Multiplication operators defined by a class of polynomials on L_a^2(D^2)

TL;DR

This paper investigates multiplication operators on the Bergman space over the bidisk defined by specific polynomials, analyzing their reducing subspaces, the von Neumann algebra they generate, and the structure of their commutant.

Contribution

It provides a complete characterization of the von Neumann algebra and reducing subspaces associated with polynomial-defined multiplication operators on the bidisk Bergman space.

Findings

Structure of V^*(p) is fully determined

Reducing subspaces of M_p are characterized

The von Neumann algebra W^*(p) is explicitly described

Abstract

In this paper, we consider those multiplication operators M_p on the Bergman space L_a^2(D^2) over the bidisk, defined by a class of polynomials p. Also, this paper consider the reducing subspaces of M_p, the von Neumann algebra W^*(p) generated by M_p, and its commutant V^*(p)=W^*(p)'. The structure of V^*(p) is completely determined, along with those reducing subspaces of M_p.