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

Abstract

Let T be an adjointable operator on a Hilbert \(C^*\)-module such that T has the polar decomposition \(T=U\vert T\vert \). For each natural number n, T is called an \((n+1)\)-centered operator if \(T^k=U^k\vert T^k\vert \) 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.