Toeplitz operators on generalized Bergman spaces

Details Toeplitz operators on generalized Bergman spaces Toeplitz operators on generalized Bergman spaces

2009-03-03

arxiv.org/abs/0903.0651v3

Integral Equations and Operator Theory, Vol. 66 (2010), 53-77

Mathematics

Complex Variables

Final version. To appear in Integral Equations and Operator Theory

Scientific paper

10.1007/s00020-009-1734-6

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.

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