Weak compactness in the space of operator valued measures \(M_{ba} (\Sigma, {\mathcal L} (X, Y))\) and its applications (Q2906303)

Let \(D\) be a compact Hausdorff space, which is an \(\mathcal{F}\)-space (i.e., any two disjoint open \(F_\sigma\)-sets have disjoint closures), and \(\Sigma\) denote the Borel \(\sigma\)-algebra of \(D\). Let \(X,\, Y\) be Banach spaces and \(\mathcal{L}(X,Y)\) denote the space of all bounded linear \(Y\)-valued operators defined on \(X\) endowed with the operator norm. \(M_{casbsv}(\Sigma,{\mathcal L}(X,Y))\) denotes the space of all \({\mathcal L}(X,Y)\)-valued measures on \(\Sigma\) which are countably additive with respect to the strong operator topology and have finite ``variation in the strong operator topology''; this variation defines a norm on \(M_{casbsv}(\Sigma,{\mathcal L}(X,Y))\). The main result gives a characterization of the relatively weakly compact subsets of \(M_{casbsv}(\Sigma,{\mathcal L}(X,Y))\) under the assumption that \(Y\) is reflexive. Moreover, some applications of this result in optimization and control theory are given.