Weighted composition operators on de Branges-Rovnyak spaces

TL;DR

This paper characterizes when weighted composition operators are bounded or compact on de Branges-Rovnyak spaces, extending previous results by linking these operators to Hardy space operators.

Contribution

It provides new characterizations of weighted composition operators on de Branges-Rovnyak spaces with rational symbols, expanding understanding beyond finite Blaschke products.

Findings

Boundedness criteria for weighted composition operators

Compactness conditions for these operators

Extension of previous results to broader class of symbols

Abstract

In this paper, we characterize the boundedness and the compactness of weighted composition operators acting on a de Branges-Rovnyak space $\mathcal H(b)$, where the symbol $b$ is a rational function in the unit ball of $H^\infty$ that is not a finite Blaschke product. Our results extend those of [2] by exploiting a close relationship between weighted composition operators on $\mathcal H(b)$ and their counterparts on the Hardy space $H^2$.