Some Classes of Absolutely Norm Attaining Weighted Shifts Operators on Directed Trees

TL;DR

The paper characterizes absolutely norm attaining quasi-$\ast$-paranormal weighted shifts on directed trees and proves a detailed block-operator structure for such operators, linking norm attainment to invariant subspaces and spectral components. It derives norm-attainment criteria for weighted shifts, establishes invariant-subspace results for quasi-$\ast$-paranormal operators, and presents a canonical decomposition of $T\in \mathcal{AN}(\mathcal{H})$ into unitary blocks and finite-dimensional perturbations, with explicit spectral descriptions $\sigma(T)=\bigcup_{\lambda\in \Lambda} \sigma(\lambda U_{\lambda})$ and $\sigma_{ess}(|T|)=\{m_e(|T|)\}$. The work provides concrete examples showing that positive $\mathcal{AN}$ operators on directed trees can have more than one eigenvalue with infinite multiplicity in the spectrum and that the essential spectrum may include multiple points, highlighting spectral richness in graph-based operator settings. Overall, the results illuminate both structural and spectral properties of AN and quasi-$\ast$-paranormal operators on directed-tree Hilbert spaces, offering a framework for further analysis of graph-structured operator classes.

Abstract

In this paper we characterise absolutely norm attaining quasi*paranormal weighted shifts on directed trees and give some examples. Moreover we give some examples which show that the spectrum of a positive absolutely norm attaining operator containing more than one eigenvalue with infinite multiplicity.

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Theorems & Definitions (55)

• Theorem 2.1
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• Theorem 2.2
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• Theorem 2.3
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• Corollary 2.4
• Theorem 2.5
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• Corollary 2.6