Some connections between Pettis integration and opertor theory
Suppose X is a Banach space, (\(\Omega\),\(\Sigma\),\(\mu\)) is a probability space, f: \(\Omega\to X\) is a scalarly measurable function such that \(x^*f\in L^ 1(\mu)\) for each \(x^*\in X^*\) and \(T_ f\) is the operator from \(X^*\) to \(L^ 1(\mu)\) whose value at \(x^*\) is \(x^*f\). \textit{R. E. Huff} [Proc. Am. Math. Soc. 96, No.3, 402-404 (1986; Zbl 0611.28005)] and \textit{E. M. Bator} [ibid. 95, No.2, 265-270 (1985; Zbl 0589.46033)] have given elegant exposés of a number of previously elusive results regarding the Pettis integral (including the Geitz- Talagrand core theorem) via a careful study of operators like \(T_ f\). Now, taking up where Bator and Huff left off, the authors generalize the ideas implicit in those studies to general operators. In so doing they obtain some very interesting (initial) characterizations of weak\({}^*\)- weak continuous operators along with further simplifications for results of Odell, Riddle, Saab, Uhl and others on when a Banach space finds itself without a subspace isomorphic to \(\ell^ 1\).
scientific article; zbMATH DE number 4159417