Cyclicity in de Branges--Rovnyak spaces

TL;DR

This paper investigates the cyclicity problem for the shift operator on de Branges--Rovnyak spaces, providing a new elementary characterization of cyclic vectors for rational functions and analyzing specific cases involving inner functions.

Contribution

It offers a new elementary proof for characterizing cyclic vectors in de Branges--Rovnyak spaces for certain rational functions and explores cases with inner functions and model spaces.

Findings

Characterization of cyclic vectors for rational functions not finite Blaschke products.

Elementary proof of invariant subspace description in these spaces.

Analysis of the case where $b=(1+I)/2$ with inner function $I$.

Abstract

In this paper, we study the cyclicity problem with respect to the forward shift operator $S_b$ acting on the de Branges--Rovnyak space $\mathscr{H}(b)$ associated to a function $b$ in the closed unit ball of $H^\infty$ and satisfying $\log(1-|b|)\in L^1(\mathbb T)$. We present a characterisation of cyclic vectors for $S_b$ when $b$ is a rational function which is not a finite Blaschke product. This characterisation can be derived from the description, given in [S. Luo, C. Gu, S. Richter, Higher order local Dirichlet integrals and de Branges--Rovnyak spaces, \emph{Adv. Math., \textbf{385} (2021), paper No. 107748, 47], of invariant subspaces of $S_b$ in this case, but we provide here an elementary proof. We also study the situation where $b$ has the form $b=(1+I)/2$, where $I$ is a non-constant inner function such that the associated model space $K_I=\mathscr{H}(I)$ has an orthonormal basis of reproducing kernels.