Non strongly bounded operators and \(\ell^1\) (Q2785234)

Let \(C(H,E)\) be the Banach space of continuous functions from a compact Hausdorff space \(H\) into a Banach space \(E\) and \(T:C(H,E)\to F\) be a bounded linear operator into a Banach space \(F\). According to the Riesz representation theorem, there is a unique representing measure \(\mu\) with values in the space \(B(E,F^{**})\) of operators such that \(T(f)=\int_H fd\mu\) for each \(f\in C(H,E)\). For a functional \(z^*\in F^*\), let \(|\mu_{z^*}|\) denote the variation of the \(E^*\)-valued measure \(\mu_{z^*}(A)(e)=\mu(A)(e)(z^*)\) where \(e\in E\), \(A\subset H\). If the space \(E\) is reflexive, then by [\textit{J. K. Brooks} and \textit{P. W. Lewis}, Trans. Am. Math. Soc. 192, 139-162 (1974; Zbl 0331.46035)] the operator \(T\) is weakly compact if and only if for any bounded sequence \((z^*_i)\subset F^*\) the family \(\{|\mu_{z^*_i}|\}\) is uniformly countably additive in the sense that for any countable family \((A_i)\) of pairwise disjoint Borel subsets of \(H\) we get \(\lim_{n\to\infty}\sup_{i} |\mu_{z^*_i}|(\cup_{m>n}A_m)=0\).NEWLINENEWLINENEWLINEThe main result of the paper under review asserts that for any bounded sequence \((z^*_i)\subset F^*\) such that the family \(\{|\mu_{z^*_i}|\}\) is not uniformly countably additive, there is a sequence \((e_i^*)\) in \(\{z_i^*-z_j^*:i,j\in\mathbb N\}\) equivalent to the unit basis of \(\ell^1\) and spanning a complemented subspace of \(F^*\) isomorphic to \(\ell^1\) upon which \(T^*\) acts as an isomorphism.NEWLINENEWLINENEWLINEThis result implies that \(T\) is strongly bounded (and weakly compact if \(E\) is reflexive) provided \(T^*\) fails to be an isomorphism on any complemented copy of \(\ell^1\) in \(F^*\).