  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.
