Operator-valued rational functions

TL;DR

This paper characterizes operator-valued inner divisors as Blaschke-Potapov factors and introduces a notion of rationality, showing such functions can be represented as finite Blaschke-Potapov products.

Contribution

It extends the classical matrix-valued rational function theory to operator-valued functions, providing a new characterization and representation framework.

Findings

Inner divisors of $zI_E$ are Blaschke-Potapov factors.

Operator-valued rational functions are characterized by finite Blaschke-Potapov products.

Extension of Potapov's result to operator-valued functions.

Abstract

In this paper we show that every inner divisor of the operator-valued coordinate function, $zI_E$, is a Blaschke-Potapov factor. We also introduce a notion of operator-valued "rational" function and then show that $\Delta$ is two-sided inner and rational if and only if it can be represented as a finite Blaschke-Potapov product; this extends to operator-valued functions the well-known result proved by V.P. Potapov for matrix-valued functions.