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

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 Šilov 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.

36 pages