Multiplication operator on the Bergman space by a proper holomorphic map

Abstract

Suppose that f := (f1, . . . , f d ) : Ω1 → Ω2 is a proper holomorphic map between two bounded domains in C d . In this paper, we find a non-trivial minimal joint reducing subspace for the multiplication operator (tuple) M f = (M f 1 , . . . , M f d ) on the Bergman space A 2 (Ω1), say M. We further show that the restriction of (M f 1 , . . . , M f d ) to M is unitarily equivalent to Bergman operator on A 2 (Ω2).