Reproducing kernels, de Branges-Rovnyak spaces, and norms of weighted composition operators

TL;DR

This paper establishes a connection between the norms of weighted composition operators on Hardy and Bergman spaces and the de Branges-Rovnyak spaces, providing new proofs and conditions for boundedness.

Contribution

It introduces a novel approach linking composition operator norms to de Branges-Rovnyak space norms, offering new proofs and boundedness criteria.

Findings

Norm of weighted composition operators is controlled by de Branges-Rovnyak space norms.

Provides a new proof of boundedness of composition operators on H^2.

Shows positivity of a generalized de Branges-Rovnyak kernel implies boundedness.

Abstract

We prove that the norm of a weighted composition operator on the Hardy space H^2 of the disk is controlled by the norm of the weight function in the de Branges-Rovnyak space associated to the symbol of the composition operator. As a corollary we obtain a new proof of the boundedness of composition operators on H^2, and recover the standard upper bound for the norm. Similar arguments apply to weighted Bergman spaces. We also show that the positivity of a generalized de Branges-Rovnyak kernel is sufficient for the boundedness of a given composition operator on the standard functions spaces on the unit ball.