Centered weighted composition operators on $L^2$-spaces revisited

TL;DR

This paper characterizes centered weighted composition operators on $L^2$-spaces without assuming they are products of multiplication and composition operators, introduces spectrally half-centered operators, and provides criteria for weighted shifts on directed trees.

Contribution

It offers a new characterization of centered weighted composition operators and introduces the concept of spectrally half-centered operators, expanding understanding of their spectral properties.

Findings

Unbounded weighted composition operators are spectrally half-centered if their powers are closed and densely defined.

Criteria for centered weighted shifts on directed trees of types I--IV are established.

Several examples illustrating the concepts are presented.

Abstract

Centered weighted composition operators on $L^2$-spaces are characterized. The characterization is obtained without the assumption that the operator is a product of a multiplication and a composition operator. The concept of spectrally half-centered operators is introduced, and it is shown that unbounded weighted composition operators are spectrally half-centered provided their powers are closed and densely defined. A criteria for centered weighted shifts on directed trees of types I--IV are provided. Various examples are presented.