Hilbert Spaces Contractively Contained in Weighted Bergman Spaces on the Unit Disk

TL;DR

This paper introduces a new approach to analyzing sub-Bergman Hilbert spaces within weighted Bergman spaces on the unit disk, strengthening existing results and utilizing reproducing kernel Hilbert space theory.

Contribution

It provides a novel method for studying sub-Bergman Hilbert spaces, extending and strengthening previous results on their norm equivalence to Hardy spaces.

Findings

Sub-Bergman Hilbert spaces are norm equivalent to Hardy spaces.

The new approach offers a stronger result than previous proofs.

Reproducing kernel Hilbert space theory is effectively applied.

Abstract

Sub-Bergman Hilbert spaces are analogues of de Branges-Rovnyak spaces in the Bergman space setting. They are reproducing kernel Hilbert spaces contractively contained in the Bergman space of the unit disk. K. Zhu analyzed sub-Bergman Hilbert spaces associated with finite Blaschke products, and proved that they are norm equivalent to the Hardy space. Later S. Sultanic found a different proof of Zhu's result, which works in weighted Bergman space settings as well. In this paper, we give a new approach to this problem and obtain a stronger result. Our method relies on the theory of reproducing kernel Hilbert spaces.