Mean transforms of unbounded weighted composition operator pairs

TL;DR

This paper characterizes the polar decomposition of unbounded weighted composition operators, introduces a new mean transform, and explores their properties, including hyponormality and quasinormality, in $L^2$-spaces.

Contribution

It introduces the $ extbf{ extlambda}$-spherical mean transform for unbounded weighted composition operators and analyzes their dense definiteness and spectral properties.

Findings

Characterization of polar decomposition of unbounded weighted composition operators.

Introduction and analysis of the $ extbf{ extlambda}$-spherical mean transform.

Identification of conditions for spherically quasinormal and $p$-hyponormal operators.

Abstract

In this paper, we first characterize the polar decomposition of unbounded weighted composition operator pairs $\textbf{C}_{\phi,\omega}$ in an $L^2$-space. Based on this characterization, we introduce the $\lambda$-spherical mean transform $\mathcal{M}_\lambda(\textbf{C}_{\phi,\omega})$ for $\lambda\in[0,1]$. We then investigate the dense definiteness of $\mathcal{M}_\lambda(\textbf{C}_{\phi,\omega})$. As an application, we provide an example of a $p$-hyponormal operator whose Aluthge transform is densely defined, while its $\lambda$-mean transform has a trivial domain. Furthermore, we establish the relationship between the dense definiteness of $\textbf{C}_{\phi,\omega}$ and $\mathcal{M}_{\lambda}(\textbf{C}_{\phi,\omega})$, based on the notion of powers for operator pairs in the sense of M{\"u}ller and Soltysiak. We also give a characterization of spherically quasinormal weighted composition operator pairs via the $\lambda$-spherical mean transform, revealing some properties that differ from the single operator case. Finally, we characterize a class of spherically $p$-hyponormal weighted composition operators on discrete measure spaces. As a corollary, we present corresponding results on the spherical $p$-hyponormality of unbounded $2$-variable weighted shifts and theirs $\lambda$-spherical mean transforms.