F. Gesztesy and E. Tsekanovskii
On Matrix-Valued Herglotz Functions
(234K, LaTeX)
ABSTRACT. 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.