Szlenk Index Research Articles

Asymptotic smoothness in Banach spaces, three-space properties and applications

We study four asymptotic smoothness properties of Banach spaces, denoted T p , A p , N p \mathsf {T}_p,\mathsf {A}_p, \mathsf {N}_p , and P p \mathsf {P}_p . We complete their description by proving the missing renorming characterization for A p \mathsf {A}_p . We show that asymptotic uniform flattenability (property T ∞ \mathsf {T}_\infty ) and summable Szlenk index (property A ∞ \mathsf {A}_\infty ) are three-space properties. Combined with the positive results of the first-named author, Draga, and Kochanek, and with the counterexamples we provide, this completely solves the three-space problem for this family of properties. We also derive from our characterizations of A p \mathsf {A}_p and N p \mathsf {N}_p in terms of equivalent renormings, new coarse Lipschitz rigidity results for these classes.

The Szlenk index of 𝐿_{𝑝}(𝑋) and 𝐴_{𝑝}

Given a Banach space X X , a w ∗ w^* -compact subset of X ∗ X^* , and 1 > p > ∞ 1>p>\infty , we provide an optimal relationship between the Szlenk index of K K and the Szlenk index of an associated subset of L p ( X ) ∗ L_p(X)^* . As an application, given a Banach space X X , we prove an optimal estimate of the Szlenk index of L p ( X ) L_p(X) in terms of the Szlenk index of X X . This extends a result of Hájek and Schlumprecht to uncountable ordinals. More generally, given an operator A : X → Y A:X\to Y , we provide an estimate of the Szlenk index of the “pointwise A A ” operator A p : L p ( X ) → L p ( Y ) A_p:L_p(X)\to L_p(Y) in terms of the Szlenk index of A A .

Szlenk index of [formula omitted

We compute the Szlenk index of the projective tensor product C(K)⊗ˆπC(L) of spaces C(K),C(L) of continuous functions on arbitrary scattered, compact, Hausdorff spaces. In particular, we show that it is simply equal to the maximum of the Szlenk indices of the spaces C(K),C(L). We deduce several results regarding non-isomorphism of C(K)⊗ˆπC(L) and C(M) or C(M)⊗ˆπC(N) for particular choices of K,L,M,N.

On Kalton’s interlaced graphs and nonlinear embeddings into dual Banach spaces

We study the nonlinear embeddability of Banach spaces and the equi-embeddability of the family of Kalton’s interlaced graphs [Formula: see text] into dual spaces. Notably, we define and study a modification of Kalton’s property [Formula: see text] that we call property [Formula: see text] (with [Formula: see text]). We show that if [Formula: see text] equi-coarse Lipschitzly embeds into [Formula: see text], then the Szlenk index of [Formula: see text] is greater than [Formula: see text], and that this is optimal, i.e. there exists a separable dual space [Formula: see text] that contains [Formula: see text] equi-Lipschitzly and so that [Formula: see text] has Szlenk index [Formula: see text]. We prove that [Formula: see text] does not coarse Lipschitzly embed into a separable dual space by a map with distortion strictly smaller than [Formula: see text]. We also show that neither [Formula: see text] nor [Formula: see text] coarsely embeds into a separable dual by a weak-to-weak[Formula: see text] sequentially continuous map.

Non-three space properties and Szlenk index

We prove that for any non-zero, countable ordinal ξ which is not additively indecomposable, the property of having Szlenk index not exceeding ωξ is not a three space property. This complements a result of Brooker and Lancien, which states that if ξ is additively indecomposable, then having Szlenk index not exceeding ωξ is a three space property.

On the embeddability of the family of countably branching trees into quasi-reflexive Banach spaces

In this note we extend to the quasi-reflexive setting the result of F. Baudier, N. Kalton and G. Lancien concerning the non-embeddability of the family of countably branching trees into reflexive Banach spaces whose Szlenk index and Szlenk index from the dual are both equal to the first infinite ordinal ω. In particular we show that the family of countably branching trees does neither embed into the James space Jp nor into its dual space Jp⁎ for p∈(1,∞).

Factorization of Asplund operators

We give necessary and sufficient conditions for an operator A:X→Y on a Banach space having a shrinking FDD to factor through a Banach space Z such that the Szlenk index of Z is equal to the Szlenk index of A. We also prove that for every ordinal ξ∈(0,ω1)∖{ωη:η<ω1a limit ordinal}, there exists a Banach space Gξ having a shrinking basis and Szlenk index ωξ such that for any separable Banach space X and any operator A:X→Y having Szlenk index less than ωξ, A factors through a subspace and through a quotient of Gξ, and if X has a shrinking FDD, A factors through Gξ.

CONCERNING SUMMABLE SZLENK INDEX

AbstractWe generalize the notion of summable Szlenk index from a Banach space to an arbitrary weak*-compact set. We prove that a weak*-compact set has summable Szlenk index if and only if its weak*-closed, absolutely convex hull does. As a consequence, we offer a new, short proof of a result from Draga and Kochanek [J. Funct. Anal. 271 (2016), 642–671] regarding the behavior of summability of the Szlenk index under c0 direct sums. We also use this result to prove that the injective tensor product of two Banach spaces has summable Szlenk index if both spaces do, which answers a question from Draga and Kochanek [Proc. Amer. Math. Soc. 145 (2017), 1685–1698]. As a final consequence of this result, we prove that a separable Banach space has summable Szlenk index if and only if it embeds into a Banach space with an asymptotic c0 finite dimensional decomposition, which generalizes a result from Odell et al. [Q. J. Math. 59, (2008), 85–122]. We also introduce an ideal norm $\mathfrak{s}$ on the class $\mathfrak{S}$ of operators with summable Szlenk index and prove that $(\mathfrak{S}, \mathfrak{s})$ is a Banach ideal. For 1 ⩽ p ⩽ ∞, we prove precise results regarding the summability of the Szlenk index of an ℓp direct sum of a collection of operators.

ξ-asymptotically uniformly smooth, ξ-asymptotically uniformly convex, and (β) operators

For each ordinal ξ, we define the notions of ξ-asymptotically uniformly smooth and w⁎–ξ-asymptotically uniformly convex operators. When ξ=0, these extend the notions of asymptotically uniformly smooth and w⁎-asymptotically uniformly convex Banach spaces. We give a complete description of renorming results for these properties in terms of the Szlenk index of the operator, as well as a complete description of the duality between these two properties. We also define the notion of an operator with property (β) of Rolewicz which extends the notion of property (β) for a Banach space. We characterize those operators the domain and range of which can be renormed so that the operator has property (β) in terms of the Szlenk index of the operator and its adjoint.

Concerning $q$-summable Szlenk index

For each ordinal $\xi$ and each $1\leqslant q<\infty$, we define the notion of $\xi$-$q$-summable Szlenk index. When $\xi=0$ and $q=1$, this recovers the usual notion of summable Szlenk index. We define for an arbitrary weak$^{*}$-compact set a transfinite, asymptotic analogue $\alpha_{\xi,p}$ of the martingale type norm of an operator. We prove that this quantity is determined by norming sets and determines $\xi$-Szlenk power type and $\xi$-$q$-summability of Szlenk index. This fact allows us to prove that the behavior of operators under the $\alpha_{\xi,p}$ seminorms passes in the strongest way to injective tensor products of Banach spaces. Furthermore, we combine this fact with a result of Schlumprecht to prove that a separable Banach space with good behavior with respect to the $\alpha_{\xi,p}$ seminorm can be embedded into a Banach space with a shrinking basis and the same behavior under $\alpha_{\xi,p}$, and in particular it can be embedded into a Banach space with a shrinking basis and the same $\xi$-Szlenk power type. Finally, we completely elucidate the behavior of the $\alpha_{\xi,p}$ seminorms under $\ell_{r}$ direct sums. This allows us to give an alternative proof of a result of Brooker regarding Szlenk indices of $\ell_{p}$ and $c_{0}$ direct sums of operators.

A metric interpretation of reflexivity for Banach spaces

We define two metrics $d_{1,\alpha}$ and $d_{\infty,\alpha}$ on each Schreier family $\mathcal{S}_\alpha$, $\alpha<\omega_1$, with which we prove the following metric characterization of reflexivity of a Banach space $X$: $X$ is reflexive if and only if there is an $\alpha<\omega_1$, so that there is no mapping $\Phi:\mathcal{S}_\alpha\to X$ for which $$ cd_{\infty,\alpha}(A,B)\le \|\Phi(A)-\Phi(B)\|\le C d_{1,\alpha}(A,B) \text{ for all $A,B\in\mathcal{S}_\alpha$.}$$ Secondly, we prove for separable and reflexive Banach spaces $X$, and certain countable ordinals $\alpha$ that $\max(\text{ Sz}(X),\text{ Sz}(X^*))\le \alpha$ if and only if $({\mathcal S}_\alpha, d_{1,\alpha})$ does not bi-Lipschitzly embed into $X$. Here $\text{Sz}(Y)$ denotes the Szlenk index of a Banach space $Y$.

A note on the relationship between the Szlenk and $$w^*$$ w ∗ -dentability indices of arbitrary $$w^*$$ w ∗ -compact sets

We prove the optimal estimate between the Szlenk and \(w^*\)-dentability indices of an arbitrary \(w^*\)-compact subset of the dual of a Banach space. For a given \(w^*\)-compact, convex subset K of the dual of a Banach space, we introduce a two player game the winning strategies of which determine the Szlenk index of K. We give applications to the \(w^*\)-dentability index of a Banach space and of an operator.

Concerning the Szlenk index

We discuss pruning and coloring lemmas on regular families. We discuss several applications of these lemmas to computing the Szlenk index of certain $w^*$ compact subsets of the dual of a separable Banach space. Applications include estimates of the Szlenk index of Minkowski sums, infinite direct sums of separable Banach spaces, constant reduction, and three space properties. We also consider using regular families to construct Banach spaces with prescribed Szlenk index. As a consequence, we give a characterization of which countable ordinals occur as the Szlenk index of a Banach space, prove the optimality of a previous universality result, and compute the Szlenk index of the injective tensor product of separable Banach spaces.

The Szlenk index of injective tensor products and convex hulls

Given any Banach space X and any weak⁎-compact subset K of X⁎, we compute the Szlenk index of the weak⁎-closed, convex hull of K as a function of the Szlenk index of K. Also as an application, we compute the Szlenk index of any injective tensor product of two operators. In particular, we compute the Szlenk index of an injective tensor product X⊗ˆεY in terms of Sz(X) and Sz(Y). As another application, we give a complete characterization of those ordinals which occur as the Szlenk index of a Banach space, as well as those ordinals which occur as the Bourgain ℓ1 or c0 index of a Banach space.

The Szlenk power type and tensor products of Banach spaces

We prove a formula for the Szlenk power type of the injective tensor product of Banach spaces with Szlenk index at most ω \omega . We also show that the Szlenk power type as well as summability of the Szlenk index are separably determined, and we extend some of our recent results concerning direct sums.

Szlenk indices of convex hulls

We study the general measures of non-compactness defined on subsets of a dual Banach space, their associated derivations and their ω-iterates. We introduce the notions of convexifiable and sublinear measure of non-compactness and investigate the properties of its associated fragment and slice derivations. We apply our results to the Kuratowski measure of non-compactness and to the study of the Szlenk index of a Banach space. As a consequence, we obtain that the Szlenk index and the convex Szlenk index of a separable Banach space are always equal. We also give, for any countable ordinal α, a characterization of the Banach spaces with Szlenk index bounded by ωα+1 in terms of the existence of an equivalent renorming. This extends a result by Knaust, Odell and Schlumprecht on Banach spaces with Szlenk index equal to ω.

Direct sums and summability of the Szlenk index

We prove that the c0-sum of separable Banach spaces with uniformly summable Szlenk index has summable Szlenk index, whereas this result is no longer valid for more general direct sums. We also give a formula for the Szlenk power type of the E-direct sum of separable spaces provided that E has a shrinking unconditional basis whose dual basis yields an asymptotic ℓp structure in E⁎. As a corollary, we show that the Tsirelson direct sum of infinitely many copies of c0 has power type 1 but non-summable Szlenk index.

Equivalent norms with the property $$(\beta )$$ of Rolewicz

We extend to the non separable setting many characterizations of the Banach spaces admitting an equivalent norm with the property \((\beta )\) of Rolewicz. These characterizations involve in particular the Szlenk index and asymptotically uniformly smooth or convex norms. This allows to extend easily to the non separable case some recent results from the non linear geometry of Banach spaces.

Operators on two Banach spaces of continuous functions on locally compact spaces of ordinals

Denote by $[0,\omega_1)$ the set of countable ordinals, equipped with the order topology, let $L_0$ be the disjoint union of the compact ordinal intervals $[0,\alpha]$ for $\alpha$ countable, and consider the Banach spaces $C_0[0,\omega_1)$ and $C_0(L_0)$ consisting of all scalar-valued, continuous functions which are defined on the locally compact Hausdorff spaces $[0,\omega_1)$ and $L_0$, respectively, and which vanish eventually. Our main result states that a bounded operator $T$ between any pair of these two Banach spaces fixes a copy of $C_0(L_0)$ if and only if the identity operator on $C_0(L_0)$ factors through $T$, if and only if the Szlenk index of $T$ is uncountable. This implies that the set $\mathscr{S}_{C_0(L_0)}(C_0(L_0))$ of $C_0(L_0)$-strictly singular operators on $C_0(L_0)$ is the unique maximal ideal of the Banach algebra $\mathscr{B}(C_0(L_0))$ of all bounded operators on $C_0(L_0)$, and that $\mathscr{S}_{C_0(L_0)}(C_0[0,\omega_1))$ is the second-largest proper ideal of $\mathscr{B}(C_0[0,\omega_1))$. Moreover, it follows that the Banach space $C_0(L_0)$ is primary and complementably homogeneous.

Estimation of the Szlenk index of reflexive Banach spaces using generalized Baernstein spaces

For each ordinal $\alpha <\omega _1$, we prove the existence of a separable, reflexive Banach space $W$ with a basis so that $ {\rm Sz}(W), {\rm Sz}(W^*)\leq \omega ^{\alpha +1}$ which is universal for the class of separable, reflexive Banach spaces $X$