Ces\`{a}ro convergent sequences in the Mackey topology

Abstract

A Banach space $X$ is said to have property ($\mu^s$) if every weak$^*$-null sequence in $X^*$ admits a subsequence such that all of its subsequences are Ces\`{a}ro convergent to $0$ with respect to the Mackey topology. This is stronger than the so-called property (K) of Kwapie\'{n}. We prove that property $(\mu^s)$ holds for every subspace of a Banach space which is strongly generated by an operator with Banach-Saks adjoint (e.g. a strongly super weakly compactly generated space). The stability of property $(\mu^s)$ under $\ell^p$-sums is discussed. For a family $\mathcal{A}$ of relatively weakly compact subsets of $X$, we consider the weaker property $(\mu_\mathcal{A}^s)$ which only requires uniform convergence on the elements of $\mathcal{A}$, and we give some applications to Banach lattices and Lebesgue-Bochner spaces. We show that every Banach lattice with order continuous norm and weak unit has property $(\mu_\mathcal{A}^s)$ for the family of all $L$-weakly compact sets. This sharpens a result of de Pagter, Dodds and Sukochev. On the other hand, we prove that $L^1(\nu,X)$ (for a finite measure $\nu$) has property $(\mu_\mathcal{A}^s)$ for the family of all $\delta\mathcal{S}$-sets whenever $X$ is a subspace of a strongly super weakly compactly generated space.