Mean Transforms of Unbounded Weighted Composition Operator Pairs

ABSTRACT In this paper, we first characterize the polar decomposition of unbounded weighted composition operator pairs in an ‐space. Based on this characterization, we introduce the ‐spherical mean transform for . We then investigate the dense definiteness of . As an application, we provide an example of a ‐hyponormal operator whose Aluthge transform is densely defined, while its ‐mean transform has a trivial domain. Furthermore, we establish the relationship between the dense definiteness of and , 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 ‐spherical mean transform, revealing some properties that differ from the single operator case. Finally, we characterize a class of spherically ‐hyponormal weighted composition operators on discrete measure spaces. As a corollary, we present corresponding results on the spherical ‐hyponormality of unbounded 2‐variable weighted shifts and theirs ‐spherical mean transforms.