arXiv subjects

Jing-Bin Zhou

Publications and source records attributed to Jing-Bin Zhou.

3 recordsLinked to original sources

arXiv2026

Power quasinormal operators and the root problem

In this paper, we construct an $n$-power quasinormal operator $T$ such that $T^n$ is not quasinormal for some positive integer $n$, thereby providing a counterexample to \cite[Lemma 3.1]{ko-filomat-2023}. We then investigate the relationships among the $n$-power quasinormality of $T$, the normality of $T^n$, and the quasinormality of $T^n$, and show that these three conditions are equivalent in finite-dimensional spaces. We also provide a new proof that $n$-power quasinormal operators have the single-valued extension property \cite[Theorem 3.2]{ko-filomat-2023}; unlike the original proof, our argument does not rely on \cite[Lemma 3.1]{ko-filomat-2023} and thus closes the gap in the original argument. In addition, for a fixed operator $T$, we characterize all positive integers $n$ for which $T$ is $n$-power quasinormal. As consequences, several results of Sid Ahmed \cite{ahmed-bmaa-2011} are extended. Finally, we prove that every paranormal $n$-power quasinormal operator is quasinormal. Closely related to this, we also give an affirmative answer to the root problem of Stanković and Kubrusly \cite[Question 2.11]{stankovic-afa-2025}.

math.FA↗

arXiv2026

Power mean transforms of operators

In this paper, we introduce the power mean transform $P_λ(T)$ of an operator $T$ on a Hilbert space, which is a convex combination of some classical operator transforms such as the mean transform $M(T)$, the Aluthge transform $Δ(T)$, and the Duggal transform $T^D$. In particular, when $T$ is invertible, this transform coincides with the induced Aluthge transform $Δ_{\mathsf{m}_{f}}(T)$ recently defined by Yamazaki \cite{yamazaki-laa-2021} with $f(x)=(λ+(1-λ)\sqrt{x})^2$ for $x\in(0,\infty)$ and $λ\in(0,1)$. We study basic properties of $P_λ(T)$ including its spectrum, norm and numerical radius. Moreover, we use the power mean transform to give new characterizations of normal, quasinormal and binormal operators. The questions of Golla et al. \cite{yamazaki-laa-2023} and some new results on the Duggal transform are also mentioned. We obtain a result close to the recent one of Osaka and Yamazaki \cite[Theorem 3.3]{yamazaki-tams-2025} on the iteration of the induced Aluthge transform for centered operators. Finally, we describe the form of bijective maps commuting with the power mean transform of the product of matrices.

math.FA↗

arXiv2025

Mean transforms of unbounded weighted composition operator pairs

In this paper, we first characterize the polar decomposition of unbounded weighted composition operator pairs $\textbf{C}_{ϕ,ω}$ 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.

math.FA↗