Quantifying properties (K) and (μs$\mu ^{s}$)

Dongyang Chen, Tomasz Kania and Yingbin Ruan

Mathematische Nachrichten, 2023, vol. 296, issue 3, 996-1012

Abstract: A Banach space X has property (K), whenever every weak* null sequence in the dual space admits a convex block subsequence (fn)n=1∞$(f_{n})_{n=1}^\infty$ so that ⟨fn,xn⟩→0$\langle f_{n},x_{n}\rangle \rightarrow 0$ as n→∞$n\rightarrow \infty$ for every weakly null sequence (xn)n=1∞$(x_{n})_{n=1}^\infty$ in X; X has property (μs)$(\mu ^{s})$ if every weak* null sequence in X∗$X^{*}$ admits a subsequence so that all of its subsequences are Cesàro convergent to 0 with respect to the Mackey topology. Both property (μs)$(\mu ^{s})$ and reflexivity (or even the Grothendieck property) imply property (K). In this paper, we propose natural ways for quantifying the aforementioned properties in the spirit of recent results concerning other familiar properties of Banach spaces.

Date: 2023
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