Multiplication operator on the Bergman space by a proper holomorphic map

Abstract

Suppose that $f:=(f_1,\dots ,f_d):{\Omega }_1\to {\Omega }_2$ is a proper holomorphic map between two bounded domains in ${\mathbb{C}}^d.$ In this paper, we find a non-trivial minimal joint reducing subspace for the multiplication operator (tuple) $M_f=(M_{f_1},\dots ,M_{f_d})$ on the Bergman space ${\mathbb{A}}^2({\Omega }_1)$, say $\mathcal{M}.$ We further show that the restriction of $(M_{f_1},\dots ,M_{f_d})$ to $\mathcal{M}$ is unitarily equivalent to Bergman operator on ${\mathbb{A}}^2({\Omega }_2).$

15 pages