Multiplication operator on the Bergman space by a proper holomorphic map

TL;DR

This paper investigates the structure of multiplication operators induced by proper holomorphic maps on Bergman spaces, revealing minimal joint reducing subspaces and unitary equivalences to Bergman operators on target domains.

Contribution

It identifies a non-trivial minimal joint reducing subspace for the multiplication operator tuple and establishes a unitary equivalence to Bergman operators on the target domain.

Findings

Existence of a non-trivial minimal joint reducing subspace.

Unitary equivalence of the restricted operator to Bergman operator.

Structural insight into multiplication operators induced by proper holomorphic maps.

Abstract

Suppose that $f := (f_1,\ldots,f_d):\Omega_1\to\Omega_2$ 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) $M_f=(M_{f_1},\ldots, M_{f_d})$ on the Bergman space $\mathbb A^2(\Omega_1)$, say $\mathcal M.$ We further show that the restriction of $(M_{f_1},\ldots,M_{f_d})$ to $\mathcal M$ is unitarily equivalent to Bergman operator on $\mathbb A^2(\Omega_2).$