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Bilinear integration and applications to operator and scattering theory. (English) Zbl 1351.28003

The author studies the integration of operator valued functions with respect to a spectral or orthogonally scattered measure, and many examples are given to illustrate the results. In particular, a simple version of Fubini’s theorem in the operator context is given in Theorem 2.3; Proposition 3.4 considers the problem of approximation by operator valued simple functions; Theorem 3.5 characterizes Pettis’ measurability, and Theorem 3.7 gives several characterizations, in the case of a finite measure space \((\Omega,\mu)\) and for Banach spaces \(X\) (which is separable) and \(Y\), of when \(f\rightarrow {\mathcal L}_s(X,Y)\) is strongly \(\mu\)-measurable. Finally, it is proved in Theorem 4.5 (among other results) that if \(\mathcal H\) is a separable Hilbert space, \(A: {\mathcal D}(A) \rightarrow {\mathcal H}\) is a selfadjoint operator with spectral measure \(P_A\), \(E={\mathcal L}({\mathcal D}(A),{\mathcal H})\widehat\otimes_\tau{\mathcal D}(A)\), \((\Gamma,{\mathcal E},\mu)\) is a \(\sigma\)-finite measure space, \(u:\mathbb R\times \Gamma\rightarrow\mathbb C\) has the property that, for every \(h\in{\mathcal D}(A)\), the function \(u(\cdot,\gamma)\) is \(P_Ah\)-integrable in \({\mathcal D}(A)\) and \(f:\Gamma\rightarrow {\mathcal L}({\mathcal D}(A),{\mathcal H})\) is a suitable strongly \(\mu\)-measurable function, then the \({\mathcal L}({\mathcal D}(A),{\mathcal H})\)-valued function \[ \sigma\rightarrow\int_Tu(\sigma,\gamma)f(\gamma)d\mu(\gamma),\qquad\sigma\in\mathbb R \] is strongly measurable in \({\mathcal L}({\mathcal D}(A),{\mathcal H})\) and \((P_Ah)\)-integrable in \(E\) with respect to the \({\mathcal D}(A)\)-valued measure \(P_Ah\), for each set \(T\in{\mathcal E}\).

MSC:

28A25 Integration with respect to measures and other set functions
46A32 Spaces of linear operators; topological tensor products; approximation properties
46N50 Applications of functional analysis in quantum physics
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