Mean Transforms of Unbounded Weighted Composition Operator Pairs

Abstract

In this paper, we first characterize the polar decomposition of unbounded weighted composition operator pairs $C_{\varphi ,\omega }$ in an $L^2$-space. Based on this characterization, we introduce the $\lambda$-spherical mean transform $M_{\lambda }\left(C_{\varphi ,\omega }\right)$ for $\lambda \in [0,1]$. We then investigate the dense definiteness of $M_{\lambda }\left(C_{\varphi ,\omega }\right)$. 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 $C_{\varphi ,\omega }$ and $M_{\lambda }\left(C_{\varphi ,\omega }\right)$, based on the notion of powers for operator pairs in the sense of Müller 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.