Multiplication operators on the Bergman space by proper holomorphic mappings

Abstract

Suppose that f:=(f1,…,fd):Ω1→Ω2 is a proper holomorphic map between two bounded domains in Cd. We show that the multiplication operator (tuple) Mf=(Mf1,…,Mfd) on the Bergman space A2(Ω1) admits a non-trivial minimal joint reducing subspace, say M and the restriction of Mf to M is unitarily equivalent to the Bergman operator on A2(Ω2). A number of interesting consequences of this result have been observed.