On cyclicity in de Branges-Rovnyak spaces

TL;DR

This work characterizes cyclic vectors in non-extreme de Branges-Rovnyak spaces ${\mathcal H}(b)$ by exploiting the canonical split ${\mathcal H}(b) = \overline{aH^{2}} \oplus (aH^{2})^{\perp}$ and constructing a model for the shift-operator adjoint on $(aH^{2})^{\perp}$ via $J_{\phi}$ and the operator $L$. In the finite-defect setting it yields a complete criterion: a nonzero outer vector $f$ is cyclic precisely when $f(\lambda_j) \neq 0$ at each eigenvalue $\overline{\lambda_j}$ of $M_z^{*}$ on $(aH^{2})^{\perp}$. For infinite defect, the paper develops sufficient and necessary conditions tied to the boundary geometry of $H^{1}$ via the set $\sigma(\phi)$ and Aleksandrov-Clark measures, and connects cyclicity to exposed points of $H^{1}$, with applications to generalized Dirichlet spaces and explicit model-space decompositions such as $b=(1+\theta)/2$. Overall, it advances the understanding of shift-invariant structures in ${\mathcal H}(b)$ and links cyclicity to boundary-analytic phenomena and Clark-measure theory.

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 $\mathcal{D}(μ)$. 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.

Theorems & Definitions (42)

• Theorem 1
• Remark
• Theorem 2
• Theorem 3
• Theorem 4
• Proposition 1
• proof
• Remark
• Theorem 5
• Proposition 2