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Some applications of operator-valued Herglotz functions. (English) Zbl 0991.30020

Alpay, D. (ed.) et al., Operator theory, system theory and related topics. The Moshe Livšic anniversary volume. Proceedings of the conference in operator theory in honour of Moshe Livčic 80th birthday held in Beer-Sheva and Rehovot, Israel, June 29-July 4, 1997. Basel: Birkhäuser. Oper. Theory, Adv. Appl. 123, 271-321 (2001).
The authors define a Herglotz function/operator to be a mapping \(M\) from the open upper half plane \(\mathbb{C}_{+}\) to the bounded operators \(B(\mathcal K)\) on a complex Hilbert space \(\mathcal K\) such that \(\left\langle M(z)k,k\right\rangle \geq 0\) for all \(k\in K\) and \(z\in \mathbb{C}_{+}\). A detailed study of (i) selfadjoint perturbations of self-adjoint operators in the context of Herglotz operators, (ii) selfadjoint extensions of densely defined closed symmetric operators with deficiency, and concrete applications to Schrödinger operators upon specialization to certain one-dimensional realizations are provided. By a self-adjoint perturbation of \(H_{0}\) defined on a separable Hilbert space \(H\) the authors mean an operator \(H_{L}=H_{0}+KLK^{\ast }\) where \(L\) is a bounded selfadjoint operator on a separable Hilbert space \(\mathcal K\) and \(K:\mathcal K\rightarrow \mathcal H\) is bounded. \(H_{0}\) itself is not assumed bounded. To such a perturbation on associates the Herglotz operator \(M_{L}(z)=K^{\ast }(H_{L}-z)^{-1}K\) where \(z\) has nonzero imaginary part satisfying a certain extra hypothesis. The authors then construct a realization of \(H_{L}\), that is, a model Hilbert space \(L^{2}(R,\mathcal K,wd\Omega _{L})\) and an associated operator \(\widehat{H}_{L}\) on it such that \(H_{L}\) on \(\mathcal H\) is unitarily equivalent to \(\widehat{H}_{L}\). The purpose of this realization is to provide a more concrete spectral decomposition of \(H_{L}\) via that of \(\widehat{H}_{L}\). Known representations of the Herglotz operator are then extended to this more concrete setting, availing the applications given later in the paper.
For the entire collection see [Zbl 0959.00055].

MSC:

30D50 Blaschke products, etc. (MSC2000)
30E20 Integration, integrals of Cauchy type, integral representations of analytic functions in the complex plane
47A10 Spectrum, resolvent
47A45 Canonical models for contractions and nonselfadjoint linear operators
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