## 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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### References:

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