Inner factors of Dirichlet space functions

TL;DR

This paper investigates the inner factors of Dirichlet space functions, focusing on Blaschke products and singular inner functions, providing new insights into zero sets and inner factorization properties.

Contribution

It characterizes which inner functions appear in Dirichlet space factorizations and extends classical theorems to this setting, including conditions for zero sets.

Findings

Characterization of Blaschke products as zero sets in Dirichlet space

Extension of Shapiro-Shields theorem to singular inner factors

New sufficient conditions for zero sets in Dirichlet space

Abstract

Every function in the Dirichlet space on the unit disc has an inner/outer factorization. We study which inner functions occur in this way. For Blaschke products, this is the well known question of which subsets of the disc are zero sets for the Dirichlet space. We also consider singular inner factors, and in particular prove an analogue of the Shapiro-Shields theorem in this setting. Our results on singular inner factors also yield new sufficient conditions for zero sets.