On the Deddens algebras of a class of bounded operators

TL;DR

This paper explores the relationships between Deddens and spectral radius algebras of bounded operators, characterizes these algebras for specific operator classes, and applies findings to weighted conditional type operators on Hilbert spaces.

Contribution

It provides new characterizations of Deddens and spectral radius algebras for rank one, similar, majorized, and quasi-isometry operators, and extends results to weighted conditional operators.

Findings

Established relations between Deddens and spectral radius algebras.

Characterized these algebras for rank one and related operators.

Applied results to weighted conditional type operators.

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 $L^2(\mu)$.