On matrix-valued Herglotz functions

We provide a comprehensive analysis of matrix-valued Herglotz functions and illustrate their applications in the spectral theory of self-adjoint Hamil tonian systems including matrix-valued Schrodinger and Dirac-type operators . Special emphasis is devoted to appropriate matrix-valued extensions of th e well-known Aronszajn-Donoghue theory concerning support properties of mea sures in their Nevanlinna-Riesz-Herglotz representation. In particular, we study a class of linear fractional transformations M-A(z) of a given n x n Herglotz matrix M(z) and prove that the minimal support of the absolutely c ontinuous 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-ran k perturbations of selfadjoint operators, self-adjoint extensions of densel y defined symmetric linear operators (especially, Friedrichs and Krein exte nsions), model operators for these two cases, and associated realization th eorems for certain classes of Herglotz matrices.