We introduce two new local l_1-indices of the same type as the Bourgain l_1 index; the l_1^+-index and the l_1^+-weakly null index. We show that the l_1^+-weakly null index of a Banach space X is the same as the Szlenk index of X, provided X does not contain l_1. The l_1^+-weakly null index has the same form as the Bourgain l_1 index: if it is countable it must take values omega^alpha for some alpha<omega_1. The different l_1-indices are closely related and so knowing the Szlenk index of a Banach space helps us calculate its local l_1-index, via the l_1^+-weakly null index. We show that I(C(omega^{omega^alpha}))=omega^{1+alpha+1}.
We prove some rather precise renorming theorems for Banach spaces with Szlenk index $ω_0$. We use these theorems to show the invariance of certain quantitative Szlenk-type indices under uniform homeomorphisms.
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.
Given any compact, Hausdorff space $K$ and $1<p<\infty$, we compute the Szlenk and $w^*$-dentability indices of the spaces $C(K)$ and $L_p(C(K))$. We show that if $K$ is compact, Hausdorff, scattered, $CB(K)$ is the Cantor-Bendixson index of $K$, and $ξ$ is the minimum ordinal such that $CB(K)\leqslant ω^ξ$, then $Sz(C(K))=ω^ξ$ and $Dz(C(K))=Sz(L_p(C(K)))= ω^{1+ξ}.$
Let $α$ be an infinite ordinal and $γ$ the unique ordinal satisfying $ω^{ω^γ}\leq α< ω^{ω^{γ+1}}$. We show that the Banach space $C([0,\,α])$ of all continuous scalar-valued functions on the compact ordinal interval $[0,\,α]$ has Szlenk index equal to $ω^{γ+1}$ and $w^\ast$-dentability index equal to $ω^{1+γ+1}$.
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 $α$, …
Given a Banach space $X$, a $w^*$-compact subset of $X^*$, and $1<p<\infty$, we provide an optimal relationship between the Szlenk index of $K$ and the Szlenk index of an associated subset of $L_p(X)^*$. As an application, given a Banach space $X$, we prove an optimal estimate of the Szlenk index of $L_p(X)$ in terms of the Szlenk index of $X$. This extends a result of Hájek and Schlumprecht to uncountable ordinals. More generally, given an operator $A:X\to Y$, we provide an estimate of the Szlenk index of the "pointwise $A$" operator $A_p:L_p(X)\to L_p(Y)$ in terms of the Szlenk index of $A$.
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 …
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 $\tens$ 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 $\ell_1$ or $c_0$ in…
Let $X$ be a Banach space and $K$ an absolutely convex, weak$^\ast$-compact subset of $X^\ast$. We study consequences of $K$ having a large or undefined Szlenk index and subsequently derive a number of related results concerning basic sequences and universal operators. We show that if $X$ has a countable Szlenk index then $X$ admits a subspace $Y$ such that $Y$ has a basis and the Szlenk indices of $Y$ are comparable to the Szlenk indices of $X$. If $X$ is separable, then $X$ also admits subspace $Z$ such that the quotient $X/Z$ has a basis and the Szlenk indices of $X/Z$ are comparable to the…
We prove that the $c_0$-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 $\mathfrak{E}$-direct sum of separable spaces provided that $\mathfrak{E}$ has a shrinking unconditional basis whose dual basis yields an asymptotic $\ell_p$ structure in $\mathfrak{E}^\ast$. As a corollary, we show that the Tsirelson direct sum of infinitely many copies of $c_0$ has power type $1$ but non-summable Szlenk index.
For each ordinal $α< ω_1$, we prove the existence of a separable, reflexive Banach space with a basis and Szlenk index $ω^{α+1}$ which is universal for the class of separable, reflexive Banach spaces $X$ such that the Szlenk indices $Sz(X), Sz(X^*)$ do not exceed $ω^α$.
For each ordinal $α<ω_1$, we prove the existence of a space with a basis and Szlenk index $ω^{α+1}$ which is universal for the class of spaces with Szlenk index not exceeding $ω^α$. Our proof involves developing a characterization of which Banach spaces embed into spaces with an FDD with upper Schreier space estimates.
We discuss an alternate method for computing the Szlenk index of an arbitrary $w^*$ compact subset of the dual of a Banach space. We discuss consequences of this method as well as offer simple, alternative proofs of a number of results already found in the literature.
We 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 2016] regarding the behavior of summability of Szlenk index under $c_0$ 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 2017]. As a f…
We prove a formula for the Szlenk power type of the injective tensor product of Banach spaces with Szlenk index at most $ω$. 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.
For $α$ an ordinal and $1<p<\infty$, we determine a necessary and sufficient condition for an $\ell_p$-direct sum of operators to have Szlenk index not exceeding $ω^α$. It follows from our results that the Szlenk index of an $\ell_p$-direct sum of operators is determined in a natural way by the behaviour of the $ε$-Szlenk indices of its summands. Our methods give similar results for $c_0$-direct sums.
For a countable ordinal a we denote by C_a the class of separable, reflexive Banach spaces whose Szlenk index and the Szlenk index of their dual are bounded by a. We show that each C_a admits a separable, reflexive universal space. We also show that spaces in the class C_{omega^{a*omega}} embed into spaces of the same class with a basis. As a consequence we deduce that each C_a is analytic in the Effros-Borel structure of subspaces of C[0,1].
We compute the Szlenk index of an arbitrary projective tensor product $C(K)\widehat{\otimes}_πC(L)$ of spaces $C(K), C(L)$ of continuous functions on 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 $C(K)\widehat{\otimes}_πC(L)$ and $C(M)$ or $C(M)\widehat{\otimes}_πC(N)$ for particular choices of $K,L,M,N$.
We find an optimal upper bound on the values of the weak$^*$-dentability index $Dz(X)$ in terms of the Szlenk index $Sz(X)$ of a Banach space $X$ with separable dual. Namely, if $\;Sz(X)=ω^α$, for some $α<ω_1$, and $p\in(1,\infty)$, then $$Sz(X)\le Dz(X)\le Sz(L_p(X))\le {cases} ω^{α+1} &\text{if $α$ is a finite ordinal,} ω^α &\text{if $α$ is an infinite ordinal.} {cases}$$
For $α$ an ordinal, we investigate the class $\mathscr{SZ}_α$ consisting of all operators whose Szlenk index is an ordinal not exceeding $ω^α$. Our main result is that $\mathscr{SZ}_α$ is a closed, injective, surjective operator ideal for each $α$. We also study the relationship between the classes $\mathscr{SZ}_α$ and several well-known closed operator ideals.
For $α$ an ordinal, we investigate the class $\mathscr{SZ}_α$ consisting of all operators whose Szlenk index is an ordinal not exceeding $ω^α$. We show that each class $\mathscr{SZ}_α$ is a closed operator ideal and study various operator ideal properties for these classes. The relationship between the classes $\mathscr{SZ}_α$ and several well-known closed operator ideals is investigated and quantitative factorization results in terms of the Szlenk index are obtained for the class of Asplund operators.
For each ordinal $ξ$ and each $1\leqslant q<\infty$, we define the notion of $ξ$-$q$-summable Szlenk index. When $ξ=0$ and $q=1$, this recovers the usual notion of summable Szlenk index. We define for an arbitrary weak$^*$-compact set a transfinite, asymptotic analogue $α_{ξ,p}$ of the martingale type norm of an operator. We prove that this quantity is determined by norming sets and determines $ξ$-Szlenk power type and $ξ$-$q$-summability of Szlenk index. This fact allows us to prove that the behavior of operators under the $α_{ξ,p}$ seminorms passes in the strongest way to injective tensor pr…
In a previous work, the first named author described the set $\cal P$ of all values of the Szlenk indices of separable Banach spaces. We complete this result by showing that for any integer $n$ and any ordinal $α$ in $\cal P$, there exists a separable Banach space $X$ such that the Szlenk of the dual of order $k$ of $X$ is equal to the first infinite ordinal $ω$ for all $k$ in $\{0,..,n-1\}$ and equal to $α$ for $k=n$. One of the ingredients is to show that the Lindenstrauss space and its dual both have a Szlenk index equal to $ω$. We also show that any element of $\cal P$ can be realized as a…
For every $α<ω_1$ we establish the existence of a separable Banach space whose Szlenk index is $ω^{αω+1}$ and which is universal for all separable Banach spaces whose Szlenk-index does not exceed $ω^{αω}$. In order to prove that result we provide an intrinsic characterization of which Banach spaces embed into a space admitting an FDD with upper estimates.
We give necessary and sufficient conditions for an operator $A:X\to Y$ 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 $ξ\in (0, ω_1)\setminus\{ω^η: η<ω_1\text{\ a limit ordinal}\}$, there exists a Banach space $\mathfrak{G}_ξ$ having a shrinking basis and Szlenk index $ω^ξ$ such that for any separable Banach space $X$ and any operator $A:X\to Y$ having Szlenk index less than $ω^ξ$, $A$ factors through a subspace and through a quotient of $\mathfrak{G}_ξ$, a…
We prove that for any non-zero, countable ordinal $ξ$ which is not additively indecomposable, the property of having Szlenk index not exceeding $ω^ξ$ 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 $ω^ξ$ is a three space property.
It is shown that a script L_infty-space with separable dual constructed by Bourgain and Delbaen has small Szlenk index and thus does not have a quotient isomorphic to C(omega^omega). It follows that this is a script L_infty-space which is the same size as c_0 in the sense of the Szlenk index but does not contain c_0. This has some consequences in the theory of uniform homeomorphism of Banach spaces.
We show that if the Szlenk index of a Banach space $X$ is larger than the first infinite ordinal $ω$ or if the Szlenk index of its dual is larger than $ω$, then the tree of all finite sequences of integers equipped with the hyperbolic distance metrically embeds into $X$. We show that the converse is true when $X$ is assumed to be reflexive. As an application, we exhibit new classes of Banach spaces that are stable under coarse-Lipschitz embeddings and therefore under uniform homeomorphisms.
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 $\mathcal{J}_p$ nor into its dual space $\mathcal{J}_p^*$ for $p\in(1,\infty)$.
We calculate ordinal $L_p$ index defined in "An ordinal L_p index for Banach spaces with an application to complemented subspaces of L_p" authored by J. Bourgain, H. P. Rosenthal and G. Schechtman, for Rosenthal's space $X_p$, $\ell_p$ and $\ell_2$. We show a subspace of $L_p$ $(2 < p < \infty)$ non isomorphic to $\ell_2$ embeds in $\ell_p$ if and only if its ordinal index is minimum possible. We also give a sufficient condition for a $\mathcal{L}_p$ subspace of $\ell_p\oplus\ell_2$ to be isomorphic to $X_p$.
The Atiyah-Singer index theorem gives a topological formula for the index of an elliptic differential operator. Enlightening from Alain Connes' tangent groupoid proof of the index theorem and van Erp's research for the Heisenberg index theory on contact manifolds, we give an explicit construction of a series of maps, whose induced map in K-theory is the Heisenberg Atiyah-Singer index map on contact manifolds. Our methods derive from Higson's construction for symbol class in K-theory.
