On cyclicity in de Branges-Rovnyak spaces

TL;DR

This paper investigates the characterization of cyclic vectors in de Branges-Rovnyak spaces, providing a complete description for finite-dimensional cases and exploring conditions for infinite-dimensional cases related to invariant subspaces and measure theory.

Contribution

It offers a complete function-theoretic description of cyclic vectors in finite-dimensional cases and advances understanding of the infinite-dimensional case through measure-theoretic conditions.

Findings

Complete description for finite-dimensional case

Conditions for cyclicity in infinite-dimensional case

Relation to Aleksandrov-Clark measures

Abstract

We study the problem of characterizing the cyclic vectors in de Branges-Rovnyak spaces. Based on a description of the invariant subspaces we show that the difficulty lies entirely in understanding the subspace $(aH^{2})^{\perp}$ and give a complete function theoretic description of the cyclic vectors in the case $\dim (aH^{2})^{\perp} < \infty$. Incidentally, this implies analogous results for certain generalized Dirichlet spaces $\mathcal{D}(\mu)$. Most of our attention is directed to the infinite case where we relate the cyclicity problem to describing the exposed points of $H^{1}$ and provide several sufficient conditions. A necessary condition based on the Aleksandrov-Clark measures of $b$ is also presented.