Cyclicity and invariant subspaces in the Dirichlet spaces

TL;DR

This paper proves the generalized Brown-Shields conjecture for Dirichlet spaces with measures supported on countable sets and provides a complete characterization of invariant subspaces in this case.

Contribution

It confirms the conjecture for countable support measures and explicitly characterizes invariant subspaces in these Dirichlet spaces.

Findings

Conjecture holds for measures with countable support.

Complete characterization of invariant subspaces for these measures.

Provides explicit criteria based on capacity and zero sets.

Abstract

Let $\mu$ be a positive finite measure on the unit circle and $\mathcal{D} (\mu)$ the associated Dirichlet space. The generalized Brown-Shields conjecture asserts that an outer function $f \in \mathcal{D} (\mu )$ is cyclic if and only if $c\_\mu (Z (f))= 0$, where $c\_\mu$ is the capacity associated with $\mathcal{D} (\mu)$ and $Z(f)$ is the zero set of $f$. In this paper we prove that this conjecture is true for measures with countable support. We also give in this case a complete and explicit characterization of invariant subspaces.