Real Symmetric, Unitary, And Complex Symmetric Weighted Composition Operators On Bergman Spaces Of Polydisk

TL;DR

This paper characterizes various symmetry properties of weighted composition operators on Bergman spaces of polydisks, including conditions for boundedness and complex symmetry, expanding understanding of their algebraic structure.

Contribution

It provides a comprehensive characterization of real symmetric, unitary, and complex symmetric weighted composition operators on Bergman spaces of polydisks, including unbounded cases.

Findings

Weighted composition operators are fully characterized by their symbols.

Boundedness is necessary for symmetric structures.

Real symmetric operators are also complex symmetric with a specific conjugation.

Abstract

In this paper, we study weighted composition operators on Bergman spaces of analytic functions which are square integrable on polydisk. We develop the study in full generality, meaning that the corresponding weighted composition operators are not assumed to be bounded. The properties of weighted composition operators such as real symmetry, unitariness, complex symmetry, are characterized fully in simple algebraic terms, involving their symbols. As it turns out, a weighted composition operator having a symmetric structure must be bounded. We also obtain the interesting consequence that real symmetric weighted composition operators are complex symmetric corresponding an adapted and highly relevant conjugation.