Composition Operators On De Branges-rovnyak Spaces Associated To A Rational (Not Inner) Function

TL;DR

This paper characterizes key properties of composition operators on de Branges-Rovnyak spaces associated with rational, non-inner functions, extending previous results to broader classes of functions.

Contribution

It provides new characterizations of boundedness, compactness, and Hilbert-Schmidt properties for these operators when acting on specific de Branges-Rovnyak spaces.

Findings

Characterization of boundedness conditions

Criteria for compactness of composition operators

Conditions for Hilbert-Schmidt property

Abstract

In this paper, we characterize the boundedness, the compactness and the Hilbert-Schmidt property for composition operators acting from a de Branges-Rovnyak space $\mathcal H(b)$ into itself, when $b$ is a rational function in the closed unit ball of $H^\infty$ (but not a finite Blaschke product). In particular, we extend some of the results obtained by D. Sarason and J.N. Silva in the context of local Dirichlet spaces.