Toeplitz operators on the weighted Bergman spaces of quotient domains

Abstract

Let $G$ be a finite pseudoreflection group and $\Omega \subseteq {\mathbb{C}}^d$ be a bounded domain which is a $G$-space. We establish identities involving Toeplitz operators on the weighted Bergman spaces of $\Omega$ and $\Omega /G$ using invariant theory and representation theory of $G.$ This, in turn, provides techniques to study algebraic properties of Toeplitz operators on the weighted Bergman space on $\Omega /G.$ We specialize on the generalized zero-product problem and characterization of commuting pairs of Toeplitz operators. As a consequence, more intricate results on Toeplitz operators on the weighted Bergman spaces on some specific quotient domains (namely symmetrized polydisc, monomial polyhedron, Rudin's domain) have been obtained.

23 pages

Cited by in corpus (1)

• Function theory on quotient domains related to the polydisc