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 $\langle f_{n},x_{n}\rangle\to 0$ 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$^{*}$ null sequence in $X^{*}$ admits a subsequence so that all of its subsequences are Ces\`{a}ro 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.