Cesàro convergent sequences in the Mackey topology (Q2325705)

Let \(X\) be a Banach space and let us denote by \(\mu(X^*, X)\) the Mackey topology on \(X^*\) (uniform convergence on weakly compact subsets of \(X\)). A Banach space \(X\) is said to have property \((\mu^s)\) if every \(w^*\)-null sequence in \(X^*\) admits a subsequence such that all of its subsequences are Cesàro convergent to 0 with respect to \(\mu(X^*, X)\). This is a stronger notion of property \((K)\) introduced by Kwapień. In this paper, this notion is mainly studied. For instance, in reflexive spaces \((\mu^s)\) is equivalent to the Banach-Saks property of \(X^*\). For \(C(K)\) spaces, \((\mu^s)\) is equivalent to the Grothendieck property. For Banach spaces strongly generated by a reflexive space, the \((\mu^s)\) property is equivalent to verifying that every \(w^*\)-null sequence in \(X^*\) admits just a subsequence which is Cesàro convergent to 0 with respect to \(\mu(X^*, X)\). This property is also investigated in the \(\ell^1\)- and \(\ell^p\)-sum. On the other hand, an example of a sequence of finite-dimensional spaces whose \(\ell^\infty\)-sum fails property \((K)\) is provided. In the final part of the paper, the author defines property \((\mu^s_{\mathcal{A}})\) for a family \(\mathcal{A}\) of subsets of \(X\) which is studied mainly in Banach lattices and in Lebesgue-Bochner spaces.