On rank invariance of moment matrices of nonnegative Hermitian-valued Borel measures on the unit circle

Abstract

This paper provides first tools for generalizing the theory of orthogonal rational functions on the unit circle 𝕋 created by Bultheel, González-Vera, Hendriksen and Njåstad to the matrix case. A crucial part in this generalization is the definition of the spaces of matrix-valued rational functions for which an orthogonal basis is to be constructed. An important feature of the matrix case is that these spaces will be considered simultaneously as left and right modules over the algebra ℂq×q. In this modules we will define simultaneously left and right matrix-valued inner products with the aid of a nonnegative Hermitian-valued q × q Borel measure on the unit circle. Given a sequence (αj)j∈ℕ of complex numbers located in ℂ\𝕋 (especially in “good position” with respect to the unit circle) we will introduce a concept of rank for nonnegative Hermitian-valued q × q Borel measures on the unit circle which is based on the Gramian matrix of particular rational matrix-valued functions with prescribed pole structure. A main result of this paper is that this concept of rank is universal. More precisely, it turns out that the rank of a matrix measure does not depend on the given sequence (αj)j∈ℕ. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)