On the Brown--Shields conjecture for cyclicity in the Dirichlet space

TL;DR

This paper introduces a new sufficient condition for cyclicity in the Dirichlet space, advancing the understanding of the Brown-Shields conjecture by linking cyclicity to outer functions and zero set capacity.

Contribution

It provides a novel criterion for cyclicity in the Dirichlet space and proves a special case of the Brown-Shields conjecture.

Findings

Established a new sufficient condition for cyclicity in D

Proved a special case of the Brown-Shields conjecture relating cyclicity to zero set capacity

Connected cyclicity with outer functions in the Dirichlet space

Abstract

Let $\cD$ be the Dirichlet space, namely the space of holomorphic functions on the unit disk whose derivative is square-integrable. We establish a new sufficient condition for a function $f\in\cD$ to be {\em cyclic}, i.e. for $\{pf: p\text{a polynomial}\}$ to be dense in $\cD$. This allows us to prove a special case of the conjecture of Brown and Shields that a function is cyclic in $\cD$ iff it is outer and its zero set (defined appropriately) is of capacity zero.