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

Abstract

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

19 pp

References in corpus (2)

• On quantification of weak sequential completeness
• Quantification of the reciprocal Dunford-Pettis property