Toeplitz operators on generalized Bergman spaces

TL;DR

This paper studies Toeplitz operators on generalized weighted Bergman spaces, characterizing symbols that produce bounded or Hilbert-Schmidt operators, extending classical theory to new function spaces.

Contribution

It introduces Toeplitz operators on generalized Bergman spaces with Sobolev-type norms and characterizes their boundedness and Hilbert-Schmidt properties.

Findings

Identified classes of symbols for bounded Toeplitz operators.

Characterized symbols for Hilbert-Schmidt Toeplitz operators.

Extended classical Toeplitz operator theory to generalized Bergman spaces.

Abstract

We consider the weighted Bergman spaces HL^2(B^d,\mu_{\lambda}), where d\mu_\lambda(z)=c_{\lambda}(1-|z|^2)^lambda d\tau, \tau being the hyperbolic volume measure. These spaces are nonzero if and only if \lambda>d. For 0<\lambda\leq d, spaces with the same formula for the reproducing kernel can be defined using a Sobolev-type norm. We define Toeplitz operators on these generalized Bergman spaces and investigate their properties. Specifically, we describe classes of symbols for which the corresponding Toeplitz operators can be defined as bounded operators or as a Hilbert--Schmidt operators on the generalized Bergman spaces.