Hilbert spaces contractively contained in weighted Bergman spaces on the unit disk

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.