Reproducing kernels and contractive divisors in bergman spaces

Abstract

In the Hardy spaces H^p of holomorphic functions, Blaschke products are applied to factor out zeros. However, for Bergman spaces, the zero sets of which do not necessarily satisfy the Blaschke condition, the study of divisors is a more recent development. Hedenmalm proved the existence of a canonical contractive zero-divisor which plays the role of a Blascke product in the Bergman space$L_\alpha ^2 \left( \mathbb{D} \right)$. Duren, Khavinson, Shapiro, and Sundberg later extended Hedenmalm's result to$L_\alpha ^p \left( \mathbb{D} \right)$, 0<p<∞. In this paper, an explicit formula for the contractive divisor is given for a zero set that consists of two points of arbitrary multiplicities. There is a simple one-to-one correspondence between contractive divisors and reproducing kernels for certain weighted Bergman spaces. The divisor is obtained by calculating the associated reproducing kernel. The formula is then applied to obtain the contractive divisor for a certain regular zero set, as well as the contractive divisor associated with an inner function that has singular support on the boundary. Bibliography: 13 titles.