Toeplitz operators on the Hardy spaces of quotient domains

Abstract

Let $\Omega$ be either the unit polydisc $\mathbb D^d$ or the unit ball $\mathbb B_d$ in $\mathbb C^d$ and $G$ be a finite pseudoreflection group which acts on $\Omega.$ Associated to each one-dimensional representation $\varrho$ of $G,$ we provide a notion of the (weighted) Hardy space $H^2_\varrho(\Omega/G)$ on $\Omega/G.$ Subsequently, we show that each $H^2_\varrho(\Omega/G)$ is isometrically isomorphic to the relative invariant subspace of $H^2(\Omega)$ associated to the representation $\varrho.$ For $\Omega=\mathbb D^d,$ $G=\mathfrak{S}_d,$ the permutation group on $d$ symbols and $\varrho = $ the sign representation of $\mathfrak{S}_d,$ the Hardy space $H^2_\varrho(\Omega/G)$ coincides to well-known notion of the Hardy space on the symmetrized polydisc. We largely use invariant theory of the group $G$ to establish identities involving Toeplitz operators on $H^2(\Omega)$ and $H^2_\varrho(\Omega/G)$ which enable us to study algebraic properties (such as generalized zero product problem, characterization of commuting Toeplitz operators, compactness etc.) of Toeplitz operators on $H^2_\varrho(\Omega/G).$