Brown-Halmos type Theorems on the proper images of bounded symmetric domains

TL;DR

This paper extends Brown-Halmos theorems to proper images of bounded symmetric domains, analyzing Toeplitz operators via group actions and reproducing kernel Hilbert spaces, revealing new structural insights.

Contribution

It introduces a family of reproducing kernel Hilbert spaces linked to finite complex reflection groups and establishes a Brown-Halmos type theorem for Toeplitz operators on these spaces.

Findings

Characterization of Toeplitz operators on proper images of bounded symmetric domains.

Identification of reproducing kernel Hilbert spaces from group representations.

Multiplicative properties of Toeplitz operators on these spaces.

Abstract

Let $\Omega\subseteq\mathbb C^n$ be a bounded symmetric domain and $f :\Omega \to \Omega^\prime\subseteq \mathbb C^n$ be a proper holomorphic mapping which is factored by a finite complex reflection group $G.$ We identify a family of reproducing kernel Hilbert spaces on $\Omega^\prime$ arising naturally from the isotypic decomposition of the regular representation of $G$ on the Hardy space $H^2(\Omega).$ Each element of this family can be realized as a closed subspace of some $L^2$-space on the \v{S}ilov boundary of $\Omega^\prime$. The reproducing kernel Hilbert space associated to the sign representation of $G$ is the Hardy space $H^2(\Omega^\prime).$ We establish a Brown-Halmos type characterization for the Toeplitz operators on $H^2(\Omega^\prime),$ where $\Omega^\prime$ is the image of the open unit polydisc $\mathbb D^n$ in $\mathbb C^n$ under a proper holomorphic mapping factored by the finite complex reflection group $G(m,p,n).$ Moreover, we prove various multiplicative properties of Toeplitz operators on $H^2(\Omega^\prime)$, where $\Omega^\prime$ is a proper holomorphic image of a bounded symmetric domain.