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Boundary values of operator-valued functions and trace class perturbations. (English) Zbl 1101.47301

Summary: The problem of the existence of boundary values for some classes of operator-valued functions of the form \(\alpha^*(S-z)^{-1}\beta\) is considered, where \(\alpha,\beta\) belong to the Schatten–von Neumann classes. The well-known result of Birman and Entina \((S=S^*)\) is generalized to the case when \(\nu^*(S-z)^{-1}\nu\) is meromorphic and possesses boundary values, where \(I-S^*S=\nu J\nu^*,\;J=J^*=J^{-1}\). The existence of a “scalar multiple” is proved for the function \(\alpha^*(S-z)^{-1}\beta\), when \(S\) is a trace class perturbation of a normal operator with spectrum on a smooth curve. These results can be applied to perturbation theory, scattering theory, functional models, and others.

MSC:

47A56 Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones)
47A55 Perturbation theory of linear operators
47A40 Scattering theory of linear operators
47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)

Citations:

Zbl 0116.32502