Free outer functions in complete Pick spaces

TL;DR

This paper extends the inner-outer factorization concept to complete Pick spaces, demonstrating the essential uniqueness of free outer cyclic factors and providing intrinsic characterizations and applications.

Contribution

It introduces the notion of free outer cyclic factors in complete Pick spaces, establishing their essential uniqueness and intrinsic characterization.

Findings

Unique factorization with free outer cyclic factors

Intrinsic characterization of factors

Applications to computing examples

Abstract

Jury and Martin establish an analogue of the classical inner-outer factorization of Hardy space functions. They show that every function $f$ in a Hilbert function space with a normalized complete Pick reproducing kernel has a factorization of the type $f=\varphi g$, where $g$ is cyclic, $\varphi$ is a contractive multiplier, and $\|f\|=\|g\|$. In this paper we show that if the cyclic factor is assumed to be what we call free outer, then the factors are essentially unique, and we give a characterization of the factors that is intrinsic to the space. That lets us compute examples. We also provide several applications of this factorization.