
TL;DR
This paper extends the concept of summable Szlenk index to weak$^*$-compact sets, explores its properties under various operations, and introduces an ideal norm, providing new insights into Banach space theory.
Contribution
It generalizes summable Szlenk index to arbitrary weak$^*$-compact sets and establishes its stability under sums, tensor products, and embeddings, also defining a Banach ideal norm.
Findings
Summable Szlenk index characterized for weak$^*$-compact sets.
Stability of summable Szlenk index under $c_0$-sums and tensor products.
Introduces a Banach ideal norm for operators with summable Szlenk index.
Abstract
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 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 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 finite dimensional decomposition, which generalizes a result from [Odell et al 2008]. We also introduce an ideal norm…
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Taxonomy
TopicsAdvanced Banach Space Theory · Intensive Care Unit Cognitive Disorders
