A Characterization of Generalized Zeros of Negative Type of Matrix Functions of the Class N κ n×n
Muhamed Borogovac, Heinz Langer · Birkhäuser Basel eBooks · 1988
An n × n-matrix function Q belongs to the class N κ n×n if it is defined and meromorphic in the upper half plane C +, and for an arbitrary k ε Z, z1,z2,…, zk ε D Q (the domain of holomorphy of Q) and ξ1,ξ2,…,ξk 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,…,zk, it has exactly κ negative eigenvalues. We always assume that Q ε N κ n×n 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].