Mean transforms of unbounded weighted composition operator pairs

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.