Mean transforms of unbounded weighted composition operator pairs
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912659766509568 |
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| author | Zhou, Jing-Bin Yang, Shihai |
| author_facet | Zhou, Jing-Bin Yang, Shihai |
| contents | In this paper, we first characterize the polar decomposition of unbounded weighted composition operator pairs $\textbf{C}_{ϕ,ω}$ in an $L^2$-space. Based on this characterization, we introduce the $λ$-spherical mean transform $\mathcal{M}_λ(\textbf{C}_{ϕ,ω})$ for $λ\in[0,1]$. We then investigate the dense definiteness of $\mathcal{M}_λ(\textbf{C}_{ϕ,ω})$. As an application, we provide an example of a $p$-hyponormal operator whose Aluthge transform is densely defined, while its $λ$-mean transform has a trivial domain. Furthermore, we establish the relationship between the dense definiteness of $\textbf{C}_{ϕ,ω}$ and $\mathcal{M}_λ(\textbf{C}_{ϕ,ω})$, 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 $λ$-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 $λ$-spherical mean transforms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_17195 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Mean transforms of unbounded weighted composition operator pairs Zhou, Jing-Bin Yang, Shihai Functional Analysis In this paper, we first characterize the polar decomposition of unbounded weighted composition operator pairs $\textbf{C}_{ϕ,ω}$ in an $L^2$-space. Based on this characterization, we introduce the $λ$-spherical mean transform $\mathcal{M}_λ(\textbf{C}_{ϕ,ω})$ for $λ\in[0,1]$. We then investigate the dense definiteness of $\mathcal{M}_λ(\textbf{C}_{ϕ,ω})$. As an application, we provide an example of a $p$-hyponormal operator whose Aluthge transform is densely defined, while its $λ$-mean transform has a trivial domain. Furthermore, we establish the relationship between the dense definiteness of $\textbf{C}_{ϕ,ω}$ and $\mathcal{M}_λ(\textbf{C}_{ϕ,ω})$, 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 $λ$-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 $λ$-spherical mean transforms. |
| title | Mean transforms of unbounded weighted composition operator pairs |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2510.17195 |