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Weak compactness in the space of operator valued measures. (English) Zbl 1240.28039

Let \(X, Y\) be separable Hilbert spaces with orthonormal bases \(\{x_{i} \}, \; \{y_{j} \}\), \(\mathcal{L} (X, Y)\) be bounded linear operators from \(X\) to \(Y\), and \(\mathcal{L}_{1} (X, Y) \) be nuclear operators from \(X\) to \(Y\) with the norm \(\| L \| _{1}= \sum_{i,j} | (L(x_{i}), y_{j}) | \). Denote by \(I\) a finite interval of the real line with a sigma algebra \(\Sigma\) of subsets of \(I\), and let \(M_{ba}(\Sigma)\) be the space of all real-valued, bounded, finitely additive measures on \(\Sigma\). Let \(B_{\infty} (I, \mathcal{L}_{1} (X, Y))\) be the space of \(\mathcal{L}_{1} (X, Y)\)-valued, bounded, \(\Sigma\)-measurable functions on \(I\) and \(M_{ba}(\Sigma, \mathcal{L}_{1} (Y, X))\) be all finitely additive \(\mathcal{L}_{1} (Y, X)\)-valued vector measures on \(\Sigma\) with finite total variation norms.
The main theorem of the paper is the following: a subset \( \Gamma \subset M_{ba}(\Sigma, \mathcal{L}_{1} (Y, X))\) is relatively (or conditionally) weakly compact if and only if the following three conditions are satisfied:
(i) \(\Gamma\) is bounded;
(ii) for any \(A \in \Sigma\), \(\sum_{i=1}^{\infty} | (K(A)x_{i}, y_{i})_{Y}| \) is uniformly convergent for \(K \in \Gamma\);
(iii) for each \(i \in N\), the set of scalar-valued measures \( \{ (K(.)x_{i}, y_{i}), K \in \Gamma \}\) is a conditionally weakly compact subset of \(M_{ba}(\Sigma)\) .
This result is applied to some feedback control problems of stochastic systems in Hilbert spaces perturbed by centered Poisson counting measures.

MSC:

28B05 Vector-valued set functions, measures and integrals
46G10 Vector-valued measures and integration
93B52 Feedback control
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