Mean transforms of unbounded weighted composition operator pairs

Abstract

In this paper, we first characterize the polar decomposition of unbounded weighted composition operator pairs ${\text{C}}_{\ddot{I}•,\ddot{I}‰}$ in an $L^2$-space. Based on this characterization, we introduce the $\hat{I}»$-spherical mean transform ${\mathcal{M}}_{\hat{I}}»({\text{C}}_{\ddot{I}•,\ddot{I}‰})$ for $\hat{I}»\in [0,1]$. We then investigate the dense definiteness of ${\mathcal{M}}_{\hat{I}}»({\text{C}}_{\ddot{I}•,\ddot{I}‰})$. As an application, we provide an example of a $p$-hyponormal operator whose Aluthge transform is densely defined, while its $\hat{I}»$-mean transform has a trivial domain. Furthermore, we establish the relationship between the dense definiteness of ${\text{C}}_{\ddot{I}•,\ddot{I}‰}$ and ${\mathcal{M}}_{\hat{I}}»({\text{C}}_{\ddot{I}•,\ddot{I}‰})$, based on the notion of powers for operator pairs in the sense of M{ü}ller and Soltysiak. We also give a characterization of spherically quasinormal weighted composition operator pairs via the $\hat{I}»$-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 $\hat{I}»$-spherical mean transforms.