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On the Deddens algebras of a class of bounded operators

Z. Huang, Y. Estaremi, S. Shimi

TL;DR

The paper investigates the relationship between Deddens algebras $\mathcal{D}_A$ and spectral radius algebras $\mathcal{B}_T$ for bounded operators, emphasizing similarity, rank-one majorization, quasi-isometry, and weighted conditional type (WCT) operators on $L^2(\mu)$. It shows that similarity by an invertible $A$ intertwines the algebras via $A\mathcal{D}_T A^{-1}=\mathcal{D}_C$ and $A\mathcal{B}_T A^{-1}=\mathcal{B}_C$, and provides detailed characterizations when $T$ is comparable to rank-one operators or majorized by them. The work extends to quasi-isometry operators, giving criteria for membership in $\mathcal{D}_T$ and $\mathcal{B}_T$ based on radius $r(T)$ and the auxiliary sequence $\alpha_m$, and culminates with applications to WCT operators on $L^2(\mathcal{F})$, where the algebras admit explicit block-structure descriptions relative to $\mathcal{N}(EM_u)^{\perp}$ and the operators $M_{E(|u|^2)}$ and $M_{E(|w|^2)}$. The results yield practical criteria for invariant/hyperinvariant subspaces in the WCT setting and clarify how quasi-normal and quasi-isometry properties influence the associated algebras.

Abstract

In this paper, we investigate the relation between the Deddens and spectral radius algebras of two bounded linear operators, noting a similarity between them. Additionally, we characterize the Deddens and spectral radius algebras related to rank one operators, operators that are similar to rank one operators, operators that are majorized by rank one operators, and quasi-isometry operators. Furthermore, we apply these results to the class of weighted conditional type operators on the Hilbert space $L^2(μ)$.

On the Deddens algebras of a class of bounded operators

TL;DR

The paper investigates the relationship between Deddens algebras and spectral radius algebras for bounded operators, emphasizing similarity, rank-one majorization, quasi-isometry, and weighted conditional type (WCT) operators on . It shows that similarity by an invertible intertwines the algebras via and , and provides detailed characterizations when is comparable to rank-one operators or majorized by them. The work extends to quasi-isometry operators, giving criteria for membership in and based on radius and the auxiliary sequence , and culminates with applications to WCT operators on , where the algebras admit explicit block-structure descriptions relative to and the operators and . The results yield practical criteria for invariant/hyperinvariant subspaces in the WCT setting and clarify how quasi-normal and quasi-isometry properties influence the associated algebras.

Abstract

In this paper, we investigate the relation between the Deddens and spectral radius algebras of two bounded linear operators, noting a similarity between them. Additionally, we characterize the Deddens and spectral radius algebras related to rank one operators, operators that are similar to rank one operators, operators that are majorized by rank one operators, and quasi-isometry operators. Furthermore, we apply these results to the class of weighted conditional type operators on the Hilbert space .
Paper Structure (4 sections, 18 theorems, 57 equations)

This paper contains 4 sections, 18 theorems, 57 equations.

Table of Contents

  1. Introduction and Preliminaries
  2. Deddens and spectral radius algebras of similar operators
  3. Applications to WCT operators
  4. Declarations

Key Result

Proposition 2.1

Let $T, A, C\in\mathcal{B}(\mathcal{H})$ and $A$ be invertible such that $T$ is similar to $C$ by $A$. Then $T^n$ is similar to $C^n$ by $A$, for every $n\in \mathbb{N}$ and $A\mathcal{D}_T=\mathcal{D}_CA$ (or equivalently $A\mathcal{D}_TA^{-1}=\mathcal{D}_C$, $\mathcal{D}_T=A^{-1}\mathcal{D}_CA$).

Theorems & Definitions (33)

  • Proposition 2.1
  • proof
  • Theorem 2.2
  • Theorem 2.3
  • proof
  • Proposition 2.4
  • proof
  • Lemma 2.5
  • proof
  • Theorem 2.6
  • ...and 23 more