Weak Banach-Saks property in the space of compact operators (Q462969)

A Banach space \(E\) has the Banach-Saks property if there exist subsequences (also called Banach-Saks sequences) having norm-convergent arithmetic means for each bounded sequence in \(E\). If every weakly null sequence in \(E\) admits Banach-Saks sequences, then we say that \(E\) satisfies the weak Banach-Saks property. In this paper, the authors introduce the strong Banach-Saks property for operators between Banach spaces \(E\) and \(F\) and establish conditions for a closed linear subspace \({\mathcal M}\) of \(K(E;F)\) or \(K_{w*}(E^*;F)\) to have the weak Banach-Saks property, where \(K(E;F)\) and \(K_{w*}(E^*;F)\) are the spaces of all compact linear operators from \(E\) to \(F\) and from \(E^*\) to \(F\), respectively. Here, the evaluation operators \(\phi_x: {\mathcal M} \rightarrow F\) at \(x \in E\) defined, for each \(T \in {\mathcal M}\), by \(\phi_x(T) = T(x)\), and \(\psi_{y^*}: {\mathcal M} \rightarrow E^*\) at \(y^* \in F^*\) defined, for each \(T \in {\mathcal M}\), by \(\psi_{y^*}(T) = T^*(y^*)\), play a key role in this outcome.