A bounded linear operator T on a complex Hilbert space H is called complex symmetric if T = CT*C, where C is a conjugation (an isometric, antilinear involution of H). We prove that T = CJ\T|, where J is an auxiliary conjugation commuting with |T| = √T*T. We consider numerous examples, including the Poincare-Neumann singular integral (bounded) operator and the Jordan model operator (compressed shift). The decomposition T = CJ\T| also extends to the class of unbounded C-selfadjoint operators, originally introduced by Glazman. In this context, it provides a method for estimating the norms of the resolvents of certain unbounded operators.

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Complex Symmetric Operators and Applications II

We provide a comprehensive analysis of matrix–valued Herglotz functions and illustrate their applications in the spectral theory of self–adjoint Hamiltonian systems including matrix–valued Schrodinger and Dirac–type operators. Special emphasis is devoted to appropriate matrix–valued extensions of the well–known Aronszajn–Donoghue theory concerning support properties of measures in their Nevanlinna–Riesz–Herglotz representation. In particular, we study a class of linear fractional transformations MA(z) of a given n × n Herglotz matrix M(z) and prove that the minimal support of the absolutely continuous part of the measure associated to MA(z) is invariant under these linear fractional transformations.



Additional applications discussed in detail include self–adjoint finite–rank perturbations of self–adjoint operators, self–adjoint extensions of densely defined symmetric linear operators (especially, Friedrichs and Krein extensions), model operators for these two cases, and associated realization theorems for certain classes of Herglotz matrices.

On Matrix–Valued Herglotz Functions

Unitary Colligations, Reproducing Kernel Hilbert Spaces, and Nevanlinna–Pick Interpolation in Several Variables

https://www.math.tugraz.at/~behrndt/BehrndtLanger06.pdf

Boundary value problems for elliptic partial differential operators on bounded domains

The transformations of all the Schrodinger operators with point interactions in dimension one under space reflection P, time reversal T and (Weyl) scaling Wλ are presented. In particular, those operators which are invariant (possibly up to a scale) are selected. Some recent papers on related topics are commented upon.

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Symmetries of Schrödinger operator with point interactions

Using the theory of linear relations in Pontryagin spaces we extend to the nonpositive case the theory of reproducing kernel spaces associated with contractions in Hilbert spaces.

Schur Functions, Operator Colligations, and Reproducing Kernel Pontryagin Spaces

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Symmetric and selfadjoint relations in krein spaces .2.

LetA be a selfadjoint operator in a Hilbert space ℌ. Its rank one perturbations A+τ(·,ω)ω, ℝ, are studied when ω belongs to the scale space ℌ−2 associated with ℌ+2=domA and (·,·) is the corresponding duality. IfA is nonnegative and ω belongs to the scale space ℌ−1, Gesztesy and Simon [4] prove that the spectral measures ofA(τ), ℝ, converge weakly to the spectral measure of the limiting perturbationA(∞). In factA(∞) can be identified as a Friedrichs extension. Further results for nonnegative operatorsA were obtained by Kiselev and Simon [14] by allowing ω∈ℌ−2, Our purpose is to show that most results of Gesztesy, Kiselev, and Simon are valid for rank one perturbations of selfadjoint operators, which are not necessarily semibounded. We use the fact that rank one perturbations constitute selfadjoint extensions of an associated symmetric operator. The use of so-calledQ-functions [6, 8] facilitates the descriptions. In the special case that ω belongs to the scale space ℌ−1 associated with ℌ+2=dom |A|1/2 the limiting perturbationA(∞) is shown to be the generalized Friedrichs extension [5].

On rank one perturbations of selfadjoint operators

For a class of closed symmetric operators S with defect numbers (1, 1) it is possible to define a generalization of the Friedrichs extension, which coincides with the usual Friedrichs extension when S is semibounded. In this paper we provide an operator-theoretic interpretation of this class of symmetric operators. Moreover, we prove that a selfadjoint operator A is semibounded if and only if each one-dimensional restriction of A has a generalized Friedrichs extension.

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Hsv Desnoo

Papers

Schur Functions, Operator Colligations, and Reproducing Kernel Pontryagin Spaces

Symmetric and selfadjoint relations in krein spaces .2.

On rank one perturbations of selfadjoint operators

A characterization of semibounded selfadjoint operators

A note on watson transforms