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On Matrix–Valued Herglotz Functions

✍ Scribed by Fritz Gesztesy; Eduard Tsekanovskii

Publisher
John Wiley and Sons
Year
2000
Tongue
English
Weight
744 KB
Volume
218
Category
Article
ISSN
0025-584X

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✦ Synopsis

We provide a comprehensive analysis of matrix -valued Herglotz functions and illustrate their applications in the spectral theory of self -adjoint Hamiltonian systems including matrixvalued Schrödinger and Dirac -type operators. Special emphasis is devoted to appropriate matrixvalued 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 × n Herglotz matrix M (z) and prove that the minimal support of the absolutely continuous 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 selfadjoint 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.

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