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On Matrix-Valued Herglotz Functions by Gesztesy, Fritz; Tsekanovskii, Eduard is a scholarly article available to read on EtoBox.

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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 Schr\"odinger 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 M_A(z) of a given n \times n Herglotz matrix M(z) and prove that the minimal support of the absolutely continuos part of the measure associated to M_A(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.

Author

Gesztesy, Fritz; Tsekanovskii, Eduard

Published

1997

Language

EN