The Bartle bilinear integration and Carleman operators
Let \(X\) and \(Y\) be Banach spaces, \(\Sigma\) a \(\sigma\)-algebra of subsets of a set \(S\) and \(v:\Sigma\to Y\) a countably additive vector measure. If \(f:S\to X\) is strongly measurable, the authors consider the bilinear Bartle integral of \(f\) with respect to \(v\) and the bilinear map \(X\times Y\to X\otimes_\varepsilon Y\), the inductive tensor product. They give several characterizatons of the integrability of \(f\), several which involve the compactness of the operator \(X'\to L^1(v)\), \(x'\to x'f\). The authors also relate the integrability of the function \(f\) with abstract Carleman operators. An operator \(u\) from \(X\) into an order complete Köthe function space \(L\) is a Carleman operator if there exists a strongly measurable function \(f: S\to X'\) such that for every \(x\in X\) we have \(u(x)(\cdot)= f(\cdot)x\) a.e. It is shown that the space of \(X\)-valued Bartle integrable functions can be identified with the space of compact Carleman operators from \(X'\) into \(L^1(v)\) which are weak\(^*\) weak continuous. A similar characterization for \(X'\)-valued strongly measurable functions is also given. Examples are given which show that analogous results fail if the inductive tensor product is replaced by the projective tensor product. The paper concludes with several interesting Fubini results for the injective tensor product of two vector measures.
scientific article; zbMATH DE number 1417567
scientific article