} ∈ L_{1}^{n_{j}} for j = 1, …, k, with n_{j+1} very large compared to ε^{−1}, to the size of x_{1}, …, x_{j} and to n_{1}, …, n_{j}, then

\frac{k}{f(k)} ≤ ∥x_{1} + ⋯ + x_{k}∥ ≤ (1 + ε) \frac{k}{f(k)}.

A sequence such as x_{1}, …, x_{k} will be called a rapidly increasing sequence (of ℓ_{1}^{n}s), in short RIS. Schlumprecht’s Lemma states that for a RIS, we almost get an equality between the norm of the sum ∥ ∑_{i=1}^{k} x_{i}∥ and the lower bound k/f(k) that was imposed by the definition of the space. Recall the observation made before the statement of the Lemma, that implies that for every k ≥ 1 and every subspace Y ⊂ S, we may find a RIS of length k consisting of small perturbations of vectors in Y. In other words, the normalized sums of RIS of length k form a nearly inevitable set A_{k} in S.

Since f(k) tends to infinity with k, it is clear that we get now infinitely many different inevitable classes in S. This was the first known example of this situation, and it was used (implicitly) by Schlumprecht in order to show that S is arbitrarily distortable. We obtain in this way what is for us the most important feature of Schlumprecht’s example: on one hand, we can find ℓ_{1}^{n} in every subspace; on the other hand, we can always combine very different ℓ_{1}^{n_{i}} in a RIS and get a behaviour arbitrarily far from the ℓ_{1} behaviour. There is an endless interplay between these classes of vectors: we see that S_{k} deviates more and more from the ℓ_{1}^{k} behaviour; if x_{1}, …, x_{k} is as above then clearly it is not a good ℓ_{1}^{k} basis, since f(k) can get as big as we want, but by Lemma 1 it can be blocked to give a good ℓ_{1}^{√\overline{k}}, say. Applying this blocking procedure with successive RIS of increasing lengths k_{j} will give a new RIS built from pieces in L_{1}^{\sqrt{k}_{j}}; the sum of this RIS can again be blocked to ℓ_{1}^{n}, and so on… This remark is crucial for the construction of the example X_{gm} in [GM1], and it is used intensively in section 7.

We can precise what we mean by “well separated” inevitable sets, in the framework of S. Schlumprecht’s Lemma does not only say that RIS deviate more and more from the ℓ_{1}^{n} behaviour, but it gives a precise estimate for the norm of the sum. With this information, it is possible to associate a class of functionals that almost norm the sums of RIS. For every n > 1, let A_{n} denote the set of normalized vectors which are multiple of the sum of a RIS sequence of length n; we define a class A_{n}^{∗} of functionals on S, consisting of all functionals of the form f(n)^{−1}(x_{1}^{∗} + ⋯ + x_{n}^{∗}), with ∥x_{i}^{∗}∥ ≤ 1, i = 1, …, n. If x ∈ A_{n}, then we may easily find x^{∗} ∈ A_{n}^{∗} such that x^{∗}(x) > 1/2. Indeed, this vector x has the form c n^{−1}f(n) ∑_{i=1}^{n} x_{i}, where (x_{i}) is a RIS and \frac{1}{2} < c ≤ 1 (apply Lemma 2 with ε < 1); we select for each i a norming functional x_{i}^{∗} for x_{i}, with supp(x_{i}^{∗}) = supp(x_{i}), and we set x^{∗} = f(n)^{−1}(x_{1}^{∗} + ⋯ + x_{n}^{∗}). We have ∥x^{∗}∥ ≤ 1 by the definition of S, and x^{∗}(x) ≥ c > 1/2. Furthermore, it can be shown (see [GM1,OS], or [BLi, Theorem 13.30]) that |x_{m}^{∗}(x_{n})| is small –depending upon min(m, n)– when x_{m}^{∗} ∈ A_{m}^{∗}, x_{n} ∈ A_{n} and m, n are very different. This gives a weak form of orthogonality, which is what we name almost biorthogonal inevitable system in the next section.

4 Coding with inevitable sets

Let X be a normed space and let S(X) be its unit sphere. We shall say that a subset A ⊂ S(X) is inevitable if A ∩ S(Y) ≠ ∅ for every infinite-dimensional (not necessarily closed) subspace Y ⊂ X. It is sometimes more convenient to work with the notion of a nearly inevitable set A ⊂ S(X), that has the property that inf{d(y, A) : y ∈ Y } = 0 for every infinite-dimensional subspace Y ⊂ X.

When X has a basis, it is easy to check that every subspace contains a further subspace which is spanned by a perturbation of a block sequence (see [LT, 1.a.11]). It follows that A is nearly inevitable in X when the above condition is true for Y an arbitrary block subspace. Observe that if we replace a nearly inevitable set A by the enlargement A_{ε} consisting of all x ∈ S(X) such that dist(x, A) < ε, we get an inevitable set A_{ε}.

The most obvious example of an inevitable set is the unit sphere S(Y) of a finite-codimensional subspace Y. On the sphere of ℓ_{2}, I would not be able to show any interesting example that can be described and proved inevitable using only pre-’90s ideas. The discussion of the preceding section shows that in a space with basis, close to ℓ_{1} in the sense above, all the classes L_{1}^{n} are inevitable; we also said that in Schlumprecht’s space S, we can even find a sequence of distinct inevitable sets.

A key observation made in [GM1] is that if a space X contains infinitely many inevitable sets that are all well disjoint from one another, then these can be used to construct an equivalent norm on X such that no sequence is C-unconditional in this norm. The idea is to use a certain coding, that has some similarity with what was done in [MR] for getting a statement about every subsequence of a given sequence; this is extended now to general vectors (general, as opposed to vectors from a given weakly null sequence); here, the coding action will be related to the numbering of a sequence of inevitable sets, while the coding action in [MR] was simply related to the numbering of the given sequence; we shall get in this way a statement about every subspace of a given space. Let us explain this obvious coding principle in a simplified setting: let Δ be a countable set, and let (B_{n})_{n≥0} be a sequence of disjoint non empty subsets of Δ; a coding function is an injective map σ, from the countable set of finite sequences of elements of Δ, to the natural numbers; a coding number N = σ(d_{1}, …, d_{m}) is thus associated in a 1 − 1 way to every finite sequence (d_{1}, …, d_{m}) of elements of Δ, where m varies in ℕ; the coding action builds a tree of finite sequences, by saying that (d_{1}, …, d_{m}, d) is a successor of (d_{1}, …, d_{m}) if and only if d belongs to B_{N} with N = σ(d_{1}, …, d_{m}), and saying that (d_{1}, …, d_{m}) is a node of the tree if (d_{1}, …, d_{j}) is a successor of (d_{1}, …, d_{j−1}) for j = 2, …, m. If (d_{1}, …, d_{m}) and (e_{1}, …, e_{n}) are two nodes of this tree and if d_{j}, e_{k} belong to the same set B_{ℓ}, then it follows that j = k and that d_{1} = e_{1}, …, d_{j−1} = e_{j−1}. If C is a class of subsets of Δ such that every C ∈ C intersects every set B_{n}, then clearly, for every C ∈ C we can construct arbitrarily long nodes (d_{1}, …, d_{n}) such that d_{j} ∈ C for j = 1, …, n. In this way, we have a tool that can affect every C ∈ C. Below, Δ will be a rich enough countable subset of the unit ball of the dual of a separable Banach space X and each C ∈ C will be a family of functionals which are norming for some subspace Y_{C} of X.

Let A_{1}, A_{2}, … be a sequence of subsets of the unit sphere of a normed space X and let A_{1}^{∗}, A_{2}^{∗}, … be a sequence of subsets of the unit ball of X^{∗}. We shall say that A_{1}, A_{2}, … and A_{1}^{∗}, A_{2}^{∗}, … are an almost biorthogonal inevitable system with constant δ > 0 if the following conditions hold for every integer n ≥ 1:

(i) the set A_{n} is inevitable; (ii) for every x ∈ A_{n} there exists x^{∗} ∈ A_{n}^{∗} such that x^{∗}(x) > 1/2; (iii) for every m ≥ 1 with n ≠ m, every x ∈ A_{n} and every x^{∗} ∈ A_{m}^{∗}, we have

|x^{∗}(x)| < δ.

The definition is interesting only when δ > 0 is small. It is not at all obvious that any Banach space contains an almost biorthogonal inevitable system with constant δ < 0.01 say. As far as I know, no example was known before the Schlumprecht space S appeared. The main result of this section is the following theorem, whose proof is taken almost verbatim from [GM1].

Theorem 3 Let r be an integer > 9 and let X be a separable normed space containing an almost biorthogonal inevitable system with constant δ = r^{−2}. Then there is an equivalent norm on X such that no sequence is r/9-uncon-ditional.

PROOF. We shall write the proof in the real case. Let ∥.∥ be the original norm on X and let A_{1}, A_{2}, … and A_{1}^{∗}, A_{2}^{∗}, … be the almost biorthogonal inevitable system with constant δ = r^{−2}. For each n ≥ 0 let Z_{n}^{∗} be a countable subset of ⋃_{0<λ<1} λA_{n}^{∗} such that for every x ∈ A_{n} there exists x^{∗} ∈ Z_{n}^{∗} with 0 < x^{∗}(x) − 1/2 < δ. Obviously the almost orthogonality relations (iii) still hold between the sets Z_{m}^{∗} and A_{n} when m ≠ n. Let Δ = ⋃_{n=1}^{∞} Z_{n}^{∗}. Next, let σ be an injection to the natural numbers from the countable collection of finite sequences of elements of Δ.

We shall now define a collection of functionals which we call special functionals. A special sequence of functionals is a sequence of the form z_{1}^{∗}, z_{2}^{∗}, …, z_{r}^{∗}, where z_{1}^{∗} ∈ Z_{1}^{∗} and, for 1 ≤ i < r, z_{i+1}^{∗} ∈ Z_{σ(z_{1}^{∗},...,z_{i}^{∗})}^{∗}. A special functional is simply the sum z^{∗} = z_{1}^{∗} + ⋯ + z_{r}^{∗} of a special sequence of functionals. We shall let Γ stand for the collection of special functionals. Let us define an equivalent norm |||.||| on X by

|||x||| = max(∥x∥, r sup{|z^{∗}(x)| : z^{∗} ∈ Γ}).

Let x_{1}, x_{2}, … be any sequence of linearly independent vectors in X. We shall show that it is not r/9-unconditional in the norm |||.|||. We shall do this by constructing a sequence of vectors z_{1}, …, z_{r}, generated by x_{1}, x_{2}, … and disjointly supported with respect to these vectors, with the property that

r |||∑_{i=1}^{r}(−1)^{i}z_{i}||| < 9 |||∑_{i=1}^{r} z_{i}|||.

Let X_{1} be the algebraic subspace generated by (x_{i})_{i=1}^{∞}. Since A_{1} is an inevitable set, we can find z_{1} ∈ A_{1} ∩X_{1}. This implies that z_{1} has norm 1 and is generated by finitely many of the x_{i}. Next we can find z_{1}^{∗} ∈ Z_{1}^{∗} such that 0 < z_{1}^{∗}(z_{1}) − 1/2 < δ. Now let X_{2} be the algebraic subspace generated by all the x_{i} not used to generate z_{1}. Since A_{σ(z_{1}^{∗})} is inevitable, we can find z_{2} ∈ A_{σ(z_{1}^{∗})} ∩ X_{2} of norm 1. We can then find z_{2}^{∗} ∈ Z_{σ(z_{1}^{∗})}^{∗} such that 0 < z_{2}^{∗}(z_{2}) − 1/2 < δ.

Continuing this process, we obtain sequences z_{1}, …, z_{r} and z_{1}^{∗}, …, z_{r}^{∗} with the following properties. Let n_{1} = 1, and n_{i+1} = σ(z_{1}^{∗}, …, z_{i}^{∗}) for 1 ≤ i < r. First, z_{i} ∈ A_{n_{i}} (thus ∥z_{i}∥ = 1) for each i. Second, z_{i}^{∗} ∈ Z_{n_{i}}^{∗} for each i (i.e. z_{1}^{∗}, …, z_{r}^{∗} is a special sequence of functionals). Third, z_{i}^{∗}(z_{i}) ∼ 1/2 for each i. Fourth, since σ is an injection, the z_{i}^{∗} belong to different Z_{n}^{∗}s, so |z_{i}^{∗}(z_{j})| < δ when i ≠ j since z_{i}^{∗} ∈ Z_{n_{i}}^{∗} and z_{j} ∈ A_{n_{j}}. Let us now estimate |||∑_{i=1}^{r} z_{i}|||. Since z_{1}^{∗}, …, z_{r}^{∗} is a special sequence, the triple norm is at least

r(∑_{i=1}^{r} z_{i}^{∗})(∑_{i=1}^{r} z_{i}) > r(r/2 − δr^{2}) = r(r/2 − 1) > r^{2}/3

(since r > 6). On the other hand, if (w_{i}^{∗})_{i=1}^{r} is any special sequence, let t be the maximal index i such that w_{i}^{∗} = z_{i}^{∗} (or let t = 0 if w_{1}^{∗} ≠ z_{1}^{∗}). Then

|∑_{i=1}^{r}(−1)^{i}w_{i}^{∗}(z_{i})| ≤ |∑_{i=1}^{t}(−1)^{i}w_{i}^{∗}(z_{i})| + |w_{t+1}^{∗}(z_{t+1})| + ∑_{i=t+2}^{r} |w_{i}^{∗}(z_{i})|.

Since σ is an injection, w_{i}^{∗} and z_{j}^{∗} are chosen from different sets Z_{n}^{∗} whenever i ≠ j or i = j > t + 1. By property (iii) this tells us that |w_{i}^{∗}(z_{j})| < δ. In particular, ∑_{i=t+2}^{r} |w_{i}^{∗}(z_{i})| < δr = 1/r. When i ≤ t we know that w_{i}^{∗} = z_{i}^{∗}, hence |1/2 − w_{i}^{∗}(z_{i})| ≤ δ, so |∑_{i=1}^{t}(−1)^{i}w_{i}^{∗}(z_{i})| ≤ 1/2 + t/r^{2}. It follows that

|∑_{i=1}^{r}(−1)^{i}w_{i}^{∗}(z_{i})| ≤ 1/2 + 1 + 2/r < 2.

We also know that ∑_{i≠j} |w_{i}^{∗}(z_{j})| ≤ δr^{2} = 1. Putting all these estimates together, we find that

r sup_{w^{∗}∈Γ}|w^{∗}(∑_{i=1}^{r}(−1)^{i}z_{i})| ≤ 3r.

Finally, by the triangle inequality, ∥ ∑_{i=1}^{r}(−1)^{i}z_{i}∥ ≤ r and ||| ∑_{i=1}^{r}(−1)^{i}z_{i}||| ≤ 3r, from which it follows that the sequence x_{1}, x_{2}, … was not r/9-unconditional in the equivalent norm.

5 HI spaces, spectral properties and consequences

Definition 4 Let X be an infinite-dimensional Banach space, real or complex. We say that X is indecomposable if X cannot be written as the topological direct sum of two infinite-dimensional closed subspaces Y_{1} and Y_{2}. We say that X is hereditarily indecomposable (in short, HI) if every closed infinitedimensional subspace Y of X is indecomposable, that is if no subspace Y of X can be written as the topological direct sum of two infinite-dimensional closed subspaces Y_{1} and Y_{2} of X.

Obviously, if X is HI then every subspace Y ⊂ X is HI. It is easy to see that a Banach space X is HI if and only if for all subspaces Y and Z of X, we have

inf{∥y − z∥ : y ∈ Y, z ∈ Z, ∥y∥ = ∥z∥ = 1} = 0.

We see that X is HI when the “angle” between any two subspaces Y and Z of X is equal to 0. Since every Banach space contains a subspace with basis, it is formal from the existence of any HI space that there exist HI spaces with monotone basis; the example X_{gm} in [GM1] has a basis, and it is also reflexive; much more difficult is another example due to Gowers [G3] of a space X_{g} without any reflexive subspace, and not containing c_{0} or ℓ_{1}; it follows by James [J1] that X_{g} does not contain any subspace with unconditional basis, hence by the dichotomy theorem, X_{g} must contain a HI subspace Y_{g}; of course Y_{g} has no reflexive subspace; saying that

there exist a HI space Y_{g} with no reflexive subspace

seems to be a clean way to present things, but [G3] left open the point of deciding whether the space X_{g} is already HI.

Definition 5 A bounded linear operator T ∈ L(X, Y) is strictly singular when there is no infinite-dimensional subspace X_{0} ⊂ X such that T restricted to X_{0} is an into isomorphism; in other words, for every X_{0} and ε > 0 there exists x ∈ X_{0} such that ∥T x∥ < ε ∥x∥.

It is standard to deduce that when T is strictly singular, we can construct in any subspace X_{0} of X a further subspace X_{1} ⊂ X_{0}, spanned by a normalized basic sequence, such that ∥T_{|X_{1}}∥ ≤ ε (see [LT, 2.c.4]). Observe that X is HI if and only if for every subspace Y ⊂ X, the quotient map π_{Y} : X → X/Y is strictly singular.

Let X be a complex Banach space and let T be a bounded linear operator on X. Let Ω_{T} be the set of all μ ∈ ℂ such that there exist c > 0 and a finite-codimensional subspace Z ⊂ X such that ∥T x − μx∥ ≥ c ∥x∥ for every x ∈ Z. Clearly Ω_{T} is open in ℂ and contains the complement of the spectrum Sp(T) of T. Saying that μ ∈ Ω_{T} implies that ker(T −μId) is finite-dimensional and that the range of T − μId is closed. This indicates that T_{1} = T − μId is semi-Fredholm when μ ∈ Ω_{T}, with generalized index ind(T_{1}) = dim(ker T_{1}) − codim(T_{1}(X)) finite or equal to −∞.

Let F_{T} = ℂ \ Ω_{T}; it is not difficult to show that this closed subset of Sp(T) is not empty. Let us give a rather simple argument for this. It is well known that U ∈ L(X) is Fredholm if and only if the class Û of U in the quotient algebra L(X)/K(X) is invertible (where K(X) denotes the ideal of compact operators); this quotient Banach algebra (the Calkin algebra) is not trivial when X is infinite-dimensional. It follows that the spectrum of T̂ in the Calkin algebra is not empty; this spectrum is the essential spectrum σ_{ess}(T) of T. Let λ be a point of σ_{ess}(T) with maximal modulus. Then T − μId is Fredholm for every μ ∈ ℂ such that |μ| > |λ|; by continuity of the index, all operators T − μId for |μ| > |λ| have index 0, because this is true when μ is large enough to make T −μId invertible. If T −λId was semi-Fredholm, then all T −μId for μ close to λ would be also semi-Fredholm, with the same (generalized) index; but this index must be 0 by the preceding argument, and T − λId would then be Fredholm, which was excluded from the beginning by the fact that λ belongs to the essential spectrum. This show that λ ∈ F_{T}.

If λ ∈ F_{T}, that is if λ ∉ Ω_{T}, we may construct by induction a normalized basic sequence (x_{n}) such that ∥T x_{n}∥ ≤ 2^{−n}ε, and get a subspace Y_{ε} ⊂ X such that ∥T y − λy∥ ≤ ε ∥y∥ for every y ∈ Y_{ε} (see again [LT, 2.c.4]).

Suppose now that X is a complex HI space. It is easy to see that F_{T} contains exactly one element λ in this case: indeed, if λ, μ ∈ F_{T}, we may find two subspaces Y and Z in X such that T ∼ λId on Y and T ∼ μId on Z; since the unit spheres of Y and Z almost meet, it follows that λ = μ. Let λ_{0} be this unique value, and let U = T − λ_{0}Id. Let Z be any subspace of X. Since λ_{0} ∈ F_{T}, there exists for every ε > 0 a subspace Y such that ∥U_{|Y} ∥ < ε. Since X is HI, the unit spheres of Z and Y almost meet, thus we may find z in the unit sphere of Z such that ∥Uz∥ < ε. This shows that U is strictly singular. We already obtained most of the next theorem.

Theorem 6 Let X be a complex HI space. Then every T ∈ L(X) can be written as T = λId + S, where λ ∈ ℂ and S is strictly singular. Thus every T ∈ L(X) is either strictly singular or Fredholm with index 0. Furthermore, the spectrum of T is either finite, or consists of a sequence (λ_{n}) converging to λ. In this second case, each λ_{n} ≠ λ is an eigenvalue of T with finite multiplicity.

PROOF. We proved that there exists a number λ ∈ ℂ such that T −λId = S is strictly singular. If λ ≠ 0, it is classical that T = λId + S is Fredholm with index 0, because it is a strictly singular perturbation of λId (see for example [LT, 2.c.10]). The property of eigenvalues is also well known. By the discussion above, if μ ≠ λ belongs to the spectrum of T, then μ is not in the essential spectrum of T, hence T − μId is Fredholm and not invertible, therefore μ is an eigenvalue of T with finite multiplicity.

In the real case, we have:

Corollary 7 Let X be a real HI space. Every T ∈ L(X) is either strictly singular or Fredholm with index 0.

PROOF. Let X be a real HI space and consider its complexification X_{ℂ} = X ⊕X. This space X_{ℂ} need not be a complex HI space, but it has the property that given three subspaces, the sphere of one of them almost meets the sum of the two other subspaces. This implies that the essential spectrum of any U ∈ L(X_{ℂ}) contains at most two points, say λ and μ (using the three subspaces property from above).

Let T ∈ L(X), and consider its complexification U = T_{ℂ} on X_{ℂ}. The spectrum of the complexified operator T_{ℂ} is invariant under complex conjugation, therefore μ = \overline{λ}. Either λ is real, and T − λId is strictly singular; then T = λId + (T − λId) is Fredholm with index 0 if λ ≠ 0, or T is strictly singular when λ = 0. Otherwise, we have λ ∈ ℂ \ ℝ; the part of the spectrum of T_{ℂ} contained in the upper half plane is finite or consists of a convergent sequence of eigenvalues with finite multiplicity, together with its limit λ or \overline{λ}. Then T_{ℂ} − νId is Fredholm with index 0 for every ν ∉ \{λ, \overline{λ}\}; in particular, T_{ℂ} is Fredholm with index 0, and the same holds for T.

Corollary 8 Let X be a HI space, real or complex. Then X is not isomorphic to any proper subspace. In particular, X is not isomorphic to its hyperplanes.

PROOF. Let T be an isomorphism from X into itself; then T is not strictly singular, hence it must be Fredholm with index 0 by Theorem 6 or Corollary 7 and thus T X = X.

These properties of operators on HI spaces were not immediately seen by the authors of [GM1]; Gowers actually constructed, before [GM1] was written, a modification X_{u} of X_{gm} and he showed that X_{u} is not isomorphic to its hyperplanes [G2]; this space X_{u} was thus the first declared example of a space not isomorphic to its hyperplanes. This example X_{u} is a very intriguing example of space with unconditional basis.

If X is a reflexive HI space, then the facts that the spectrum is countable and that T − μId is Fredholm with index 0 for all but one value μ = λ ∈ ℂ hold true for operators on X^{∗}, although X^{∗} need not be HI. As a consequence, the dual of a reflexive HI space is not isomorphic to any proper subspace, and a reflexive HI space is not isomorphic to any proper quotient. However, these results do not seem to answer the following question: does there exist a HI space X isomorphic to its dual X^{∗}? We can say that such an X must be reflexive, because X^{∗∗} ≃ X is HI and X ⊂ X^{∗∗}. An obvious remark is that the usual way to get simple examples of spaces isomorphic to their dual, namely X = Y ⊕ Y^{∗}, Y reflexive, has no chance to yield a HI space X!

Ferenczi [F2] has shown that, given a complex HI space X and a bounded linear operator T from a subspace Y of X to X, one can write T = λi_{Y,X} + S, where λ ∈ ℂ, S is strictly singular and i_{Y,X} denotes the inclusion map from Y to X. Clearly, Ferenczi’s result is a characterization of complex HI spaces.

Let us sketch Ferenczi’s proof. In some sense all subspaces of a HI space X intersect; we may define a net of subspaces of X that captures a good part of the structure of this HI space; the order of this net is not the inclusion, as it is not strictly true that any two subspaces have an infinite-dimensional intersection, but almost. … We say that Y ≤ Z if there exists a compact operator K : Y → X such that (i_{Y,X} + K)(Y) ⊂ Z. Given Z_{1}, Z_{2} ⊂ X there exists Y such that Y ≤ Z_{1} and Y ≤ Z_{2}. We could call “germ” of HI space an equivalence class of such nets, in a way to be made precise. An interesting class of examples is the family of spaces containing no infinite unconditional sequence but having only a finite set of germs of HI spaces [F3]. The example X_{g} of Gowers [G3] of a HI space with no reflexive subspace gives an example of “non reflexive germ”.

Ferenczi’s proof consists essentially in showing that the space of “germs” of operators on X is a Banach field, hence isomorphic to ℂ. A germ of operator is an equivalence class for the relation where T_{1} ∈ L(Z_{1}, X) and T_{2} ∈ L(Z_{2}, X) are equivalent if we can find a subspace Y ≤ Z_{1}, Z_{2} such that the operator T_{1} ∘ (i_{Y,X} + K_{1}) − T_{2} ∘ (i_{Y,X} + K_{2}) is compact on Y, where K_{1} and K_{2} are the compact operators that appear in the definition of the order.

6 Sequence spaces

The scalar field is 𝕂 = ℝ or ℂ. Since we want to include some of the results of Argyros-Felouzis, we shall work in an extended setting where the space of sequences is a space of sequences of vectors taken from some separable space V, or even more generally, from a sequence V = (V_{n}) of separable spaces. We let K denote the sequence (V_{n}) where V_{n} = 𝕂 for every n ≥ 0. The reader may decide that V = K until he wants to study the section about [AF]. The norm on V_{n} is denoted ∥ . ∥_{n}. The exposition is slightly simpler if we assume that each V_{n} is reflexive, but one can modify the construction in order to take care of the case V_{n} separable but not reflexive.

Let us denote by c_{00}(V) the space of vector sequences x = (v_{n})_{n≥0}, where v_{n} ∈ V_{n} for every n ≥ 0, and such that v_{n} ≠ 0 for only finitely many values of n. If x ∈ c_{00}(V), we call support of x the (finite) set supp x of integers n such that v_{n} ≠ 0. If x, y ∈ c_{00}(V), we say that x < y if supp(x) < supp(y). We also write n < x when n ∈ ℕ and n < min supp(x). If x_{1} < … < x_{n}, then we say that the vectors x_{1}, …, x_{n} are successive. An infinite sequence of successive non-zero vectors is also called a block basis and a subspace generated by a block basis is a block subspace. An interval of integers is a set of the form {n, n + 1, …, m} and the range of a vector x, written ran(x), is the smallest interval containing supp(x). It is convenient to write x = (v_{n}) ∈ c_{00}(V) as x = ∑_{n=0}^{+∞} v_{n} ⊗ e_{n}. Given a subset E ⊂ ℕ and a vector x as above, we write Ex for the vector ∑_{n∈E} v_{n}⊗e_{n}. We let P_{n} denote the projection corresponding to the set E_{n} = {0, …, n}; thus P_{n} is the natural projection from c_{00}(V) onto ∑_{k=0}^{n} V_{k} ⊗ e_{k}.

If V^{∗} is the sequence of duals (V_{n}^{∗}), we have a natural duality between c_{00}(V) and c_{00}(V^{∗}); the second space will be considered as space of functionals; we extend our terminology to functionals, for example we shall talk about successive functionals.

Let X (V) stand for the set of Banach spaces X obtained as the completion of c_{00}(V) for a norm ∥ . ∥ such that ∥Ex∥ ≤ ∥x∥ for every interval E, and ∥v_{n} ⊗ e_{n}∥ = ∥v_{n}∥_{n} for every n ≥ 0 and v_{n} ∈ V_{n}; when V = K, this means that (e_{n}) is a normalized bimonotone basis for X.

The reader must pay attention to the following fact: when V = K, every subspace Y of X contains a normalized sequence (y_{n}) which is a perturbation of a successive sequence (x_{n}) ⊂ X. This is clearly not the case in the vector case, as Y could be equal to V_{0} ⊗ e_{0} for example. However, if we assume that Y is a subspace of X such that for every n ≥ 0, the projection P_{n} is not an into isomorphism from Y to ∑_{k=0}^{n} V_{k} ⊗ e_{k}, then the standard gliding hump procedure can be extended to the vector setting in an obvious way. We shall say in this case that Y is a GH-subspace. We shall obtain interesting information only about GH-subspaces. This is a vacuous limitation in the scalar case, but it is an important one in the vector case. When the restriction to Y of every P_{n}, n ≥ 0, is strictly singular, we get that every subspace of Y is a GH-subspace.

Given a “slowly increasing” function f from [1, +∞) to [1, +∞) and a space X ∈ X (V), we shall say that X satisfies a lower f-estimate if, given any vector x ∈ X and any sequence of intervals E_{1} < … < E_{n}, we have that ∥x∥ ≥ f(n)^{−1} ∑_{i=1}^{n} ∥E_{i}x∥. In the dual formulation, this property means that whenever x_{1}^{∗}, …, x_{n}^{∗} are successive functionals with norm ≤ 1, then

∥(x_{1}^{∗} + ⋯ + x_{n}^{∗})/f(n)∥_{X^{∗}} ≤ 1.

Let X ∈ X (V) and x ∈ X. For every n ≥ 1, let

∥x∥_{(n)} = sup ∑_{i=1}^{n} ∥E_{i}x∥

where the supremum is extended to all families E_{1} < … < E_{n} of successive intervals. This quantity is clearly increasing with n, and ∥x∥ = ∥x∥_{(1)} by the monotonicity property of (e_{i}), since X ∈ X (V). Observe that the basis (e_{i}) satisfies ∥v_{i} ⊗ e_{i}∥_{(n)} = ∥v_{i}∥_{i} for every n ≥ 1. Clearly, ∥ . ∥_{(n)} is an equivalent norm on X, with ∥x∥ ≤ ∥x∥_{(n)} ≤ n ∥x∥; also, ∥Ex∥_{(n)} ≤ ∥x∥_{(n)} for every interval E.

The value of this norm at x is related to the fact that x can be broken into blocks that look like the unit vector basis of ℓ_{1}^{n}, or more accurately, into blocks that look like a ℓ_{1+}^{n} basis; this norm will be used in the definition of RIS as a substitute for the notion of ℓ_{1+}^{n}-average used in [GM1]. It is easy to check that when (x_{i})_{i=1}^{n^{2}} are successive and have norm ≤ 1, then x = n^{−2} ∑_{i=1}^{n^{2}} x_{i} satisfies ∥x∥_{(n)} ≤ 1 + 1/n. When (x_{i}) is a ℓ_{1+}^{n^{2}}-sequence with constant C, we get the additional fact that ∥x∥ ≥ 1/C. This type of vectors x for which the original norm is well equivalent to ∥x∥_{(n)} for some large n will play an essential rôle later. Not surprisingly, the proof of the next lemma, which states the existence of such vectors x when X ∈ X (V), is essentially identical to that of Lemma 1.

Lemma 9 Let X ∈ X (V) satisfy a lower f-estimate. Given a positive integer n and ε > 0, there exists an integer N(n, ε) such that for every sequence x_{1}, …, x_{N} of successive norm one vectors with N ≥ N(n, ε), we can find x of the form x = λ ∑_{i∈A} x_{i}, where A is some subinterval of {1, …, N}, such that ∥x∥ = 1 and ∥x∥_{(n)} ≤ 1 + ε.

Corollary 10 Let X ∈ X (V) satisfy a lower f-estimate. Then for every n ≥ 1 and ε > 0, every GH-subspace Y of X contains a vector y such that ∥y∥ = 1, ∥y∥_{(n)} ≤ 1 + ε and ∥P_{n}(y)∥ < ε.

PROOF. By the gliding hump procedure, we may select for every integer N a normalized sequence y_{1}, …, y_{N} of vectors in Y and successive vectors n < x_{1} < … < x_{N} in X such that ∥y_{i} − x_{i}∥ < ε/nN. The result follows from Lemma 9 and an obvious approximation argument.

7 A class of examples

The contents of this section come from [GM1] and [GM2]. The general strategy is as follows: we want to use the coding principle from section 4 and build special functionals that will somehow distinguish between sums like x + y + z and x − y + z; the difference in this section is that the inevitable sets are not given in advance, but must be constructed together with the norm, by an inductive procedure similar to the construction of Schlumprecht’s space; in short, our example is a Schlumprecht space with special functionals.

Another difference with section 4 is that in order to kill unconditionality, we want to push the unconditionality constant beyond C, not only for a given big C, but for every C. In section 4, we used special functionals of a fixed length, depending upon C. Here, we shall need special sequences with different lengths k, tending to infinity. Each length k will be used to prove that every basic sequence has unconditionality constant ≥ C_{k}, with lim_{k} C_{k} = +∞. In order to make the behaviour of the space easier to understand, we try to ensure that the different types of special functionals, of lengths k_{1} and k_{2} ≠ k_{1}, have as little interaction as possible; for this we shall prove various Lemmas that give almost orthogonality of several classes of vectors and functionals. These Lemmas are easy. Also, special functionals should not ruin the possibility of having some form of Schlumprecht’s Lemma in our space X; this will require some harder work.

The definition of special sequences is essentially taken from section 4, except that we must use sets (A_{ℓ}^{∗}) which are not given in advance, but are enriched step by step as the induction proceeds; we have to guess from the beginning that the chosen sets (A_{ℓ}^{∗}) will satisfy the needed properties at the end of the construction. However our choice is simple and inspired by what we saw about Schlumprecht’s space S at the end of section 3: the set A_{ℓ}^{∗} will consist of functionals of the form f(ℓ)^{−1}(x_{1}^{∗} + ⋯ + x_{ℓ}^{∗}), where f is some logarithmic function, fixed in the whole chapter. But in order to make sure that some set A_{ℓ} (consisting of normalized sums of RIS of length ℓ) will be inevitable and normed by A_{ℓ}^{∗}, we have to have some sort of Schlumprecht’s Lemma here (as explained at the end of section 3, it is important to have a precise estimate for the norm of the sum of a RIS, in order to be able to predict a class of almost norming functionals for the vectors in A_{ℓ}). An obvious start is to force a lower f-estimate, which ensures that the scheme L_{1} is inevitable in X, as well as all sets A_{ℓ} obtained from RIS of length ℓ. But the presence of the special functionals will ruin the possibility that X has a behaviour as regular as that of the Schlumprecht space, and this regular behaviour seemed important for getting Schlumprecht’s Lemma.

This difficulty will be solved in the following way: the usual Schlumprecht Lemma will hold for RIS that have length ℓ in some thin subset L of ℕ, and the corresponding norming functionals will be of the form f(ℓ)^{−1}(x_{1}^{∗} +⋯+x_{ℓ}^{∗}) as before. On the other hand, the special sequences y_{1}^{∗}, …, y_{k}^{∗} will have lengths k in another thin subset K, chosen “very far” from L. Then, the “special normalization” puts the special functional f(k)^{−1/2}(y_{1}^{∗} + ⋯ + y_{k}^{∗}) in the dual unit ball, so that a RIS sequence of length k ∈ K may have a sum whose norm is as big as k/\sqrt{f(k)} (to be compared to the “usual” smaller value k/f(k)).

The failure of unconditionality will appear exactly as in section 4, by constructing together two sequences (x_{j})_{j=1}^{k}, (x_{j}^{∗})_{j=1}^{k}, in such a way that x_{j} ∈ A_{n_{j}}, where n_{j} is the coding number assigned to the beginning x_{1}^{∗}, …, x_{j−1}^{∗} of the special sequence x_{1}^{∗}, …, x_{k}^{∗}, and (x_{j}) is also a RIS vector (a notion to be defined, essentially the normalized sum of a RIS); the special functional f(k)^{−1/2}(x_{1}^{∗} + ⋯ + x_{k}^{∗}) will then give to the sum ∑_{j=1}^{k} x_{j} an abnormally large norm, that would not be achieved by the alternate sum ∑_{j=1}^{k}(−1)^{j}x_{j}, for which the special functionals will fail to push the norm far beyond the usual k/f(k) estimate. In this way we show that the unconditionality constant is larger than \sqrt{f(k)}, and this can be done with arbitrarily large k ∈ K.

In order to generalize Schlumprecht’s Lemma, we analyze what properties of a slowly increasing function g are needed. We shall show that the class of possible functions is flexible enough to allow the construction of a function g which agrees with f on L and with \sqrt{f} on K (or on part of K), and stays between \sqrt{f} and f everywhere. To this end we introduce the family F of functions g : [1, ∞) → [1, ∞) satisfying the following conditions:

(i) g(1) = 1 and g(t) < t for every t > 1; (ii) g is strictly increasing and tends to infinity; (iii) lim_{t→∞} t^{−ε}g(t) = 0 for every ε > 0; (iv) the function t/g(t) is concave and non-decreasing;

(v) g(st) ≤ g(s)g(t) for every s, t ≥ 1.

We shall give a convenient representation formula for a subclass F_{0} of F. Let us denote by L the class of real functions on [0, +∞) that are non-decreasing and 1-Lipschitz on [0, +∞[, and tend to +∞ at +∞. Suppose that M belongs to L; the reader will easily check that the formula

(F)  g_{M}(t) = exp(∫_{0}^{ln t} \frac{du}{1 + e^{M(u)}} )

defines a function g_{M} ∈ F provided g_{M} tends to infinity at infinity, which means that ∫_{0}^{+∞}(1+e^{M(u)})^{−1} du = +∞. The only unpleasant point is to check that t/g(t) is concave, see the appendix. The idea of using this subclass F_{0} is taken from Habala’s paper [Ha].

One nice point about this subclass F_{0} is that it is extremely easy to glue together different functions from the class L: if we divide [0, +∞) into successive intervals (I_{n}), it is obvious that M belongs to L if and only if M is continuous, tends to +∞ at infinity and coincides on each interval I_{n} with a function M_{n} ∈ L. Therefore, a function g is in F_{0} when it is C^{1}, tends to +∞ at infinity and coincides on each interval e^{I_{n}} with a function g_{n} ∈ F_{0}.

If we let M(u) = ln(a + bu), with 0 < b ≤ a, we get a function in L, for which g_{M}(t) = (1 + \frac{b}{1+a} ln t)^{1/b}. We shall use the two special cases M_{0}(u) = ln(1 + u), with f_{0}(t) = 1 + \frac{1}{2} ln t (corresponding to a = b = 1) and M_{1}(u) = ln(3 + 2u), with f_{1}(t) = (1 + \frac{1}{2} ln t)^{1/2} = \sqrt{f_{0}(t)} (corresponding to b = 2, a = 3). For the rest of the paper we set f(t) = f_{0}(t) = 1 + \frac{1}{2} ln t; then \sqrt{f} = f_{1} also belongs to the class F_{0}. Notice that t^{−1/4} ln(t) decreases when t > e^{4}. Checking the value at t = e^{16}, we obtain that

(F_{1})  4t^{−1/4}f(t) ≤ 1 when t ≥ e^{16} .

We need a technical lemma.

Lemma 11 For every t_{0} = e^{u_{0}} with u_{0} ≥ 5 and t_{1} = e^{4u_{0}^{2}} there exists a function g ∈ F_{0} such that \sqrt{f} ≤ g ≤ f on [1, +∞), g = f on [1, t_{0}] and g = \sqrt{f} on [t_{1}, +∞). Similarly, there exists a function g ∈ F_{0} such that \sqrt{f} ≤ g ≤ f on [1, +∞), g = \sqrt{f} on [1, t_{0}] and g = f on [t_{1}, +∞).

For the proof, see the appendix.

We begin by gathering some lemmas that do not make use of any “special” construction. Amongst them is a version of Schlumprecht’s lemma (Lemma 14); Lemma 16 shows how to rebuild an ℓ_{1+}^{n} from a RIS, and Lemma 12 deals with almost orthogonality.

The first ingredient is the notion of a rapidly increasing sequence, in short RIS; it was already mentioned in sections 2 and 3, but not in precise terms. Let X ∈ X (V). We say that a sequence x_{1}, …, x_{r} of successive non-zero vectors in X satisfies the RIS condition if there is a sequence n_{1} < … < n_{r} of integers such that e^{2r^{3}} < n_{1}, ∥x_{i}∥_{(n_{i})} ≤ 1 for each i = 1, …, r and

\sqrt{f(n_{i})} > |ran(∑_{j=1}^{i−1} x_{j})|, i = 2, …, r.

If E is an interval, then the non-zero vectors of the sequence Ex_{1}, …, Ex_{r} clearly form a new RIS of length r_{1} ≤ r. Also, λx_{1}, …, λx_{r} is a RIS when 0 < |λ| ≤ 1 and every subsequence of a RIS is again a RIS.

Given g ∈ F, q ≥ 1 and X ∈ X (V), a (q, g)-form on X is defined to be a functional x^{∗} of norm at most one which can be written as ∑_{j=1}^{q} x_{j}^{∗} for a sequence x_{1}^{∗} < … < x_{q}^{∗} of successive functionals all of which have norm at most g(q)^{−1}. Observe that if x^{∗} is a (q, g)-form then ∥x^{∗}∥_{∞} ≤ g(q)^{−1} and |x^{∗}(x)| ≤ g(q)^{−1}∥x∥_{(q)} for any x ∈ X. Observe the obvious fact that a (q, g_{1})- form is a (q, g_{2})-form when g_{1} ≥ g_{2}.

We are looking for two kinds of orthogonalities: the first says that a long qform x^{∗} has a small action on a much longer ℓ_{1+}^{n}-average x, where q << n; this is simply given by the preceding relation |x^{∗}(x)| ≤ g(q)^{−1}∥x∥_{(q)}. The second kind says that a very long form has a moderate action on the sum of a RIS (thus, by Lemma 14 below, a small action on the normalized sum of this RIS).

Lemma 12 Suppose that (x_{1}, …, x_{r}) satisfies the RIS condition in a space X ∈ X (V). Let g ∈ F satisfy g ≥ \sqrt{f}. If x^{∗} is a (q, g)-form on X and q ≥ e^{2r^{2}}, then we have

|x^{∗}(x_{1} + ⋯ + x_{r})| ≤ 3.

PROOF. Let n_{1} < n_{2} < … < n_{r} be the sequence of integers associated to the RIS property of the sequence x_{1}, …, x_{r}. Let i ∈ {0, …, r} be such that n_{i} < q ≤ n_{i+1} (consider that n_{0} = 0 and that n_{r+1} is larger than q and n_{r}). Observe that the RIS condition implies ∥x_{j}∥_{∞} ≤ ∥x_{j}∥ ≤ 1 for each j = 1, …, r. The result follows from three easy inequalities,

|x^{∗}(∑_{j=1}^{i−1} x_{j})| ≤ ∥x^{∗}∥_{∞} |ran(∑_{j=1}^{i−1} x_{j})| ≤ g(q)^{−1} f(n_{i})^{1/2} ≤ 1,

|x^{∗}(x_{i})| ≤ ∥x_{i}∥ ≤ 1, and for j ≥ i + 1,

|x^{∗}(x_{j})| ≤ g(q)^{−1} ∥x_{j}∥_{(q)} ≤ g(q)^{−1} ∥x_{j}∥_{(n_{j})} ≤ g(q)^{−1},

so that |x^{∗}(x_{1} + ⋯ + x_{r})| ≤ 2 + rg(q)^{−1}. When we have q ≥ e^{2r^{2}}, we get that g(q) ≥ \sqrt{f(q)} = (1 + \frac{1}{2} ln q)^{1/2} ≥ r.

Remark 13 It is possible to prove a (1+ε)-version of Lemma 12 (see [GM2]), as well as (1 + ε)-versions of all lemmas that follow; we make the deliberate choice of giving simpler proofs, to the cost of introducing ridiculous constants 5, 15, 45, 75… in what follows.

The next Lemma is a variation on Schlumprecht’s Lemma. We need a more general statement than the one from section 3, that allows to play with different functions from the family F. In our example, the next lemma will be applied either to g = f or to g = \sqrt{f}, or actually to a variety of functions g between \sqrt{f} and f.

Lemma 14 Variant of Schlumprecht’s lemma. Let X ∈ X (V), g ∈ F, and let p ≥ 2 be an integer. Suppose that x_{1} < … < x_{r} in X satisfy ∥x_{i}∥_{(p^{r})} ≤ 1 for every i = 1, …, r. Let x = ∑_{i=1}^{r} x_{i} and suppose that

∥Ex∥ ≤ 1 ∨ sup{|x^{∗}(Ex)| : x^{∗} is a (q, g)-form, 2 ≤ q ≤ p}

for every interval E. Then

∥x∥ ≤ rg(r)^{−1}.

The painful proof is deferred to the appendix.

Corollary 15 Suppose that V = K. Let X ∈ X (K) and g ∈ F, g ≥ \sqrt{f}; suppose that X satisfies a lower f-estimate and that

∥x∥ ≤ ∥x∥_{c_{0}} ∨ sup{|x^{∗}(x)| : x^{∗} is a (q, g)-form, 2 ≤ q}

for every x ∈ X. Then X is reflexive. When V ≠ K, this is also true if each space V_{n} is reflexive.

PROOF. It follows immediately from the fact that X satisfies a lower festimate that the standard basis e_{0}, e_{1}, … is boundedly complete. Now suppose that it is not a shrinking basis. Then we can find ε > 0, a norm-1 functional x^{∗} ∈ X^{∗} and a sequence of successive normalized blocks y_{1}, y_{2}, … such that x^{∗}(y_{n}) ≥ 2ε for every n ≥ 1. It is easy to see that if x = \frac{1}{2} |A|^{−1} ∑_{i∈A} y_{i}, then ∥x∥_{(n)} ≤ 1/2 + 1/n for every A ⊂ ℕ such that |A| ≥ n^{2}, and x^{∗}(x) ≥ ε. It is clear that for every r, we may construct a RIS x_{1}, …, x_{r} with vectors x_{i} of the preceding form. By Lemma 12, we have x^{∗}(∑_{i=1}^{r} x_{i}) ≤ 3 for every long form, therefore Lemma 14 can be applied to the sequence \frac{1}{3}(x_{i}), proving that ∥ ∑_{i=1}^{r} x_{i}∥ ≤ 3 r/g(r). For r sufficiently large, this contradicts x^{∗}(∑_{i=1}^{r} x_{i}) > r ε. For the vector setting V, the reader may consult [AF].

The next simple Lemma is useful in conjunction with Lemma 14. It explains how to “rebuild” a good ℓ_{1+}^{n} basis from a RIS.

Lemma 16 Let X ∈ X (V), let m, r be integers such that e^{16} < m < r ≤ m^{4}; let x_{1} < … < x_{r} in X be such that

∥x_{i}∥ ≤ 1, i = 1, …, r, and ∥∑_{i∈A} x_{i}∥ ≤ \frac{|A|}{f(|A|)}

for every interval A ⊂ {1, …, r} such that |A| ≥ m. Then for every integer n such that mn ≤ 2r^{3/4} we have

∥∑_{i=1}^{r} x_{i}∥_{(n)} ≤ 5 \frac{r}{f(r)}.

PROOF. Let x = ∑_{i=1}^{r} x_{i} and let (E_{h})_{h=1}^{n} be a sequence of n successive intervals. For every h, let E′_{h} be the largest interval contained in E_{h} such that E′_{h}x = ∑_{i∈A_{h}} x_{i} for some interval A_{h} ⊂ {1, …, r}; then, E_{h} is the union of E′_{h} and two small intervals at both ends of E_{h}, and ∥E_{h}x∥ ≤ 2 + ∥E′_{h}x∥. Let H = {h : |A_{h}| ≥ m}. We get ∥E_{h}x∥ ≤ 2 + (m − 1) when h ∉ H, and

∑_{h=1}^{n} ∥E_{h}x∥ ≤ n(m + 1) + ∑_{h∈H} ∥E_{h}x∥ ≤ 2nm + ∑_{h∈H} \frac{|A_{h}|}{f(|A_{h}|)}
≤ 4r^{3/4} + \frac{∑_{h} |A_{h}|}{f(m)} ≤ 4r^{3/4} + \frac{r}{f(m)} ≤ 4r^{3/4} + 4 \frac{r}{f(r)}

since f(r) ≤ f(m^{4}) ≤ 4 f(m) (using the conditions for the family F). The result follows, because the condition r ≥ e^{16} implies that 4r^{3/4} ≤ r/f(r), by formula (F_{1}).

The construction of our examples uses a lacunary subset J of ℕ. Let us write J in increasing order as {j_{1}, j_{2}, …}. We assume that each j_{n} is the fourth power of some integer. This set J should be such that between two successive elements j_{n} and j_{n+1} of J, there is enough room to apply Lemma 11; to be on the safe side, let us assume that

(L1)  j_{1} ≥ e^{256} and ∀n ≥ 1, e^{256 j_{n}^{2}} < j_{n+1}.

This implies by induction that

(L2)  j_{n} ≥ e^{n^{3}}

for every n ≥ 1, because j_{1} > e and e^{(n+1)^{3}} ≤ e^{8n^{3}} ≤ e^{8j_{n}} < j_{n+1} for n ≥ 1. Now we check that Lemma 11 can be applied between j_{n}^{4} and j_{n+1}^{1/4}. If we let u = 4j_{n}, then u ≥ 5 and j_{n}^{4} ≤ e^{4j_{n}} = e^{u}; by (L1), j_{n+1}^{1/4} ≥ e^{64j_{n}^{2}} = e^{4u^{2}}. With Lemma 11 and the conditions on J it is rather clear that

Lemma 17 Let K_{0} ⊂ K. There exists a function g ∈ F_{0} such that f ≥ g ≥ \sqrt{f}, g(k) = \sqrt{f(k)} whenever k ∈ K_{0} and g(t) = f(t) whenever j ∈ J \ K_{0} and t is in the interval [j^{1/4}, j^{4}].

Since we want to include some of the results of Argyros-Felouzis, we shall work in the extended setting V. Also, we want to present a part of [GM2], so we are trying to build a space with a given algebra of operators generated by some set S of basic operators on our space. Here again, the reader may decide to consider that S = {Id} (the trivial case), which is what is needed for constructing a HI space.

Given two infinite sets A, B ⊂ ℕ, define the spread from A to B to be the map on c_{00} defined as follows. Let the elements of A and B be written in increasing order respectively as {a_{0}, a_{1}, …} and {b_{0}, b_{1}, …}. Then e_{n} maps to zero if n ∉ A, and e_{a_{i}} maps to e_{b_{i}} for every i ≥ 0. Denote this map by S_{A,B}. Note that S_{B,A} is (formally) the adjoint of S_{A,B}. Observe that for every interval projection E and any U ∈ S, there exist two intervals F_{1} and F_{2} such that EU = UF_{1}, UE = F_{2}U.

Given any set S of spreads containing the identity map, we shall say that it is a semi-group if it is closed under composition (note that this applies to all compositions and not just those of the form S_{B,C}S_{A,B}). If we want to define extensions of the S_{A,B}’s in the V setting, we need to assume more about the relations between the different spaces V_{n}. We say that V and S are compatible if whenever U ∈ S and Ue_{m} = e_{n} for some m and n, then V_{m} ⊂ V_{n} and ∥v∥_{n} ≤ ∥v∥_{m} for every vector v ∈ V_{m}. The trivial semi-group S = {Id} is of course compatible with any family V. Given any set S of spreads, compatible with V, all maps S_{A,B} in S are extended to c_{00}(V) in the usual way, by tensoring with the identity of V_{n}, giving the set S(V). For example, S_{A,B}(v_{n} ⊗ e_{n}) = v_{n} ⊗ S_{A,B}(e_{n}). The compatibility assumption means that ∥S_{A,B}∥ ≤ 1 for the c_{00}(V) norm (or ℓ_{1}(V)), for every S_{A,B} ∈ S.

The main tool for our construction, already seen in section 4, is the notion of special sequence. We split the lacunary set J into two disjoint parts K and L: let K, L ⊂ J be the sets {j_{1}, j_{3}, j_{5}, …} and {j_{2}, j_{4}, j_{6}, …}. The set K is used for the lengths of special sequences, while m ∈ L or m “close” to L is used for lengths of “regular” RIS x_{1}, …, x_{m} that will satisfy the ordinary inequality ∥ ∑_{i=1}^{m} x_{i}∥ ≤ 3 m/f(m); the sum of such RIS will be therefore normed by (m, f)-forms; this explains our choice for the members of special sequences below. For every n ≥ 0, let us choose a countable subset Δ_{n} ⊂ V_{n}^{∗} dense in V_{n}^{∗} (this is where assuming V_{n} reflexive makes our life easier; if not, we would have to work with countable norming sets for V_{n}). Let Q ⊂ c_{00}(V^{∗}) be the (countable) set of sequences with the nth coordinate in Δ_{n} for every n and maximum at most 1 in V_{n}^{∗}-norm. Let σ be an injection from the collection of finite sequences of successive elements of Q to the set L introduced above. Given X ∈ X (V) and given an integer m ≥ 1, let A_{m}^{∗}(X) be the set of (m, f)- forms on X, i.e. the set of all functionals x^{∗} of norm at most 1 of the form x^{∗} = f(m)^{−1} ∑_{i=1}^{m} x_{i}^{∗}, where x_{1}^{∗} < … < x_{m}^{∗} and ∥x_{i}^{∗}∥ ≤ 1 for each i = 1, …, m. If k ∈ K, let Γ_{k}^{X} be the set of sequences y_{1}^{∗} < … < y_{k}^{∗} such that y_{i}^{∗} ∈ Q for each i, y_{1}^{∗} ∈ A_{j_{2k}}^{∗}(X) and y_{i+1}^{∗} ∈ A_{σ(y_{1}^{∗},...,y_{i}^{∗})}^{∗}(X) for each 1 ≤ i ≤ k − 1. We call these special sequences. Let B_{k}^{∗}(X) be the set of all functionals y^{∗} of the form

y^{∗} = \frac{1}{\sqrt{f(k)}} ∑_{j=1}^{k} y_{j}^{∗}

such that (y_{1}^{∗}, …, y_{k}^{∗}) ∈ Γ_{k}^{X} is a special sequence. These, when k ∈ K, are the special functionals (on X of size k). The idea behind this notion of special functionals is that their normalization is different from the usual normalization of functionals obtained by the “Schlumprecht operation” (x_{1}^{∗} + ⋯ + x_{n}^{∗})/f(n), so that they produce “spikes” in the unit ball of X^{∗}; but special functionals are extremely rare, and they are easy to trace, as it was explained in section 4. Now, given a semi-group S of spreads, compatible with V, we consider the smallest norm on c_{00}(V) satisfying the following equation,

∥x∥ = ∥x∥_{c_{0}} ∨ sup{f(n)^{−1} ∑_{i=1}^{n} ∥E_{i}x∥ : n ≥ 2, E_{1} < … < E_{n} intervals}
∨ sup{|x^{∗}(Ex)| : k ∈ K, x^{∗} ∈ B_{k}^{∗}(X), E ⊂ ℕ an interval}
∨ sup{∥Ux∥ : U ∈ S}.

We define the space X(S, V) as the completion of c_{00}(V) under this norm. In the case S = {Id} the fourth term drops out and the definition reduces — when V = K — to that of the space constructed in [GM1]. The fourth term is there to force every spread U = S_{A,B} ∈ S to define a bounded operator on X (actually, we get ∥S_{A,B}∥ ≤ 1. This restrictive choice could perhaps be relaxed to produce more examples). The second term ensures that X satisfies a lower f-estimate.

It is useful to understand the construction of X in a way similar to what we have said about the spaces T and S in section 2: we construct a norming subset in X^{∗} in a sequence of steps, producing an increasing sequence (B_{n}) of convex subsets of B_{c_{0}(V^{∗})}. We start with B_{0} = B_{ℓ_{1}(V^{∗})} ∩ c_{00}(V^{∗}). After B_{n} is defined, we enlarge it as follows:

— for any integer m ≥ 2, we add all functionals x^{∗} = f(m)^{−1} ∑_{j=1}^{m} x_{j}^{∗} built from elements x_{j}^{∗} ∈ B_{n},

— for every k ∈ K, we add all functionals λEU^{∗}x^{∗} where |λ| = 1, E is an interval, U ∈ S and x^{∗} is any special functional x^{∗} = f(k)^{−1/2} ∑_{j=1}^{k} y_{j}^{∗} with y_{j}^{∗} ∈ B_{n} for j = 1, …, k;

we obtain in this way an expanded set B̃_{n} ⊃ B_{n}, and we let finally B_{n+1} be the convex hull of B̃_{n}. We let B = ⋃_{n} B_{n} and we can check that the above defined norm is equal to

∥x∥ = sup{|x^{∗}(x)| : x^{∗} ∈ B}.

Observe that the images of successive functionals by a spread are still successive; observe also that the adjoint operation of a spread is again a spread, and that for every interval F and spread U, the operator UF can also be expressed as EU for some interval E; it follows by induction that B_{n} is stable under the adjoints of the spreads in S and under projections on intervals, and that B_{n} is contained in the unit ball of c_{0}(V^{∗}), for every n ≥ 0.

All this implies that X belongs to the family X (V). We summarize the preceding discussion in the following statement.

Proposition 18 Let S be a semi-group of spreads, compatible with the family V. The space X(S, V) belongs to X (V) and satisfies a lower f-estimate. Every spread U ∈ S verifies ∥U∥ ≤ 1.

If we want to compute the norm of x ∈ X = X(S, V), either ∥x∥ = ∥x∥_{c_{0}} or, given ε > 0 such that ∥x∥_{c_{0}} < ∥x∥ − ε, there exists a first n ≥ 0 such that |x^{∗}(x)| > ∥x∥ − ε for some x^{∗} ∈ B̃_{n} that was adjoined to B_{n} in the construction of B_{n+1}, namely either an (m, f)-form or some EU^{∗}y^{∗}, with y^{∗} a special functional of some length k ∈ K, E an interval and U ∈ S. Let us call surface functional any functional x^{∗} on X which is either a (m, f)-form for some m ≥ 2 or a (k, \sqrt{f})-form EU^{∗}y^{∗}, with k ∈ K and y^{∗} a special functional. We may summarize the lines above by saying that for every vector x in X, either x has the c_{0}(V)-norm, or ∥x∥ is the supremum of |x^{∗}(x)|, when x^{∗} runs over the set of surface functionals. Note that if g ∈ F and g(k) = f(k)^{1/2}, then a special functional y^{∗} of size k ∈ K and norm ≤ 1 is also a (k, g)-form, and the same is true for each EU^{∗}y^{∗}, for every interval E and every U ∈ S. A trick which will be repeated several times is that, when g ∈ F satisfies \sqrt{f} ≤ g ≤ f and g = \sqrt{f} on K, then all surface functionals are g-forms of a certain length ≥ 2, either because g ≤ f or because g = \sqrt{f} on K. This remark explains why the generalized Schlumprecht Lemma (Lemma 14), applied to suitable functions g ∈ F, will be our main tool for estimating norms in X(S, V).

The preceding paragraph illustrates an aspect of our class of examples that is simpler than what happens with the example of [AD], produced by mixed Tsirelson norms. When working with Tsirelson norms, it is often necessary to analyze how a vector was constructed, in a tree of operations corresponding to the inductive definition of the space. This is not the case here. We will not need to look “below the surface”.

Let L̂ = ⋃_{ℓ∈L}[ℓ^{1/4}, ℓ^{4}]. The next lemma talks about “regular” RIS. For them, the norm of the sum behaves essentially as in Schlumprecht’s space.

Lemma 19 Let x_{1}, …, x_{r} be a RIS in X(S, V) with r ∈ L̂. Then

∥∑_{i=1}^{r} x_{i}∥ ≤ 3 \frac{r}{f(r)}.

PROOF. Use Lemma 17 to get g ∈ F_{0} equal to f on L̂, to \sqrt{f} on K, and satisfying \sqrt{f} ≤ g ≤ f. Then all surface functionals are g-forms of a certain length. Let n_{1}, …, n_{r} be the integers associated to x_{1}, …, x_{r} by the definition of a RIS. We have e^{2r^{3}} < n_{1} and ∥x_{i}∥_{(n_{i})} ≤ 1. Let p be the integral part of e^{2r^{2}}. Then p^{r} ≤ e^{2r^{3}} < n_{1} ≤ n_{i}, thus ∥x_{i}∥_{(p^{r})} ≤ ∥x_{i}∥_{(n_{i})} ≤ 1 for every i = 1, …, r. Let x = ∑_{i=1}^{r} x_{i}; by Lemma 12, we know that |x^{∗}(Ex)| ≤ 3 for every interval E, when x^{∗} is a (q, g)-form with q > p, and by our remarks, we know that ∥Ex∥ is less than the supremum of |x^{∗}(Ex)|, when x^{∗} runs in the set of (q, g)- forms (unless Ex has the c_{0} norm, in which case ∥Ex∥ ≤ 1). We see that Lemma 14 applies to the RIS (\frac{1}{3}x_{i})_{i=1}^{r}, hence ∥x∥ ≤ 3 r g(r)^{−1} = 3 r f(r)^{−1}.

Lemma 20 Let ℓ ∈ L and let x_{1}, …, x_{r} be a RIS in X(S, V) with ℓ ≤ r ≤ ℓ^{4}. Then

∥∑_{i=1}^{r} x_{i}∥_{(\sqrt{ℓ})} ≤ 15 \frac{r}{f(r)}.

PROOF. Let m denote the smallest integer larger than r^{1/4}. Then ℓ^{1/4} ≤ m ≤ ℓ and we know that [m, r] ⊂ L̂ by construction, hence ∥ ∑_{i∈A} x_{i}∥ ≤ 3|A|/f(|A|) when |A| ≥ m, by Lemma 19. The result follows from Lemma 16, applied to the sequence \frac{1}{3} (x_{i}) with n = \sqrt{ℓ}, because e^{16} < j_{2}^{1/4} ≤ m ≤ r ≤ m^{4}, m ≤ 2 r^{1/4}, hence mn = m \sqrt{ℓ} ≤ 2 r^{3/4}.

A RIS vector is a vector x of the form

x = r^{−1}f(r) ∑_{i=1}^{r} x_{i},

where x_{1}, …, x_{r} is a RIS. We say that r is the length of the RIS vector x. By definition of a RIS, ∥x_{i}∥ ≤ ∥x_{i}∥_{(n_{i})} ≤ 1 for some n_{i} > e^{2r^{3}}. The most interesting case is when the vectors (x_{i}) have norms bounded below, by 1/2 say. In this case the lower f-estimate gives ∥x∥ ≥ 1/2. If furthermore r ∈ L̂, then we know that ∥x∥ ≤ 3 by Lemma 19.

The next Lemma is absolutely crucial for understanding what happens in the space X(S, V). This Lemma says roughly the following. Suppose that we construct together a special sequence (x_{j}^{∗})_{j=1}^{k} of length k ∈ K and a sequence (x_{j})_{j=1}^{k} of RIS vectors, in such a way that the number ℓ_{j} ∈ L such that x_{j}^{∗} is a (ℓ_{j}, f)-form, coincides with the length of the RIS vector x_{j}. Then the special functional x^{∗} = f(k)^{−1/2} ∑_{i=1}^{k} x_{i}^{∗} and its images U^{∗}x^{∗} by the adjoints of the spreads U ∈ S will be essentially the only functionals that can force the norm of the vector x = ∑_{i=1}^{k} x_{i} to exceed significantly the usual bound k/f(k). The corresponding lemma was not correctly stated in [GM1] (the condition ran(x_{i}) ⊂ ran(x_{i}^{∗}) was missing there).

Lemma 21 Let k ∈ K and let x_{1}^{∗}, …, x_{k}^{∗} be a special sequence of length k; let ℓ_{1} = j_{2k}, and for 2 ≤ i ≤ k let ℓ_{i} = σ(x_{1}^{∗}, …, x_{i−1}^{∗}). Assume that f(\sqrt{ℓ_{i}}) > | ran(∑_{j=1}^{i−1} x_{j}^{∗})|^{2} for i = 2, …, k. Let x_{1}, …, x_{k} be a sequence of successive vectors in X(S, V) such that every x_{i} is a RIS vector of length ℓ_{i} and ran(x_{i}) ⊂ ran(x_{i}^{∗}), i = 1, …, k. Suppose that

|(∑_{i=1}^{k} U^{∗}x_{i}^{∗})(∑_{i=1}^{k} Ex_{i})| ≤ 16

for every interval E and every U ∈ S. Then

∥∑_{i=1}^{k} x_{i}∥ ≤ 45 \frac{k}{f(k)}.

PROOF. We know by Lemma 20, applied to the decomposition of x_{i} into a RIS, that ∥x_{i}∥_{(m_{i})} ≤ 15, where m_{i} = \sqrt{ℓ_{i}}. Also ℓ_{1} = j_{2k} ≥ e^{8k^{3}} by the lacunarity condition (L_{2}), thus ℓ_{1}^{1/2} = m_{1} ≥ e^{2k^{3}}. Since ran x_{i} ⊂ ran x_{i}^{∗} for each i, this implies that

\sqrt{f(m_{i})} ≥ |ran(∑_{j=1}^{i−1} x_{j}^{∗})| ≥ |ran(∑_{j=1}^{i−1} x_{j})|, i = 2, …, k.

This and the lower bound for m_{1} ensure that \frac{1}{15} (x_{1}, …, x_{k}) is a RIS of length k, preparing thus to apply Lemma 12 to the sequence (x′_{i}) = (x_{i}/15). Let x = ∑_{j=1}^{k} x_{j} and x′ = ∑_{j=1}^{k} x′_{j} = x/15.

By Lemma 17, we may select a function g ∈ F equal to f on L̂ ∪ \{k\}, to \sqrt{f} on K \ \{k\} and such that \sqrt{f} ≤ g ≤ f. Then all f-forms and all ∗-spreads of special forms of length ≠ k are g-forms (by ∗-spread, we mean the adjoint of a spread in S). In order to prove Lemma 21 we shall apply Lemma 14 to the sequence (x′_{i}). We observe that all vectors Ex are either normed by (q, g)- forms or by ∗-spreads of special functionals of length k, or they have norm at most 1. The first observation is that “long” forms have a small action on x′: indeed by Lemma 12 we know that

(∗)  |z^{∗}(Ex′)| ≤ 3

whenever z^{∗} is a (q, g)-form with q > p, where p denotes the integral part of e^{2k^{2}}. On the other hand, each vector x_{i} satisfies ∥x_{i}∥_{(m_{i})} ≤ 15 with m_{i} ≥ e^{2k^{3}} ≥ p^{k}, which puts us in a position to apply Lemma 14 to x′. Before this, we need to show that ∥Ex′∥ is not given by |z^{∗}(Ex′)|, where z^{∗} is a ∗-spread of a special functional of length k (the only kind of surface functionals which are not g-forms, in the present situation).

We shall show that if z_{1}^{∗}, …, z_{k}^{∗} is any special sequence of functionals of length k and E is any interval, then |U^{∗}z^{∗}(Ex′)| ≤ 1 for every U ∈ S, where z^{∗} is the (k, \sqrt{f})-form f(k)^{−1/2} ∑_{i=1}^{k} z_{i}^{∗}. Indeed, let t be maximal such that z_{t}^{∗} = x_{t}^{∗} or zero if no such t exists. Suppose i ≠ j or one of i, j is greater than t+1. We show that |(U^{∗}z_{i}^{∗})(Ex′_{j})| < k^{−2}. Since σ is an injection, we can find λ_{1} ≠ λ_{2} ∈ L such that z_{i}^{∗} is a (λ_{1}, f)-form, x_{j} is a RIS vector of length λ_{2} and ∥x_{j}∥_{(m′_{2})} ≤ 15, where m′_{2} = \sqrt{λ_{2}}. If λ_{1} < λ_{2}, it follows from the lacunarity properties of the set J ⊃ L that λ_{1} < m′_{2}. This yields that |(U^{∗}z_{i}^{∗})(Ex′_{j})| ≤ ∥x′_{j}∥_{(λ_{1})}/f(λ_{1}) ≤ f(λ_{1})^{−1}. We know that λ_{1} ≥ j_{2k} since λ_{1} appears in a special sequence of length k. The conclusion in this case now follows from the fact that f(ℓ) ≥ k^{2} when ℓ ≥ j_{2k} (implied by the lacunarity condition (L_{2})).

If λ_{2} < λ_{1}, we apply Lemma 12 to the vector x′′_{j} = λ_{2}f(λ_{2})^{−1}x_{j} equal to the sum of the vectors in the RIS defining x_{j}. The definition of L gives us that e^{256 λ_{2}^{2}} < λ_{1}, so Lemma 12 gives |(U^{∗}z_{i}^{∗})(Ex′′_{j})| ≤ 3. It follows that |(U^{∗}z_{i}^{∗})(Ex′_{j})| ≤ f(λ_{2})/λ_{2}. The conclusion follows because ℓ ≥ j_{2k} implies that f(ℓ)/ℓ ≤ ℓ^{−3/4} ≤ e^{−6k^{3}} ≤ k^{−2} (by condition (F_{1})).

Now choose an interval F (depending upon U) such that

|(∑_{i=1}^{t} U^{∗}z_{i}^{∗})(Ex)| = |(∑_{i=1}^{k} U^{∗}x_{i}^{∗})(F x)| ≤ 16.

It follows (since ∥x_{t+1}∥ ≤ 3 by Lemma 19) that

|(∑_{i=1}^{k} U^{∗}z_{i}^{∗})(Ex)| ≤ 16 + |(U^{∗}z_{t+1}^{∗})(Ex_{t+1})| + 15 k^{2}.k^{−2} ≤ 45.

We finally obtain that |U^{∗}z^{∗}(Ex′)| ≤ (45/15)f(k)^{−1/2} < 3f(j_{1})^{−1/2} < 1 as claimed. It follows from (∗) and from what we have just shown about special sequences of length k that

∥Ex′∥ ≤ 3 ∨ sup{|x^{∗}(Ex′)| : 2 ≤ q ≤ p, x^{∗} is a (q, g)-form}

whenever E is an interval. Since ∥x′_{j}∥_{(p^{k})} ≤ 1 for each j = 1, …, k, Lemma 14 applied to x′/3 implies that ∥x′∥ ≤ 3 k/g(k) = 3 k/f(k).

7.1 We have a HI space!

In this paragraph, we assume that S = {Id}, so that any family V is compatible with S. We are primarily interested in the case V = K, and we shall prove that the resulting space X = X({Id}, K) is HI. However, we shall keep the V framework and prove at the same time that any two GH-subspaces of X(V) = X({Id}, V) almost intersect. When V = K, every subspace Y of X is a GH-subspace, and we get that X is HI. Let Y, Z be two GH-subspaces of X. Let us choose δ > 0 and let k ∈ K be an integer such that f(k)^{−1/2} < δ/360. We want to show that the distance between the unit spheres of Y and Z is less than δ.

By the gliding hump property of Y and Z, we may assume that both Y and Z are spanned by block bases. Since X satisfies a lower f-estimate, Corollary 10 tells us that every block subspace of X contains, for every n ≥ 1, a vector x such that ∥x∥_{(n)} ≤ 1 and ∥x∥ > 1/2. We already observed that every vector Ex either has the supremum norm or satisfies the inequality

∥Ex∥ ≤ sup{|x^{∗}(Ex)| : q ≥ 2, x^{∗} is a (q, g)-form}

where g is the function obtained from Lemma 17 in the case K_{0} = K. This allows us to make the following construction.

Using Corollary 10, we may find in Y (or in Z) for any given integer ℓ ∈ L, a sequence y_{1}, …, y_{ℓ} which satisfies the RIS condition, and also such that ∥y_{j}∥ > 1/2 for j = 1, …, ℓ. We are going to apply this fact k times, with increasing values of ℓ, and alternating our choice between Y and Z at each step. Let first ℓ_{1} = j_{2k} ∈ L, and let x_{1} = ℓ_{1}^{−1}f(ℓ_{1}) ∑_{j=1}^{ℓ_{1}} y_{j} ∈ Y be a RIS vector of length ℓ_{1}, with ∥y_{j}∥ > 1/2 for every j. For each j between 1 and ℓ_{1} let y_{j}^{∗} be a functional such that ∥y_{j}^{∗}∥ ≤ 1 and 0 < y_{j}^{∗}(y_{j}) − 1/2 < k^{−1}; let x′_{1}^{∗} be the (ℓ_{1}, f)-form f(ℓ_{1})^{−1} ∑_{j=1}^{ℓ_{1}} y_{j}^{∗}. Then 0 < x′_{1}^{∗}(x_{1}) − 1/2 < k^{−1}. By continuity and the density of Δ_{n} in the unit ball of V_{n}^{∗}, we may assume that there exists an (ℓ_{1}, f)-form x_{1}^{∗} ∈ Q such that 0 < x_{1}^{∗}(x_{1}) − 1/2 < k^{−1} and ran(x_{1}^{∗}) = ran(x_{1}) (in the real case; in the complex case, we ask for 1/2 < ℜx_{1}^{∗}(x_{1}) and |x_{1}^{∗}(x_{1}) − 1/2| < k^{−1}); since we have an infinite sequence of possible choices for x_{1}^{∗} ∈ Q and since σ is injective, we may choose x_{1}^{∗} such that ℓ_{2} = σ(x_{1}^{∗}) satisfies f(\sqrt{ℓ_{2}}) > | ran(x_{1}^{∗})|^{2}. Also, note that there is no difference between an (ℓ_{1}, g)-form and an (ℓ_{1}, f)-form, because g = f on L.

Now let ℓ_{2} = σ(x_{1}^{∗}) and pick a RIS vector x_{2} ∈ Z of length ℓ_{2} such that x_{1} < x_{2}, similarly to the first step. As above, we can find an (ℓ_{2}, g)-form x_{2}^{∗} ∈ Q such that 0 < x_{2}^{∗}(x_{2}) − 1/2 < k^{−1}, ran(x_{2}^{∗}) = ran(x_{2}) and f(\sqrt{ℓ_{3}}) > | ran(x_{1}^{∗} + x_{2}^{∗})|^{2}, where ℓ_{3} = σ(x_{1}^{∗}, x_{2}^{∗}).

Continuing in this manner, we obtain a pair of sequences x_{1}, …, x_{k} and x_{1}^{∗}, …, x_{k}^{∗} with various properties we shall need. First, x_{i} ∈ Y when i is odd and x_{i} ∈ Z when i is even. We also know that 0 < x_{i}^{∗}(x_{i})−1/2 < 1/k for each i. Finally, and perhaps most importantly, the sequence x_{1}^{∗}, …, x_{k}^{∗} has been carefully chosen to be a special sequence of length k. It follows immediately from the implicit definition of the norm and from the fact that ran(x_{i}^{∗}) = ran(x_{i}) for each i that

∥∑_{i=1}^{k} x_{i}∥ ≥ f(k)^{−1/2} ∑_{i=1}^{k} x_{i}^{∗}(x_{i}) > \frac{1}{2} k f(k)^{−1/2}.

The proof will be complete if we can find a suitable upper bound for the norm of the alternate sum ∑_{i=1}^{k}(−1)^{i−1}x_{i}. For this we apply Lemma 21. The conditions on f(\sqrt{ℓ_{i}}) and on the inclusions of ranges have been taken care of during the construction of the sequences (x_{i}) and (x_{i}^{∗}). It remains to show that |(∑_{i=1}^{k} x_{i}^{∗})(∑_{i=1}^{k}(−1)^{i}Ex_{i})| ≤ 16 for every interval E. This follows easily from the fact that x_{i}^{∗}(x_{i}) is almost exactly 1/2 for every i; there are possibly two incomplete terms, one at the beginning and one at the end of E, for which we use |x_{i}^{∗}(x_{i})| ≤ ∥x_{i}∥ ≤ 3 (this follows from Lemma 19 applied to the RIS corresponding to x_{i}). Lemma 21 therefore shows that ∥ ∑_{i=1}^{k}(−1)^{i−1}x_{i}∥ ≤ 45 kf(k)^{−1}.

We have now constructed two vectors y ∈ Y, the sum of the odd-numbered x_{i}s, and z ∈ Z, the sum of the even-numbered x_{i}s, such that ∥y + z∥ > (1/90) f(k)^{1/2}∥y − z∥. If a is the maximum of ∥y∥ and ∥z∥, then y/a and z/a have distance < 180/\sqrt{f(k)} < δ/2, and one of them belongs to the unit sphere of Y or Z. It follows that the distance from the unit sphere of Y to the unit sphere of Z is less that δ.

Suppose that V = K. The above proof shows that X is HI. We also observe that X is reflexive. This follows from Corollary 15. In the vector case, the conclusion is that any two GH-subspaces have distance 0. In particular, we obtain that every subspace Y of X such that all projections P_{n}, n ≥ 0 are strictly singular on Y, is a HI space (because all subspaces of Y are GHsubspaces of X).

Theorem 22 Let X = X(V) be the space constructed in section 7 when S = {Id}. Then any two GH-subspaces of X have distance 0. Every subspace Y of X such that all projections P_{n}, n ≥ 0 are strictly singular on Y, is a HI space. When V = K, then X is a reflexive HI space.

8 Factorization through a HI space

In this section we present some of the results of Argyros and Felouzis [AF, Theorems 2.3 and 2.4]. We shall use a variant of the interpolation spaces of Lions-Peetre (see [BL] for instance), following the spirit of the exposition of [DFJP] rather than that of interpolation theory. Let W be a bounded symmetric closed convex subset of a Banach space V. Suppose that a = (a_{n}) is a decreasing sequence of positive numbers, such that lim_{n} a_{n} = 0. For every n ≥ 0, we consider the bounded symmetric convex set C_{n} = 2^{n}W + a_{n}B_{V} and we let j_{a,n} be the gauge of C_{n}. For every n, the gauge j_{a,n} defines an equivalent norm on V, and we shall call V_{n} the space V equipped with the equivalent norm j_{a,n}. We shall be interested in the (usually unbounded, possibly infinite) gauge j_{a} = sup_{n≥0} j_{a,n} on V. It is clear that j_{a} is finite on W (because j_{a} is less than the gauge j_{W} of W, since j_{a,n} ≤ 2^{−n}j_{W} for every n ≥ 0). It is also clear that j_{a} is larger than a multiple of the norm of V (because C_{0} is bounded, say). The classical construction of factorization spaces in [DFJP] uses the “quadratic” gauge q_{a}(x) = (∑_{n=0}^{+∞} j_{a,n}(x)^{2})^{1/2}, but we shall instead use a non standard way of mixing the norms of the V_{n}s, namely the construction from the preceding section 7.

We need a notion of thinness of W that guarantees that there is no subspace Y of V on which the j_{a} norm is finite and equivalent to the V -norm. This was done by Neidinger (see [N1,N2]) in a similar setting (Neidinger was looking for hereditarily-ℓ_{p} interpolation spaces, while we are looking for hereditarily indecomposable interpolation spaces). Let us say that a bounded symmetric closed convex subset W is a-thin in V when there is no infinite-dimensional subspace Y of V such that j_{a} is finite and bounded on the unit ball B_{Y} = Y ∩ B_{V} of Y. This means that the inclusion from V_{1} to V is strictly singular, where V_{1} is the space of those v ∈ V such that ∥v∥_{V_{1}} = j_{a}(v) < +∞.

The set W is thin in Neidinger’s sense when for every subspace Y of V, there exists ε > 0 such that for every C, the unit ball B_{Y} of Y is not contained in C W + εB_{V}. If W is thin in Neidinger’s sense, then it is a-thin for every sequence a: there exists ε > 0 such that B_{Y} ⊄ 2^{m}W + εB_{V} for every m, in particular for any given ℓ, when a_{n} < ε2^{−ℓ} there exists y ∈ B_{Y} such that y ∉ 2^{n+ℓ}W + 2^{ℓ}a_{n}B_{V}, thus j_{a,n}(y) ≥ 2^{ℓ} and W is a-thin. On the other hand, if a is the sequence a_{n} = e^{−8^{n}}, then W = B_{L_{∞}} is a-thin in V = L_{1} = L_{1}[0, 1] (see [AF, Proposition 2.2]), but the inclusion L_{∞} → L_{1} is not thin, as mentioned by Neidinger. This shows that the statement of the factorization theorem below is slightly more general when formulated with a-thin instead of thin. We say that a bounded linear operator T : U → V is a-thin if the closure \overline{T (B_{U})} of T (B_{U}) is a-thin in V.

Theorem 23 Suppose that T : U → V is a-thin for some a. Then T factors through a HI space.

PROOF. We know that for some a, the set W = \overline{T (B_{U})} is a-thin in V. For every n ≥ 0, let V_{n} denote V with the norm j_{a,n}, and consider the family V = (V_{n})_{n≥0}. Let X = X(V) be the space constructed in section 7, and let Z denote the diagonal of this space, in other words Z is the vector subspace of V consisting of those v ∈ V such that v = (v, v, …) belongs to X. We define a norm on Z by ∥v∥_{Z} = ∥v∥_{X}. We shall check that T factors through Z, and that Z is a HI space.

For every u ∈ U such that ∥u∥ ≤ 1, the vector w = T u ∈ V belongs to W, therefore ∥w∥_{V_{n}} = j_{a,n}(w) ≤ 2^{−n}; the series ∑_{n} w ⊗ e_{n} is normally convergent in X and defines an element T_{1}(u) ∈ Z. On the other side, the norm of v in X is larger than j_{a} (by the general assumptions about X (V)), thus larger than ∥v∥_{V}, and this shows that there is a natural inclusion map i from Z to V. We have therefore obtained the factorization T = i ∘ T_{1}.

In order to prove that Z is HI, we have only to check that Z, regarded as a subspace of X via the diagonal map, is such that P_{n_{0}} is strictly singular on Z for every n_{0} ≥ 0. Given any subspace Y ⊂ Z, it is possible to find y ∈ Y such that j_{a,n}(y) < 2^{−n_{0}} for n = 0, …, n_{0} but ∥y∥_{X} = 1. This is clear since each of the first gauges j_{a,n} is equivalent to the V -norm while j_{a} is unbounded on B_{Y} and less than the norm of X. This shows that P_{n_{0}} is strictly singular on Z. The result then follows from Theorem 22 (we are a little bit cheating, since we wrote the proof under the additional hypothesis that V is reflexive).

Theorem 24 For every p ∈ (1, +∞), the space ℓ_{p} is a quotient of some HI space. This is also true for c_{0}.

It is obvious that this cannot hold for ℓ_{1}, by the lifting property of ℓ_{1}. We shall only sketch the case of ℓ_{p}, p ∈ (1, +∞); see [AF] for a much more general result, but also much more difficult to prove. The strategy for the proof is the following: we shall construct a space V and a symmetric a-thin closed convex subset W ⊂ V which is norming for an ℓ_{q}-subspace L of V^{∗}. Let U be the Banach space whose unit ball is W, and let us apply the preceding factorization result to the inclusion map T : U → V. Then T = i ∘ T_{1}, with T_{1} : U → Y and Y a HI space with an embedding i : Y → V. We need only show that i^{∗} induces an isomorphism from L to a subspace of Y^{∗}. This is easy since the set T_{1}(B_{U}) = W_{1}, which is smaller than B_{Y}, is already norming for the space i^{∗}(L).

The construction of V uses a tree; (very) roughly speaking, we introduce an infinite branch γ of the tree for every vector z^{∗} of L, and a sequence w_{γ} ∈ V supported on that branch, which norms z^{∗}. Next, we define W to be the symmetric closed convex hull of the set of all elements w_{γ}. For every n ≥ 0, let D_{n} be the subset of [−1, 1] consisting of all numbers of the form i 2^{−n−3}, |i| ≤ 2^{n+3}. Let T_{0} be the set of all ν = (d_{0}, …, d_{n}), for n ≥ 0, such that d_{j} ∈ D_{j} for j = 0, …, n. We say that |ν| = n is the length of ν. We say that ν′ ≤ ν if ν′ is an initial segment (d_{0}, …, d_{m}), m ≤ n, of the sequence (d_{0}, …, d_{n}) = ν. We shall restrict our attention to the subtree T ⊂ T_{0} consisting of those nodes ν such that ∑_{i=0}^{|ν|} |d_{i}|^{p} < 1. On the space c_{00}(T ) of finitely supported scalar sequences indexed by T , we consider the ℓ_{p}((ℓ_{1}^{k_{n}})) norm,

∥v∥_{V} = (∑_{n=0}^{+∞}( ∑_{ν∈R_{n}} |v_{ν}|)^{p})^{1/p}

where R_{n} ⊂ T is the set of nodes ν such that |ν| = n and k_{n} = |R_{n}|.

Let (f_{ν})_{ν∈T} be the natural unit vector basis for V. It is clearly 1-unconditional. For every ν = (d_{0}, …, d_{n}) ∈ T , we set c_{ν} = d_{n}; if b is a segment in the tree we set x(b) = ∑_{ν∈b} c_{ν} f_{ν}. Let W be the symmetric closed convex set generated by all vectors x(b); by the definition of T , we know that ∥x(b)∥_{V} < 1 for every segment b of T . Let L be the subspace of V^{∗} generated by the sequence g_{n} = ∑_{ν∈R_{n}} f_{ν}^{∗}, for n ≥ 0. It is clear that (g_{n}) is isometrically equivalent to the unit vector basis of ℓ_{q}, 1/q + 1/p = 1. It is easy to prove that W is \frac{1}{4}-norming for L. Indeed, given z^{∗} = ∑_{n≥0} v_{n}g_{n} such that ∥z^{∗}∥ = 1, we choose the branch γ = (ν_{0}, …, ν_{n}, …), |ν_{n}| = n, such that d_{n} = c_{ν_{n}} is as close as possible to u_{n} = |v_{n}|^{q−1} sign(v_{n}): we choose d_{n} ∈ D_{n} such that sign d_{n} = sign v_{n}, and if |u_{n}| ≥ 2^{−n−2}, we may also make sure that \frac{1}{2} |u_{n}| ≤ |d_{n}| ≤ |u_{n}|. We get

z^{∗}(x(γ)) = ∑_{n=0}^{+∞} c_{ν_{n}}v_{n} ≥ \frac{1}{2} ∑\{|v_{n}|^{q} : |v_{n}| ≥ 2^{−n−2}\} ≥ \frac{1}{4}.

We shall check that W is thin in V. If not, we can find a subspace Y of V, such that choosing 0 < ε < 1/4, we have B_{Y} ⊂ C W + εB_{V} for some C. By a standard gliding hump argument, we can find in Y a normalized sequence (y_{n}) which is a small perturbation of a sequence supported on disjoint subsets (E_{n}) of the tree, say ∥y_{n} − E_{n}y_{n}∥ < ε, where E_{n} is the set of all nodes ν such that k_{n} ≤ |ν| ≤ ℓ_{n}, with ℓ_{n} < k_{n+1} for every n ≥ 0. We need the following lemma, whose proof is sketched in the appendix. Let us call band in T any subset E consisting of all nodes t such that k ≤ |t| ≤ ℓ, for some k ≤ ℓ.

Lemma 25 Let α > 0, let (w_{n}) be a sequence of elements of W and (E_{n}) be a sequence of successive bands in T . There exists an infinite subset M ⊂ ℕ and for every m ∈ M a decomposition of E_{m} in E′_{m}, E′′_{m}, such that: ∥E′′_{m} w_{m}∥ < α, and for every m_{1} ≠ m_{2} in M, the nodes in E′_{m_{1}} and E′_{m_{2}} are incomparable.

Let us finish the proof of Theorem 24. We assumed that B_{Y} ⊂ C W + εB_{V}, with 0 < ε < 1/4; for every n, there exists w_{n} ∈ W such that ∥y_{n} −C w_{n}∥ < ε; passing to a subsequence, we may assume that E_{n} is decomposed into E′_{n} and E′′_{n} satisfying the properties cited in Lemma 25, with α = ε/C. We have ∥E′′_{n}(y_{n} − C w_{n})∥ < ε and ∥E′′_{n}(C w_{n})∥ < C α = ε, therefore ∥E′′_{n}y_{n}∥ < 2ε and ∥E′_{n}y_{n}∥ > 1 − 3ε. We may thus find a normalized sequence of functionals (y_{n}^{∗}) in V^{∗}, with y_{n}^{∗} supported on E′_{n}, and norming for E′_{n}y_{n}. This sequence (y_{n}^{∗}) is isometrically equivalent to the unit vector basis of ℓ_{q}. Consider y^{∗} = N^{−1/q}(y_{1}^{∗} + ⋯ + y_{N}^{∗}); since the supports of the (y_{j}^{∗}) are incomparable, every vector x(b) from the family generating W acts on at most one y_{i}^{∗}; it follows that sup_{w∈W} |y^{∗}(w)| ≤ N^{−1/q}. On the other hand, let y = N^{−1/p}(y_{1} +⋯+y_{n}) ∈ Y. We have that y^{∗}(y) > 1−3ε and by assumption, there exists w ∈ W such that ∥y − C w∥ < ε. This implies that |y^{∗}(C w)| > 1 − 4ε; this is a contradiction when N is large enough.

9 Additional results

In this section we present some of the results of [GM2]. We assume that V = K. Given a semi-group S of spreads, we denote by X = X(S) = X(S, K) the space constructed in section 7, with V = K. In [GM2], three examples are given, corresponding to three semi-groups of spreads. We shall concentrate here on the example X_{s}, which is a space with an isometric right shift operator S, on which every bounded operator is a strictly singular perturbation of a normally converging series of powers of S and its adjoint, the left shift on X_{s}. Before presenting this example, we need to study some general properties of a larger class of examples.

Given any set S of spreads containing the identity map, we shall say that it is a ∗-semi-group if it is a semi-group closed under taking adjoints. An example of such a set is the collection of all spreads S_{A,B} where A = {m, m+1, m+2, …} and B = {n, n + 1, n + 2, …} for some m, n ≥ 0. This is the ∗-semi-group generated by the shift operator. Given any ∗-semi-group S of spreads, we shall say that it is a proper ∗-semi-group if, for every (i, j) ≠ (k, l), there are only finitely many spreads S ∈ S for which e_{i}^{∗}(Se_{j}) ≠ 0 and e_{k}^{∗}(Se_{l}) ≠ 0. The ∗-semi-group generated by the shift operator is proper. Let x = ∑ v_{j} e_{j} and x^{∗} = ∑ v_{i}^{∗} e_{i}^{∗} be two elements of c_{00}. If S is proper, then except for finitely many S ∈ S, the sum ∑_{i,j} v_{i}^{∗} v_{j} e_{i}^{∗}(Se_{j}) = x^{∗}(Sx) has only one non zero term, and is therefore bounded by ∥x∥_{∞} ∥x^{∗}∥_{∞}. Note that a proper set S of spreads must be countable, and if we write it as {S_{1}, S_{2}, …} and set S_{m} = {S_{1}, …, S_{m}} for every m, then for any x ∈ X(S), x^{∗} ∈ X(S)^{∗}, we have

(P)  lim_{m} sup{|x^{∗}(Ux)| : U ∈ S \ S_{m}} ≤ ∥x∥_{∞}∥x^{∗}∥_{∞}.

Let X = X(S, K) be the space constructed in section 7, with V = K; by Corollary 15, we know that X is reflexive. Let y ∈ X. Recall that for every integer n ≥ 1,

∥y∥_{(n)} = sup ∑_{i=1}^{n} ∥E_{i} y∥

where the supremum is extended to all families E_{1} < … < E_{n} of successive intervals. Observe that ∥e_{i}∥_{(n)} = 1 for every n ≥ 1.

Given a subspace Y ⊂ X, we will be interested in a seminorm |||.||| defined on L(Y, X) as follows. Given T ∈ L(Y, X) let |||T ||| be the supremum of those numbers κ such that for every n ≥ 1, there exists a vector y ∈ Y such that

∥P_{n}y∥ ≤ 2^{−n}, ∥y∥_{(n)} ≤ 1 and ∥T y∥ > κ. Clearly, | ; | ; |T | ; | ; | ≤ ∥T ∥. Let us say the

same thing in a slightly different way. The number | ; | ; |T | ; | ; | is the smallest number with the following property: for every ε > 0, there exists an integer n ≥ 1 such that, for every y ∈ Y, the conditions ∥P_{n} y∥ ≤ 2^{−n} and ∥y∥_{(n)} ≤ 1 imply that ∥T y∥ ≤ | ; | ; |T | ; | ; | + ε. We may also write that for every y ∈ Y and n ≥ N(ε),

∥T y∥ ≤ (| ; | ; |T | ; | ; | + ε) ∥y∥_{(n)} + 2^{n} ∥P_{n}y∥.

We say that a bounded sequence (y_{n}) ⊂ Y is a sequence of almost successive vectors in X if there exists a sequence (x_{n}) of successive vectors such that lim_{n} ∥x_{n} − y_{n}∥ = 0. If (y_{n}) ⊂ Y is a sequence such that ∥P_{n}y_{n}∥ ≤ 2^{−n} for every n ≥ 1, then clearly we may find almost successive subsequences (y_{n_{k}}). Let M_{Y} be the set of sequences y = (y_{n})_{n=1}^{+∞} of almost successive vectors in Y such that lim sup_{n} ∥y_{n}∥_{(n)} ≤ 1. Now, given T ∈ L(Y, X) it is clear that

| ; | ; |T | ; | ; | = sup_{y∈M_{Y}} lim sup_{n} ∥T y_{n}∥.

Lemma 26 For every infinite-dimensional subspace Y of X = X(S, K) and every T ∈ L(Y, X), we have

(i) if | ; | ; |T | ; | ; | = 0, then T is strictly singular; (ii) if T is compact, then | ; | ; |T | ; | ; | = 0; (iii) if for every z in some infinite-dimensional subspace Z of Y, we have

∥T z∥ ≥ ∥z∥, then | ; | ; |T | ; | ; | ≥ 1.

PROOF. By Corollary 10, every subspace Y contains normalized sequences in M_{Y}. Hence every subspace of Y contains a norm one vector y such that ∥T y∥ ≤ | ; | ; |T | ; | ; | + ε; in particular, if |||T ||| = 0, then T is strictly singular.

It is clear that lim ∥T x_{n}∥ = 0 if T is compact and (x_{n}) almost successive (because X is reflexive), hence | ; | ; |T | ; | ; | = 0. Lastly, suppose that ∥T z∥ ≥ ∥z∥ for every z in some subspace Z of Y. We know from Corollary 10 that Z contains a normalized sequence (z_{n}) of almost successive vectors with lim ∥z_{n}∥_{(n)} = 1. By definition, | ; | ; |T | ; | ; | ≥ lim_{n} ∥T z_{n}∥ ≥ 1.

Theorem 27 Let S be a proper ∗-semi-group of spreads. The Banach space X = X(S, K) from section 7 satisfies a lower f-estimate and the following three properties.

(i) For every x ∈ X and every S_{A,B} ∈ S, ∥S_{A,B} x∥ ≤ ∥x∥, (and therefore ∥S_{A,B} x∥ = ∥x∥ if supp(x) ⊂ A);

(ii) If Y is an infinite-dimensional subspace of X, then every operator from Y to X is in the |||.|||-closure of the set of restrictions to Y of operators in the algebra A generated by S. In particular, all operators on X are |||.|||-perturbations of operators in A.

(iii) The seminorm |||.||| on L(X) satisfies the algebra inequality |||UV ||| ≤ |||U||| |||V |||.

Notice a straightforward consequence of this result. If we write G for the |||.|||- completion of A (after quotienting by operators with |||.||| zero) then G is a Banach algebra. Given T ∈ L(X), we can find by (ii) a |||.|||-Cauchy sequence (T_{n})_{n=1}^{+∞} of operators in A such that |||T − T_{n}||| → 0. Let ϕ(T) be the limit of (T_{n})_{n=1}^{+∞} in G. This map is clearly well-defined. It follows easily from (iii) that it is also a unital algebra homomorphism. The kernel of ϕ is the set of T such that |||T ||| = 0. We have K(X) ⊂ ker ϕ ⊂ S(X). The restriction of ϕ to A is the identity (or more accurately the embedding of A into G). If A is small, then, since the kernel of ϕ consists of small operators, L(X) is also small.

Let us indicate why X does not contain an infinite unconditional sequence. Let Y ⊂ X be a subspace with an unconditional basis (y_{n}). Let (M_{α})_{α} denote an uncountable family of subsets of ℕ, such that any two of them differ by an infinite set. For every α, let P_{α} denote the projection from Y on to the span of (y_{n})_{n∈M_{α}}. For α ≠ β, there exists a subspace Z ⊂ Y such that ∥(P_{α} −P_{β})z∥ ≥ ∥z∥ for every z ∈ Z. This implies that | ; | ; |P_{α} − P_{β}| ; | ; | ≥ 1 by Lemma 26, but this contradicts the separability of L(X) in the | ; | ; | . | ; | ; |-norm, that follows from (ii) and the countability of S.

Lemma 28 Let S be a proper ∗-semi-group of spreads, let X = X(S), let Y ⊂ X be an infinite-dimensional subspace and let T be a continuous linear operator from Y to X. Let S = ⋃_{m=1}^{∞} S_{m} be a decomposition of S satisfying condition (P). Then for every ε > 0 there exists m such that, for every x ∈ Y such that ∥x∥_{(m)} ≤ 1 and ∥P_{m}x∥ ≤ 2^{−m},

d(T x, m conv{λUx : U ∈ S_{m}, |λ| = 1}) ≤ ε.

PROOF. Suppose that the result is false. Then, for some ε > 0, we can find a sequence (y_{n})_{n=1}^{+∞} with y_{n} ∈ Y, ∥y_{n}∥_{(n)} ≤ 1 and ∥P_{n}y_{n}∥ ≤ 2^{−n} such that, setting C_{n} = n conv{λUy_{n} : U ∈ S_{n}, |λ| = 1}, we have d(T y_{n}, C_{n}) > 2 ε. This yields that (y_{n}) is bounded away from 0. We may pass to an almost successive subsequence, still denoted (y_{n}), such that ∑ ∥y_{n} − y′_{n}∥ < +∞ for some successive sequence (y′_{n}) ⊂ X satisfying ∥y′_{n}∥_{(n)} ≤ 1 for every integer n ≥ 1. Then for some n_{0}, (y_{n})_{n≥n_{0}} and (y′_{n})_{n≥n_{0}} are equivalent basic sequences (see [LT, 1.a.9]), with the additional property that for every α > 0, there exists n_{1} = n_{1}(α) ≥ n_{0} such that every norm one vector y = ∑_{n≥n_{1}} a_{n}y_{n} satisfies ∥y − ∑_{n≥n_{1}} a_{n}y′_{n}∥ < α. If we replace Y by the block subspace Y′ generated by (y′_{n})_{n≥n_{0}} and T by T′ defined on Y′ by T′y′_{n} = T y_{n} for n ≥ n_{0}, we still get the conclusion that d(T′y′_{n}, C′_{n}) > ε, where C′_{n} is defined from (y′_{n}) as C_{n} is defined from (y_{n}), provided n_{0} was chosen large enough. This argument shows that if the result is false, then it is already false for some block subspace Y and some operator T from Y to X.

In this case, it is not hard to show that T can be perturbed (in the operator norm) to an operator whose matrix (with respect to the natural bases of X and Y) has only finitely many non-zero entries in each row and column. We may therefore assume that T has this property. We also assume ∥T ∥ ≤ 1.

Since we assumed that the result is false for Y and T, then for some ε > 0, we can find a sequence (y_{n})_{n=1}^{+∞} with y_{n} ∈ Y, ∥y_{n}∥_{(n)} ≤ 1 and supp(y_{n}) > {n} such that d(T y_{n}, C_{n}) > ε, and also such that if z_{n} is any one of y_{n}, T y_{n} or Uy_{n} for some U ∈ S_{n} and z_{n+1} is any one of y_{n+1}, T y_{n+1} or V y_{n+1} for some V ∈ S_{n+1}, then z_{n} < z_{n+1}. By the Hahn-Banach theorem, for every n ≥ 1 there is a norm-one functional y_{n}^{∗} such that

sup{y_{n}^{∗}(x) : x ∈ C_{n} + εB(X)} < y_{n}^{∗}(T y_{n}).

It follows that y_{n}^{∗}(T y_{n}) > ε and sup |y_{n}^{∗}(C_{n})| ≤ 1. Therefore |y_{n}^{∗}(Uy_{n})| ≤ n^{−1} for every U ∈ S_{n}. We may also assume that the support of y_{n}^{∗} is contained in the smallest interval containing the supports of y_{n}, T y_{n} and Uy_{n} for U ∈ S_{n}. (The case of complex scalars requires a standard modification.)

Given ℓ ∈ L define an ℓ-pair to be a pair (x, x^{∗}) constructed as follows. Let y_{n_{1}}, y_{n_{2}}, …, y_{n_{ℓ}} be a subsequence of (y_{n})_{n=1}^{+∞} satisfying the RIS condition, which implies that n_{1} > e^{2ℓ^{3}} > ℓ^{2}. Let x = ℓ^{−1}f(ℓ)(y_{n_{1}} + ⋯ + y_{n_{ℓ}}) and let x^{∗} = f(ℓ)^{−1}(y_{n_{1}}^{∗} + ⋯ + y_{n_{ℓ}}^{∗}), where the y_{n_{i}}^{∗} are as above. Lemma 19 implies that ∥x∥ ≤ 3 and Lemma 20 that ∥x∥_{(\sqrt{ℓ})} ≤ 15.

If (x, x^{∗}) is such an ℓ-pair, then x^{∗} ∈ A_{ℓ}^{∗}(X) and, by our earlier assumptions about supports,

x^{∗}(T x) = ℓ^{−1} ∑_{i=1}^{ℓ} y_{n_{i}}^{∗}(T y_{n_{i}}) > ε.

Similarly, |x^{∗}(Ux)| ≤ n_{1}^{−1} < ℓ^{−2} for every U ∈ S_{ℓ}.

Let k ∈ K be such that ε f(k)^{1/2} > 45. We now construct sequences x_{1}, …, x_{k} and x_{1}^{∗}, …, x_{k}^{∗} as follows. Let ℓ_{1} = j_{2k} and let (x_{1}, x_{1}^{∗}) be an ℓ_{1}-pair. Let m_{2} be such that |x_{1}^{∗}(Ux_{1})| ≤ ∥x_{1}∥_{∞}∥x_{1}^{∗}∥_{∞} if U ∈ S \ S_{m_{2}}. The functional x_{1}^{∗} can be perturbed so that it is in Q and so that ran(x_{1}^{∗}) ⊃ ran(x_{1}), ℓ_{2} = σ(x_{1}^{∗}) is larger than m_{2} and f(\sqrt{ℓ_{2}}) > | ran(x_{1}^{∗})|^{2} + 4, while (x_{1}, x_{1}^{∗}) is still an ℓ_{1}- pair. In general, after x_{1}, …, x_{i−1} and x_{1}^{∗}, …, x_{i−1}^{∗} have been constructed, let (x_{i}, x_{i}^{∗}) be an ℓ_{i}-pair such that all of x_{i}, T x_{i} and x_{i}^{∗} are supported after all of x_{i−1}, T x_{i−1} and x_{i−1}^{∗}, and then perturb x_{i}^{∗} in such a way that, setting ℓ_{i+1} = σ(x_{1}^{∗}, …, x_{i}^{∗}), we have |x_{i}^{∗}(Ux_{i})| ≤ ∥x_{i}∥_{∞}∥x_{i}^{∗}∥_{∞} whenever U ∈ S \ S_{ℓ_{i+1}} and we also have ran(x_{i}^{∗}) ⊃ ran(x_{i}), f(\sqrt{ℓ_{i+1}}) > | ran(∑_{j=1}^{i} x_{j}^{∗})|^{2} + 2^{i+1}. This yields that ∥x_{i+1}^{∗}∥_{∞} ≤ f(ℓ_{i+1})^{−1} ≤ 2^{−i−1}.

Now let x = x_{1} + ⋯ + x_{k}, let z^{∗} = x_{1}^{∗} + ⋯ + x_{k}^{∗} and x^{∗} = f(k)^{−1/2} z^{∗}. Our construction guarantees that x^{∗} is a special functional, and therefore of norm at most 1. We have

∥T x∥ ≥ x^{∗}(T x) > εkf(k)^{−1/2}.

Our aim is now to get an upper bound for ∥x∥ and to deduce an arbitrarily large lower bound for ∥T ∥. For this purpose we use Lemma 21. In order to apply this Lemma, it is enough to show that |(U^{∗}z^{∗})(Ex)| = |z^{∗}(UEx)| ≤ 16 for any interval E and U ∈ S. We have x_{i}^{∗} ∈ A_{ℓ_{i}}^{∗} for i = 1, …, k. Suppose that U ∈ S_{m+1} \S_{m}, and let t be such that ℓ_{t} ≤ m < ℓ_{t+1}. If i > t, then U ∈ S_{ℓ_{i}} and |x_{i}^{∗}(Ux_{i})| < ℓ_{i}^{−2}. If i < t, then U ∉ S_{ℓ_{i+1}}, so |x_{i}^{∗}(Ux_{i})| ≤ ∥x_{i}∥_{∞}∥x_{i}^{∗}∥_{∞} ≤ 2^{−i}. If i = t, then at least we know that |x_{i}^{∗}(Ux_{i})| ≤ ∥x_{i}∥ ≤ 3.

Putting all these facts together, we get that |z^{∗}(UEx)| ≤ 16, as desired (we may have two incomplete terms at both ends of E). Hence, by Lemma 21, ∥x∥ ≤ 45kf(k)^{−1}. It follows that ∥T ∥ ≥ (ε/45)f(k)^{1/2} > 1, a contradiction.

Lemma 29 Let S, X, Y, T and ε be as in the previous lemma, let m be as given by that lemma and let A_{m} = m conv{λS_{m} : |λ| = 1}. Then there exists U ∈ A_{m} such that |||T − U||| ≤ 64 ε.

PROOF. If the statement of the lemma is false, then for every operator U ∈ A_{m} there is a sequence \underline{x}_{U} = (x_{n}) ∈ M_{Y} of vectors in Y such that lim_{n} ∥(T −U)x_{n}∥ > 64ε. We write this symbolically as ∥(T −U)\underline{x}_{U}∥ > 64ε. This yields that lim inf ∥x_{n}∥ > δ > 0, with δ depending only upon T and ε. At this point, we may argue as in Lemma 28 in order to reduce the situation to the case of a block subspace Y′. Since A_{m} is compact in operator norm, we may find a finite set (\underline{x}_{α}) ⊂ M_{Y} such that for every U ∈ A_{m}, we have ∥(T −U)\underline{x}_{α}∥ > 64 ε for some α. Passing to subsequences, we may assume that the sequences (\underline{x}_{α}) can be arranged to be subsequences of a single sequence (y_{n}) ⊂ Y such that δ ≤ ∥y_{n}∥ ≤ ∥y_{n}∥_{(n)} ≤ 1 for every n, and such that we can find a successive sequence (y′_{n}) in X satisfying ∥y′_{n}∥_{(n)} ≤ 1 for every n and ∑ ∥y_{n} − y′_{n}∥ < +∞. Let n_{0} be chosen so that (y_{n})_{n≥n_{0}} and (y′_{n})_{n≥n_{0}} are equivalent basic sequences. Recall that for every α > 0, there exists n_{1} = n_{1}(α) ≥ n_{0} such that every norm one vector y = ∑_{n≥n_{1}} a_{n}y_{n} satisfies ∥y − ∑_{n≥n_{1}} a_{n}y′_{n}∥ < α. Let Y_{1} ⊂ Y be the subspace generated by the sequence (y_{n})_{n≥n_{0}}; the conclusion of Lemma 28 is obviously still true for the restriction T_{1} of T to Y_{1}. Now, let Y′_{1} be the block subspace generated by the sequence (y′_{n})_{n≥n_{0}}; we may assume that all vectors in Y′_{1} sit after m. Let us define T′_{1} on Y′_{1} by T′_{1}y′_{n} = T y_{n} for every n ≥ n_{0}. For every vector y′ ∈ Y′_{1} such that ∥y′∥_{(m)} ≤ 1, we may find –if n_{0} was chosen large enough– a vector y ∈ Y_{1} such that T′_{1}y′ = T y, ∥y − y′∥ < ε/2 and ∥y − y′∥_{(m)} ≤ 2^{−m}, hence ∥P_{m}y∥ ≤ 2^{−m} and ∥y∥_{(m)} ≤ 3/2. We see that the conclusion of Lemma 28 is still true for T′_{1}, provided we lose an additional ε. Furthermore, for every U ∈ A_{m}, we have ∥(T′_{1} − U)\underline{x}∥ > 62 ε for some \underline{x} ∈ M_{Y′_{1}}. This shows that we may assume that Y is a block subspace such that

(∗)  ∀y ∈ Y, d(T y, {Uy : U ∈ A_{m}}) ≤ 2 ε ∥y∥_{(m)}

and that for every U ∈ A_{m}, we have ∥(T − U)\underline{x}∥ > 62 ε for some \underline{x} ∈ M_{Y}.

Let U_{i=1}^{s} be a covering of A_{m} by open sets of diameter less than ε in the operator norm. For every i = 1, …, s, let U_{i} ∈ U_{i} and let \underline{x}_{i} = (x_{i,n})_{n} be a successive sequence in M_{Y} such that ∥(T − U_{i})\underline{x}_{i}∥ > 62 ε. By the condition on the diameter of U_{i}, we have ∥(T − U)\underline{x}_{i}∥ > 60 ε for every U ∈ U_{i}. As in the last lemma, we can assume that the matrix of T has only finitely many non-zero entries in each row and column.

Our first aim is to show that the vectors \underline{x}_{U} can be chosen continuously in U. (This statement will be made more precise later.) Let (ϕ_{i})_{i=1}^{r} be a partition of unity on A_{m} with ϕ_{i} supported inside U_{i} for each i. Let ℓ ∈ L be greater than s and m^{2}. For each i ≤ s, let x_{i,n_{1}}, …, x_{i,n_{ℓ}} satisfy the RIS condition and let m < x_{i,n_{1}}. Let y_{i} = ℓ^{−1}f(ℓ)(x_{i,n_{1}} + ⋯ + x_{i,n_{ℓ}}). Let this be done in such a way that y_{1} < … < y_{s} and also (T − U)x_{i,n_{1}} < … < (T − U)x_{i,n_{ℓ}} for every i and every U ∈ A_{m}. Finally, let the x_{i,n_{j}} be chosen so that ∥(T − U)x_{i,n_{j}}∥ > 60 ε for every U ∈ U_{i}.

Now let us consider the vector y(U) = ∑_{i=1}^{s} ϕ_{i}(U)y_{i}. By Lemma 20 we know that ∥y_{i}∥_{(\sqrt{ℓ})} ≤ 15 for each i = 1, …, s, from which it follows by the triangle inequality that ∥y(U)∥_{(\sqrt{ℓ})} ≤ 15. We shall show that y(U) is a “bad” vector for U, by showing that ∥(T − U)y(U)∥ > 30 ε.

To do this, let U ∈ A_{m} be fixed and let I = {i : ϕ_{i}(U) > 0}. Recall that ∥(T − U)x_{i,n_{j}}∥ > 60 ε for every i ∈ I and j = 1, …, ℓ. For such an i and for j ≤ ℓ let z_{i,j}^{∗} be a norm-one functional such that z_{i,j}^{∗}((T − U)x_{i,n_{j}}) > 60 ε. Let these functionals be chosen to be successive. Let z_{i}^{∗} = f(ℓ)^{−1}(z_{i,1}^{∗} + ⋯ + z_{i,ℓ}^{∗}) and z^{∗} = ∑_{i∈I} z_{i}^{∗}. Then z_{i}^{∗}(T − U)y_{i} > 60 ε, so

z^{∗}((T − U)y(U)) = z^{∗}(∑_{i∈I} ϕ_{i}(U)(T − U)y_{i}) > 60 ε.

However, ∥z^{∗}∥ ≤ f(sℓ)/f(ℓ) ≤ 2, proving our claim.

The function U → y(U) is clearly continuous on A_{m}. The vector y(U) satisfies ∥y(U)∥_{(m)} ≤ 15 and ∥(T − U)y(U)∥ > 30 ε. We now apply a fixed-point theorem. For every U ∈ A_{m}, let Γ(U) denote the set of V ∈ A_{m} such that ∥(T − V)y(U)∥ ≤ 30 ε. Clearly Γ(U) is a compact convex subset of A_{m}. By property (∗), we know that Γ(U) is non-empty for every U. The continuity of U → y(U) gives that Γ is upper semi-continuous, so there exists a point U ∈ A_{m} such that U ∈ Γ(U). But this is a contradiction.

Lemma 29 shows in particular that any operator T : Y → X can be approximated arbitrarily well in the |||.|||-norm by the restriction of some operator U ∈ A. We have therefore finished the proof of property (ii). The proof of (iii) is much easier, and will complete the proof of Theorem 27.

Lemma 30 The seminorm |||.||| on L(X) satisfies the algebra inequality

|||UV ||| ≤ 75 |||U||| |||V |||.

PROOF. In [GM2], this statement is proved with the constant 1 in place of 75 (see Remark 13). Pick c > 1 and let (x_{n})_{n=1}^{+∞} ∈ M_{X} be a sequence such that ∥UV x_{n}∥ ≥ c^{−1}|||UV ||| for every n. After suitable perturbations and selections of subsequences we may assume that x_{n}, V x_{n}, UV x_{n} have supports before x_{n+1}, V x_{n+1}, UV x_{n+1}, and that x_{n+1}, …, x_{2n} is a RIS for every n ≥ 1. Let u > |||U|||, v > |||V ||| and pick ℓ ∈ L large enough so that

(∗)  ∥Ux∥ ≤ u ∥x∥_{(\sqrt{ℓ})}, ∥V x∥ ≤ v ∥x∥_{(\sqrt{ℓ})}

whenever x ∈ X and ℓ < x. We consider the vectors x_{n+1} < … < x_{n+ℓ^{4}}, for some n > ℓ^{4} such that x_{n+1}, V x_{n+1}, UV x_{n+1} have supports after ℓ. For every subset A ⊂ {1, …, ℓ^{4}} such that |A| ≥ ℓ, we know by Lemma 20 applied with r = |A| to the RIS (x_{n+i})_{i∈A} that ∥ ∑_{i∈A} x_{n+i}∥_{(\sqrt{ℓ})} ≤ 15 |A|/f(|A|), hence we get by (∗) that ∥ ∑_{i∈A} V x_{n+i}∥ ≤ 15 v |A|/f(|A|); by Lemma 16 applied with m = ℓ, n = \sqrt{ℓ} and r = ℓ^{4} to the successive sequence (κ V x_{n+i})_{i=1}^{r}, κ = (15 v)^{−1}, this yields

∥ ∑_{i=1}^{r} V x_{n+i}∥_{(\sqrt{ℓ})} ≤ 75 r f(r)^{−1} v;

by (∗) it follows that ∥ ∑_{i=1}^{r} UV x_{n+i}∥ ≤ 75 r f(r)^{−1} u v; but

∥ ∑_{i=1}^{r} UV x_{n+i}∥ ≥ c^{−1} rf(r)^{−1}|||UV |||

by the lower f-estimate, and finally |||UV ||| ≤ 75 c u v.

9.1 The shift space

Let S be the proper ∗-semi-group mentioned earlier, generated by the shift, which we denote by S. That is, S consists of all maps of the form S_{A,B} where A = [m, ∞) and B = [n, ∞). We will write L for the left shift, which is (formally) the adjoint of S. Then every operator in S is of the form S^{m}L^{n}, because LS = Id. Since SL − Id is of rank one, every operator in A is a finite-rank perturbation of an operator of the form ∑_{n=0}^{N} λ_{n}S^{n} + ∑_{n=1}^{N} μ_{n}L^{n}, so the difference is of |||.|||-norm zero. Let X_{s} denote the space obtained from Theorem 27 in this case.

Lemma 31 Let U = ∑_{n=0}^{N} λ_{n}S^{n} + ∑_{n=1}^{N} μ_{n}L^{n}. Then

∥U∥ = |||U||| = ∑_{n=0}^{N} |λ_{n}| + ∑_{n=1}^{N} |μ_{n}|.

PROOF. For notational convenience, let λ_{−n} = μ_{n} for 1 ≤ n ≤ N. Clearly it is enough to prove that |||U||| ≥ ∑_{n=−N}^{N} |λ_{n}|. In this paper, we did not write almost isometric versions of the basic Lemmas, so we will only prove this inequality up to some multiplicative constant.

Let m ≥ 1 be given. For an integer r ∈ L consider the vector x_{r} = ∑_{j=r+1}^{2r} e_{3jN}. Since every unit vector e_{i} satisfies ∥e_{i}∥_{(n)} = 1 for every n, we have ∥x_{r}∥ ≤ r/f(r) by Lemma 14; let y_{r} = r^{−1}f(r)x_{r}; by Lemma 16, there exists r such that ∥y_{r}∥_{(m)} ≤ 5 and we may choose r as big as we like, in particular r ≥ 3N. This shows that the sequence \frac{1}{5}(y_{r}) has subsequences in M, thus ∥Uy_{r}∥ ≤ (5 + ε) | ; | ; |U| ; | ; | for large r. On the other hand, splitting Ux_{r} into 3rN singleton pieces from 3rN + 1 to (6r + 1)N gives that ∥Ux_{r}∥ ≥ r f(3Nr)^{−1} ∑_{n=−N}^{N} |λ_{n}| by the lower f-estimate. This lower bound on ∥Ux_{r}∥ gives

10 |||U||| ≥ 5f(3Nr) f(r)^{−1} |||U||| ≥ ∑_{n=−N}^{N} |λ_{n}|.

Let W denote the Wiener convolution algebra ℓ_{1}(ℤ). The preceding Lemma gives an isometric embedding i from W into L(X_{s}). Indeed, since all powers of S and L have norm 1, we may associate to any a = (a_{n}) ∈ ℓ_{1}(ℤ) the operator i(a) = ∑_{n=0}^{∞} a_{n}S^{n} + ∑_{n=1}^{∞} a_{−n}L^{n} ∈ L(X_{s}). The next result gives, up to a strictly singular perturbation, the converse of this fact. We call Toeplitz operators on X_{s} the elements from the subspace T = i(W).

Corollary 32 There is an algebra homomorphism and projection ϕ from the space L(X_{s}) onto the subspace T consisting of Toeplitz operators with absolutely summable coefficients. If T ∈ L(X_{s}) then |||ϕ(T) − T ||| = 0.

PROOF. Recall the remark following the statement of Theorem 27. In this case, by Lemma 31, the algebra G, the |||.|||-completion of A, is the same as the completion in L(X_{s}) and also the completion in the operator norm on ℓ_{1}. Therefore G can be regarded as a subalgebra of L(X_{s}) consisting of Toeplitz operators with absolutely summable coefficients. If we do this, then the algebra homomorphism ϕ defined after Theorem 27 is also a projection. The equation |||ϕ(T) − T ||| = 0 follows easily from the definition of ϕ.

Theorem 33 The space X_{s} is prime.

PROOF. Let P : X_{s} → X_{s} be a projection. By the previous corollary the operator ϕ(P) is a convolution by some absolutely summable sequence (a_{n})_{n∈ℤ}. Moreover, ϕ(P)^{2} = ϕ(P). But the Fourier transform of the sequence (a_{n})_{n∈ℤ} is a continuous function on the circle squaring to itself. Hence it is constantly zero or one. It follows that a_{0} is zero or one and all the other a_{n} are zero. That is, ϕ(P) is zero or the identity. Since P − ϕ(P) is strictly singular, it follows that P is of finite rank or corank. Thus, if P X_{s} is infinite-dimensional, then it has finite codimension. Since the shift on X_{s} is an isometry, it follows that X_{s} is isometric to its range, namely an hyperplane; using powers of S, we see that X_{s} and P X_{s} are isomorphic, which proves the theorem.

A simple consequence of Corollary 32 is that, up to strictly singular perturbations, any two operators on X_{s} commute. Indeed, if V and W are two operators, then ϕ(V) and ϕ(W) commute, so ϕ(V W − W V) = 0, from which it follows that |||V W − W V ||| = 0. For the rest of this section, we assume that X_{s} has complex scalars. Let ψ : L(X_{s}) → C(𝕋) be the composition of ϕ with the Fourier transform. Then ψ is also a continuous algebra homomorphism. Given an operator T on X_{s}, let K_{T} be the compact set of μ ∈ ℂ such that μ is infinitely singular for T, which means for us that for every ε > 0 there is an infinite-dimensional subspace Y ⊂ X_{s} such that ∥T y − μy∥ ≤ ε∥y∥ for every y ∈ Y. Since T − ϕ(T) is strictly singular, K_{ϕ(T)} = K_{T}.

Lemma 34 The function ψ(T) takes the value zero at some exp(iθ) if and only if 0 is infinitely singular for T.

PROOF. If ψ(T) takes the value zero at exp iθ, we can construct an approximate eigenvector for ϕ(T) with eigenvalue zero as follows. Suppose that ϕ(T) is convolution by the sequence (a_{n})_{n∈ℤ}, and let ε > 0. We know that ψ(T)(θ) = ∑_{n∈ℤ} a_{n} exp(inθ) = 0. Let ℓ ∈ L and let

x_{ℓ} = f(ℓ^{2}) ℓ^{−2} ∑_{n=ℓ^{2}}^{2ℓ^{2}} exp(inθ)e_{n}.

By Lemma 14 we have ∥x_{ℓ}∥ = 1, because ∥e_{n}∥_{(p)} = 1 for every p ≥ 1. Let U be convolution by the sequence (a_{n})_{n=−ℓ}^{ℓ}. If ℓ is large enough, then ∥U − ϕ(T)∥ ≤ ε/2, since (a_{n})_{n∈ℤ} is absolutely summable. Moreover, all but at most 4ℓ of the possible ℓ^{2} + 2ℓ non-zero coordinates of Ux_{ℓ} are equal to f(ℓ^{2}) ℓ^{−2} ∑_{n=−ℓ}^{ℓ} a_{n} exp(inθ). Taking ℓ sufficiently large, we can therefore make ∥ϕ(T) − U∥ and ∥Ux_{ℓ}∥ as small as we like. Therefore zero is infinitely singular for ϕ(T). Since |||T − ϕ(T)||| = 0, the same is true for T.

Conversely, if ψ(T) never takes the value zero, then it can be inverted in C(𝕋). A classical result states that the Fourier transform of this inverse will also be in ℓ_{1}(ℤ), so in particular ϕ(T) has an inverse U which is continuous when considered as an operator on X_{s} and satisfies U = ϕ(U). Therefore ϕ(UT − Id) = 0, so UT − Id is strictly singular and 0 is not infinitely singular for T.

Corollary 35 The set K_{T} is the image under ψ(T) of the unit circle 𝕋.

PROOF. This follows from Lemma 34 applied to the operator T − λId.

Theorem 36 A subspace Y of X_{s} is isomorphic to X_{s} if and only if it has finite codimension.

PROOF. Let T : X_{s} → Y be an isomorphism. Then 0 is not infinitely singular for T, so, as in the proof of Lemma 34, we can find U such that T U, UT and Id are the same, up to a strictly singular perturbation. Since T U − Id is strictly singular, T U is Fredholm with index zero. In particular codim Y = codim T X_{s} ≤ codim T UX_{s} < ∞. As we have already mentioned, the if part follows from the existence of the isometric shift.

10 Appendix

Let us check that when g is defined by the formula (F) from section 7, then t/g(t) is concave. We have that

k(t) = t/g(t) = exp( ∫_{0}^{ln(t)}\frac{e^{M(u)}}{1 + e^{M(u)}} du),

hence after some computations

k′′(t) = \frac{k′(t)}{t(1 + e^{M(ln t)})} (M′(ln t) − 1) ≤ 0

because M is 1-Lipschitz and k′(t) ≥ 0.

Proof of Lemma 11. We shall explain how to go down from the “big” function f_{0} at t_{0} = e^{u_{0}} to the “small” function f_{1} at t_{1} = e^{4u_{0}^{2}}, or equivalently, how to go from the small function M_{0}(u) = ln(1+u) at u_{0} to the large function M_{1}(u) = ln(3 + 2u) at u_{1} = 4u_{0}^{2}. More precisely, we need to build a function M ∈ L, that coincides with M_{0} on [0, u_{0}] and with M_{1} on [u_{1}, +∞), and such that g = g_{M} satisfies the conclusions of the lemma, which reduce then to g(t_{0}) = f_{0}(t_{0}), g(t_{1}) = f_{1}(t_{1}) and f_{1} ≤ g ≤ f_{0} on [t_{0}, t_{1}]. The proof of the other case is similar.

It is clear that M ≤ N on [0, u] implies g_{N} ≤ g_{M} on [1, e^{u}], and since tg′_{N}(t)/g_{N}(t) = (1 + e^{N(ln(t))})^{−1} we see that the inequality M ≤ N on some interval [u, v] ⊂ [0, +∞) implies that g_{M}/g_{N} is non-decreasing on the interval [e^{u}, e^{v}].

We first define a function N ∈ L by letting N(u) = M_{0}(u) for u ≤ u_{0} and N(u) = M_{0}(u_{0}) + u − u_{0} = ln(1 + u_{0}) + u − u_{0} for u ≥ u_{0}. There is a unique value v_{0} > u_{0} such that N(v_{0}) = M_{1}(v_{0}) and a v_{1} such that N(v_{1}) = M_{1}(u_{1}) = ln(3+2u_{1}), given by v_{1} = ln(3+2u_{1})+u_{0}−ln(1+u_{0}); one can check that v_{1} < u_{1} since M_{1}(u_{1}) − M_{0}(u_{0}) < u_{1} − u_{0} (exercise); it follows that u_{0} < v_{0} < v_{1} < u_{1} (the reader must draw a picture). Let N_{0} ∈ L be equal to N on [0, v_{0}] and to M_{1} on [v_{0}, +∞), and N_{1} ∈ L be equal to N on [0, v_{1}], equal to the constant value M_{1}(u_{1}) on [v_{1}, u_{1}] and equal to M_{1} on [u_{1}, +∞). For every s ∈ [0, 1] let N_{s} = (1 − s)N_{0} + sN_{1}, and let g_{s} = g_{N_{s}} ∈ F_{0} be the corresponding function. Notice that g_{s}(t_{0}) = f_{0}(t_{0}) for every s. Our problem is to make sure that for some s = s_{0} ∈ (0, 1), the value of g_{s} at the other end t_{1} is what we expect, namely g_{s}(t_{1}) = f_{1}(t_{1}). It will follow that g_{s_{0}} = f_{1} on [t_{1}, +∞), and it will only remain to check that f_{1} ≤ g_{s_{0}} ≤ f_{0}. We shall first check that g_{1}(e^{u_{1}}) < f_{1}(e^{u_{1}}), and next that g_{0}(e^{u_{1}}) > f_{1}(e^{u_{1}}): we have

ln(\frac{g_{1}(e^{v_{1}})}{f_{0}(e^{u_{0}})}) = ln(\frac{g_{1}(e^{v_{1}})}{g_{1}(e^{u_{0}})}) = ∫_{u_{0}}^{v_{1}} \frac{du}{1 + e^{N(u)}} = ∫_{u_{0}}^{v_{1}} \frac{du}{1 + (1 + u_{0}) e^{u−u_{0}}}

≤ \frac{1}{1 + u_{0}} ∫_{u_{0}}^{+∞}e^{u_{0}−u} du ≤ \frac{1}{1 + u_{0}} < \frac{1}{6},

ln(\frac{g_{1}(e^{u_{1}})}{g_{1}(e^{v_{1}})}) = ∫_{v_{1}}^{u_{1}} \frac{du}{1 + e^{M_{1}(u_{1})}} = ∫_{v_{1}}^{u_{1}} \frac{du}{4 + 2u_{1}} < \frac{1}{2}

so that finally g_{1}(e^{u_{1}})/f_{0}(e^{u_{0}}) ≤ e^{2/3} < 2. On the other hand

f_{1}(e^{u_{1}})/f_{0}(e^{u_{0}}) = (1 + \frac{1}{2} ln t_{1})^{1/2}/(1 + \frac{1}{2} ln t_{0})
= (1 + 2u_{0}^{2})^{1/2}/(1 + \frac{u_{0}}{2} ) > 2,

(because u_{0} ≥ 5) so that f_{1}(e^{u_{1}}) > g_{1}(e^{u_{1}}). Next we see that g_{0}(e^{u_{1}}) > f_{1}(e^{u_{1}}): since N_{0} = M_{1} on [v_{0}, u_{1}], the quotient g_{0}/f_{1} is constant on the interval [e^{v_{0}}, e^{u_{1}}], thus g_{0}(e^{u_{1}})/f_{1}(e^{u_{1}}) = g_{0}(e^{v_{0}})/f_{1}(e^{v_{0}}) > 1 because N_{0} < M_{1} on the interval [0, v_{0}).

We have that g_{1}(e^{u_{1}}) < f_{1}(e^{u_{1}}) < g_{0}(e^{u_{1}}). By continuity there exists s ∈ (0, 1) such that g_{s}(e^{u_{1}}) = f_{1}(e^{u_{1}}). It only remains to check that f_{1} ≤ g_{s} ≤ f_{0}. Since N_{s} < M_{1} on [u_{0}, v_{0}) and N_{s} > M_{1} on (v_{0}, u_{1}] we know that g_{s}/f_{1} is increasing on [t_{0}, e^{v_{0}}) and decreasing on (e^{v_{0}}, t_{1}]. Since the quotient is > 1 at t_{0} and equal to 1 at t_{1}, it follows that f_{1} ≤ g_{s} on [t_{0}, t_{1}]. Using similar but simpler arguments one can see that g_{s} ≤ f_{0} on the same interval, because N_{s} ≥ M_{0} on [0, u_{1}].

Proof of Lemma 14. Let G(t) = t/g(t) when t ≥ 1 and G(t) = t when 0 ≤ t ≤ 1. This function G is concave and increasing on [0, +∞). For every interval E and every integer ℓ ≥ 0, let

σ_{ℓ}(E) = ∑_{i=1}^{r} ∥Ex_{i}∥_{(p^{ℓ})}.

This expression is increasing with ℓ, and

σ_{0}(E) = ∑_{i=1}^{r} ∥Ex_{i}∥_{(1)} = ∑_{i=1}^{r} ∥Ex_{i}∥ ≥ ∥Ex∥.

We shall prove by induction on κ, 1 ≤ κ ≤ r that whenever E is an interval such that Ex_{i} ≠ 0 for at most κ indices i, then

(∗)  ∥Ex∥ ≤ G(σ_{κ}(E)).

Once this is done, we obtain the result for κ = r and E = ran(x),

∥x∥ ≤ G(σ_{r}(ran(x))) = G(∑_{i=1}^{r} ∥x_{i}∥_{(p^{r})}) ≤ G(r) = \frac{r}{g(r)}.

Let κ(E) denote the number of indices i ∈ {1, …, r} such that Ex_{i} ≠ 0 (if κ(E) = 0, then Ex = 0 and this case is obvious). Observe first that when ∥Ex∥ ≤ 1, we have ∥Ex∥ = G(∥Ex∥) ≤ G(∑_{i=1}^{r} ∥Ex_{i}∥) ≤ G(σ_{ℓ}(E)) for every ℓ ≥ 0. This shows in particular that (∗) is true when κ(E) = 1, since ∥Ex∥ ≤ 1 in this case. Assume (∗) true when κ(E) ≤ ℓ < r, and suppose there exists an interval E such that κ(E) = ℓ + 1 and ∥Ex∥ > G(σ_{ℓ+1}(E)); since (∗) is not true for E we know that ℓ ≥ 1 and ∥Ex∥ > 1. By assumption there exists a (q, g)-form x^{∗} = g(q)^{−1}(∑_{j=1}^{q} A_{j}x_{j}^{∗}), 2 ≤ q ≤ p, where A_{j} = ran(x_{j}^{∗}), A_{1} < … < A_{q} and ∥x_{j}^{∗}∥ ≤ 1, such that

G(σ_{ℓ+1}(E)) < |x^{∗}(Ex)|.

Assume first that κ(A_{j}E) ≤ ℓ for every j = 1, …, q. We have ∥A_{j}Ex∥ ≤ G(σ_{ℓ}(A_{j}E)) by the induction hypothesis, and using the concavity of G and the relation q ≤ p we obtain

|x^{∗}(Ex)| ≤ \frac{1}{g(q)} ∑_{j=1}^{q} ∥A_{j}Ex∥ ≤ \frac{q}{g(q)} ∑_{j=1}^{q} \frac{1}{q} G(σ_{ℓ}(A_{j}E))

≤ \frac{q}{g(q)} G(\frac{1}{q} ∑_{j=1}^{q} σ_{ℓ}(A_{j}E)) = \frac{q}{g(q)} G(\frac{1}{q} ∑_{i=1}^{r} ∑_{j=1}^{q} ∥A_{j}Ex_{i}∥_{(p^{ℓ})})
≤ \frac{q}{g(q)} G(\frac{1}{q} ∑_{i=1}^{r} ∥Ex_{i}∥_{(p^{ℓ+1})}) = \frac{q}{g(q)} G(\frac{σ_{ℓ+1}(E)}{q}).

If σ_{ℓ+1}(E) ≤ q, this last expression is equal to

σ_{ℓ+1}(E)/g(q) ≤ σ_{ℓ+1}(E)/g(σ_{ℓ+1}(E)) = G(σ_{ℓ+1}(E)),

otherwise it is equal to

\frac{σ_{ℓ+1}(E)}{g(q)g(σ_{ℓ+1}(E)/q)} ≤ \frac{σ_{ℓ+1}(E)}{g(σ_{ℓ+1}(E))} = G(σ_{ℓ+1}(E)),

so that we have reached a contradiction.

In the remaining case there exists j_{0} ∈ {1, …, q} such that A_{j_{0}}Ex_{i} ≠ 0 for every i such that Ex_{i} ≠ 0. Assume for example j_{0} < q (otherwise 1 < j_{0} deserves a similar treatment). Let m be the last integer i such that Ex_{i} ≠ 0. Let B_{j_{0}} = A_{j_{0}} \ ran(Ex_{m}), B′_{j_{0}+1} = A_{j_{0}} ∩ ran(Ex_{m}), B′′_{j_{0}+1} = A_{j_{0}+1}, B_{j_{0}+1} = B′_{j_{0}+1} ∪ B′′_{j_{0}+1} and B_{j} = A_{j} otherwise. We see that

∥A_{j_{0}}Ex∥ + ∥A_{j_{0}+1}Ex∥ ≤ ∥B_{j_{0}}Ex∥ + ∥B′_{j_{0}+1}Ex_{m}∥ + ∥B′′_{j_{0}+1}Ex_{m}∥
≤ ∥B_{j_{0}}Ex∥ + ∥B_{j_{0}+1}Ex_{m}∥_{(2)}.

Every B_{j} satisfies κ(B_{j}E) ≤ ℓ, so that the induction hypothesis applies and since p^{ℓ} ≥ 2 we obtain

∑_{j=1}^{q} ∥A_{j}Ex∥ ≤ ∥B_{j_{0}+1}Ex_{m}∥_{(2)} + ∑_{j≠j_{0}+1} ∥B_{j}Ex∥
≤ G(σ_{ℓ}(B_{j_{0}+1}E)) + ∑_{j≠j_{0}+1} G(σ_{ℓ}(B_{j}E)),

and the conclusion follows as before.

Proof of Lemma 25. Let t be a node in ⋃_{n≥0} E_{n}. If m is the integer such that t ∈ E_{m}, let b_{t} denote the segment consisting of those nodes s ∈ E_{m} such that s ≤ t. We let F′_{m} denote the set of nodes t ∈ E_{m} such that ∥x(b_{t})∥ ≥ α/2 and we let F′′_{m} be the complement of F′_{m} in E_{m}. It is clear that b ∩ F′′_{m} is a segment, whenever b is a segment.

Let w = ∑_{b∈A} λ_{b} x(b) be an element of W, with A a finite family of segments and ∑_{b∈A} |λ_{b}| ≤ 1. Let us check that ∥F′′_{m} w∥ < α/2 for every m. For each b ∈ A, let b′′ denote the segment b ∩ F′′_{m}. If t is the longest node in b′′, then ∥x(b′′)∥ ≤ ∥x(b_{t})∥ < α/2. Therefore

∥F′′_{m} w∥ = ∥∑_{b∈A} λ_{b} x(b′′)∥ < α/2.

For every n let us write E_{n}w_{n} = ∑_{b∈A_{n}} λ_{b} x(b), where A_{n} is a finite set of segments b contained in E_{n}, and ∑_{b∈A_{n}} |λ_{b}| ≤ 1. Let Γ be the compact set of infinite branches of the tree T , with the topology of pointwise convergence at nodes. For every b ∈ A_{n}, let γ_{b} be an infinite branch containing the segment b and consider the non-negative measure μ_{n} = ∑_{b∈A_{n}} |λ_{b}| δ_{γ_{b}} on Γ. Passing to a subsequence we may assume that (μ_{n}) is weak-∗ convergent to a finite non-negative measure μ on Γ.

Let B_{m} denote the set of γ ∈ Γ that intersect F′_{m}. We want to show that μ(B_{m}) tends to 0. If not, we may find for every k a branch γ and m_{1} < … < m_{k} such that γ ∈ B_{m_{j}} for j = 1, …, k. Let b_{j} be the segment γ ∩ E_{m_{j}}. It follows from the definition of F′_{m_{j}} that ∥x(b_{j})∥ ≥ α/2 for j = 1, …, k. But this is impossible when k is large, because ∑_{j} ∥x(b_{j})∥^{p} ≤ ∥x(γ)∥^{p} ≤ 1, by the definition of the space.

Suppose that m = m_{0} satisfies μ(B_{m}) < α/4. This implies that μ_{n}(B_{m}) < α/4 when n is large, which means that most of the vectors x(b) used in the construction of E_{n} w_{n} sit on branches that do not meet F′_{m}. Taking n > m large enough, we may also have μ(B_{n}) < α/8. We let F′′′_{n} denote the set of nodes in E_{n} that are above some node in F′_{m}. Then ∥F′′′_{n}w_{n}∥ < α/4 and the nodes in E′_{n} = F′_{n} \ F′′′_{n} and in E′_{m} = F′_{m} are incomparable. Furthermore, if we let E′′_{n} = F′′_{n} ∪ F′′′_{n}, then ∥E′′_{n} w_{n}∥ < α.

We have just explained the beginning of a construction by induction of a sequence (m_{j})_{j=0}^{∞} satisfying the properties asked in Lemma 25. We let m_{0} = m and m_{1} = n, where m and n are as in the preceding paragraph. The next value of n, which will be chosen as m_{2}, must satisfy μ_{n}(B_{m_{0}}) < α/4, μ_{n}(B_{m_{1}}) < α/8 and μ(B_{n}) < α/16. The reader will easily complete the missing steps.

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