Subinner-free outer factorizations on an annulus

TL;DR

This paper explores a specialized form of function factorization called subinner-free outer factorization within the context of the annulus, extending recent theoretical developments in Pick spaces to this geometric setting.

Contribution

It investigates specific examples of subinner-free outer factorizations in the annulus, expanding understanding of these factorizations beyond the classical Hardy space setting.

Findings

Identifies particular cases of subinner-free outer factorizations on the annulus

Connects factorization theory with the structure of the annulus in Pick spaces

Provides insights into the uniqueness and structure of these factorizations

Abstract

Recent work of Aleman, Hartz, McCarthy and Richter generalizes the classical inner-outer factorization of Hardy space functions to the complete Pick space setting, establishing an essentially unique "subinner-free outer" factorization. In this note, we investigate certain special examples of such factorizations in the setting of the function space induced on the annulus $A_r=\{r<|z|<1\}$ by the complete Pick kernel $$k_{r}(\lambda,\mu):=\frac{1-r^2}{(1-\lambda\bar{\mu})(1-r^2/\lambda\bar{\mu})}.$$