Toeplitz operators on the Hardy spaces of quotient domains

TL;DR

This paper develops a framework for Hardy spaces on quotient domains under finite pseudoreflection groups, linking them to invariant subspaces and analyzing Toeplitz operators' algebraic properties.

Contribution

It introduces Hardy spaces on quotient domains associated with group representations and studies Toeplitz operators using invariant theory.

Findings

Hardy spaces on quotient domains are isometrically isomorphic to invariant subspaces.

Identities relating Toeplitz operators on original and quotient spaces are established.

Algebraic properties of Toeplitz operators, such as zero product and commutativity, are characterized.

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 $H^2_\varrho(\Omega/G).$