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

Na Liu, Qingxiang Xu, Xiaofeng Zhang

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.

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

Abstract

Let be an adjointable operator on a Hilbert -module such that has the polar decomposition . For each natural number , is called an -centered operator if is the polar decomposition for . This paper initiates the study of the -centered operator via the generalized Aluthge transform and the generalized iterative Aluthge transform. Some new characterizations of the -centered operator are provided.
Paper Structure (4 sections, 14 theorems, 70 equations)

This paper contains 4 sections, 14 theorems, 70 equations.

Table of Contents

  1. Preliminaries
  2. A new characterization of $(n+1)$-centered operators
  3. Generalized aluthge transforms and $(n+1)$-centered operators
  4. Generalized iterated aluthge transforms and $(n+1)$-centered operators

Key Result

Lemma 1.1

Liu-Luo-Xu Let $A\in\mathcal{L}(H,K)$ and $B,C\in\mathcal{L}(E,H)$ be such that $\overline{\mathcal{R}(B)}=\overline{\mathcal{R}(C)}$. Then $\overline{\mathcal{R}(AB)}=\overline{\mathcal{R}(AC)}$.

Theorems & Definitions (28)

  • Lemma 1.1
  • Lemma 1.2
  • Lemma 1.3
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Remark 2.1
  • Theorem 2.4
  • ...and 18 more