Mean transforms of unbounded weighted composition operator pairs

TL;DR

The paper develops a comprehensive framework for unbounded weighted composition operator pairs on $L^2$ spaces by establishing explicit polar decompositions and introducing the $\lambda$-spherical mean transform $\mathcal{M}_{\lambda}$ for operator pairs. It analyzes dense definiteness, closedness, and generalized normality (spherical quasinormality and spherical $p$-hyponormality), and provides concrete examples showing intricate domain behavior (e.g., a densely defined Aluthge transform with a trivial $\lambda$-mean transform domain). It further connects these notions via Müller–Soltysiak powers and explores the transform’s behavior on discrete measure spaces, including non-bijectivity results and preservation criteria for spherical $p$-hyponormality under certain weight sequences. Altogether, the work extends single-operator transform theory to unbounded operator pairs, offering tools for analyzing spherical transforms and their spectral/normality properties in multi-operator settings.

Abstract

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.

Theorems & Definitions (60)

• Theorem 1
• Theorem 2
• Proposition 3
• proof
• Remark 4
• Example 5
• Lemma 6
• proof
• Remark 7
• Theorem 8