The $(n+1)$-centered operator on a Hilbert $C^*$-module

TL;DR

This paper introduces the concept of $(n+1)$-centered operators on Hilbert $C^*$-modules, exploring their properties through generalized Aluthge transforms and providing new characterizations.

Contribution

It defines $(n+1)$-centered operators in the context of Hilbert $C^*$-modules and develops their properties using generalized Aluthge transforms, offering new insights.

Findings

Characterization of $(n+1)$-centered operators

Use of generalized Aluthge transform in analysis

New criteria for operator properties

Abstract

Let $T$ be an adjointable operator on a Hilbert $C^*$-module such that $T$ has the polar decomposition $T=UT|$. For each natural number $n$, $T$ is called an $(n+1)$-centered operator if $T^k=U^k|T^k|$ is the polar decomposition for $1\le k\le n+1$. This paper initiates the study of the $(n+1)$-centered operator via the generalized Aluthge transform and the generalized iterative Aluthge transform. Some new characterizations of the $(n+1)$-centered operator are provided.