 97626 F. Gesztesy and E. Tsekanovskii
 On MatrixValued Herglotz Functions
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Dec 11, 97

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Abstract. We provide a comprehensive analysis of matrixvalued Herglotz functions and
illustrate their applications in the spectral theory of selfadjoint
Hamiltonian systems including matrixvalued Schr\"odinger and Diractype
operators. Special emphasis is devoted to appropriate matrixvalued
extensions of the wellknown
AronszajnDonoghue theory concerning support properties of measures in their
NevanlinnaRieszHerglotz 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 selfadjoint
finiterank perturbations of selfadjoint operators, selfadjoint
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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