On rank invariance of generalized Schwarz-Pick-Potapov block matrices of matrix-valued Carathéodory functions

摘要

In view of a multiple Nevanlinna-Pick interpolation problem, we study the rank of generalized Schwarz-Pick-Potapov block matrices of matrix-valued Caratheodory functions. Those matrices are determined by the values of a Caratheodory function and the values of its derivatives up to a certain order. We derive statements on rank invariance of such generalized Schwarz-Pick-Potapov block matrices. These results are applied to describe the case of exactly one solution for the finite multiple Nevanlinna-Pick interpolation problem and to discuss matrix-valued Caratheodory functions with the highest degree of degeneracy. (c) 2005 Elsevier Inc. All rights reserved.