Toeplitz operators on the weighted Bergman spaces of quotient domains

Abstract

Let G be a finite pseudoreflection group and Ω⊆Cd be a bounded domain which is a G-space. We establish identities involving Toeplitz operators on the weighted Bergman spaces of Ω and Ω/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 Ω/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.