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

In view of a multiple Nevanlinna–Pick interpolation problem, we study the rank of generalized Schwarz–Pick–Potapov block matrices of matrix-valued Carathéodory functions. Those matrices are determined by the values of a Carathéodory 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 Carathéodory functions with the highest degree of degeneracy.