Reproducing kernels and contractive divisors in Bergman spaces

Jonas Hansbo

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 24, Tome 232 (1996), pp. 174-198

Résumé

In the Hardy spaces $H^p$ of holomorphic functions Blaschke products are used to factor out zeros. However, for the Bergman spaces, the zero sets of which do not necessarily satisfy the Blaschke condition, the study of divisors is a more recent development. In [7], Hedenmalm showed the existence of a canonical contractive zero-divisor which plays the role of a Blascke product in the Bergman space $L^2_\alpha(\mathbb D)$. Duren, Khavinson, Shapiro and Sundberg [4,5] later extended Hedenmalm's result to $L^2_\alpha(\mathbb D)$, $0$. In this paper an explicit formula for the contractive divisor is given for a zero set consisting of two points of arbitrary multiplicities. There is a simple one-to-one correspondence between the contractive divisors and reproducting kernels for certain weighted Bergman spaces. The divisor is obtained by calculating the associated reproducing kernel. The formula is then used to obtain the contractive divisor for a certain regular zero-set, as well as the contractive divisor associated with an inner function, with singular support on the boundary. Bibl. 13 titles.