Cesàro 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 Cesàro convergent to 0 with respect to the Mackey topology. This is stronger than the so-called property (K) of Kwapień. 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.