A Characterization of Generalized Zeros of Negative Type of Matrix Functions of the Class N ^n×n_κ

Abstract

An n × n-matrix function Q belongs to the class N ^n×n_κ if it is defined and meromorphic in the upper half plane C ⁺, and for an arbitrary k ε Z, z₁,z₂,…, z_k ε D _Q (the domain of holomorphy of Q) and ξ₁,ξ₂,…,ξ_k ε C ⁿ the matrix

$${\{ [\frac{{Q({z_i}) - Q({z_j})*}}{{{z_{\text{i}}} - {{\overline z }_{\text{j}}}}}{\xi _j},{\xi _j}]\} _{i,j = 1,2,...,k}}$$

has at most κ negative eigenvalues and, for at least one choice of k, z₁,z₂,…,z_k, it has exactly κ negative eigenvalues. We always assume that Q ε N ^n×n_κ has been extended to the lower half plane as follows: $Q(\overline z ) = Q(z)*$ (z ε D _Q). For the basic properties of these functions see [5], [2].

Tbhis research was supported by Republicka zajednica za naucni rad BiH.