On the Theory of Matrix Valued Functions Belonging to the Smirnov Class

TL;DR

This paper develops a comprehensive theory for matrix-valued functions in the matricial Smirnov class, extending classical scalar results to the matrix setting and analyzing their factorization properties.

Contribution

It introduces a systematic framework for matrix-valued Smirnov class functions, including factorization, maximum principles, and a parameterized family analysis with Blaschke-Potapov products.

Findings

Extension of Smirnov class theory to matrix-valued functions

Inner-outer factorization with Blaschke-Potapov products

Parameter-dependent analysis of inner factors

Abstract

A theory of matrix-valued functions from the matricial Smirnov class ${\goth N}_n^+({\Bbb D})$ is systematically developed. In particular, the maximum principle of V.I.Smirnov, inner-outer factorization, the Smirnov-Beurling characterization of outer functions and an analogue of Frostman's theorem are presented for matrix-valued functions from the Smirnov class ${\goth N}_n^+({\Bbb D})$. We also consider a family $F_{\lambda} =F-\lambda I$ of functions belonging to the matricial Smirnov class which is indexed by a complex parameter $\lambda$. We show that with the exception of a ''very small'' set of such $\lambda$ the corresponding inner factor in the inner-outer factorization of the function $F_{\lambda}$ is a Blaschke-Potapov product. The main goal of this paper is to provide users of analytic matrix-function theory with a standard source for references related to the matricial Smirnov class.