Weakly centered weighted composition operators in $L^2$-spaces

TL;DR

The paper characterizes weakly centered weighted composition operators on $L^2$ spaces through the Radon–Nikodym derivative ${\mathsf h}_{\phi,w}$ and conditional expectation ${\mathsf E}_{\phi,w}$, and extends the analysis to spectrally weakly centered unbounded operators using a spectral framework. It establishes a bounded-case characterization (Theorem $wcent01$) and an invariant-subspace criterion (Proposition $suka01$), with the Aluthge transform playing a key role in relating the operator to its weighted shift components. It further develops a spectrally centered theory for unbounded operators (Theorem $gamon01$) via spectral projections and the projection ${\mathsf P}_{\phi,w}$, and demonstrates how invariant subspaces arise in both discrete settings and directed-tree models. The results connect classical weighted shifts, composition operators, and their unbounded counterparts, providing new insights into when invariant subspaces exist and how spectral properties govern centering behavior.

Abstract

Weakly centered and spectrally weakly cenetered weighted composition operators in $L^2$-spaces are characterized. Criteria for existence of invariant subspaces are given. Additional results and examples are supplied.

Figures (2)

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Theorems & Definitions (38)

• Theorem 1
• Theorem 1
• proof
• Remark 2
• Remark 3
• Corollary 4
• Corollary 5
• Corollary 6
• proof
• Proposition 7