Self-adjoint, unitary, and normal weighted composition operators in several variables

TL;DR

This paper investigates the properties of weighted composition operators on Hilbert spaces of analytic functions in several variables, focusing on conditions for adjoints, self-adjointness, unitarity, and normality.

Contribution

It provides new necessary and sufficient conditions for the adjoint of weighted composition operators to be another such operator, and characterizes self-adjoint, unitary, and normal operators in this context.

Findings

Characterized when the adjoint of a weighted composition operator is also a weighted composition operator.

Derived conditions for self-adjoint and unitary weighted composition operators.

Explored the normality of these operators in several variables.

Abstract

We study weighted composition operators on Hilbert spaces of analytic functions on the unit ball with kernels of the form $(1-<z,w>)^{-\gamma}$ for $\gamma>0$. We find necessary and sufficient conditions for the adjoint of a weighted composition operator to be a weighted composition operator or the inverse of a weighted composition operator. We then obtain characterizations of self-adjoint and unitary weighted composition operators. Normality of these operators is also investigated.