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

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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.

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Notes

  1. In this paper, the generalized Aluthge transform is defined for every positive numbers \(\alpha \) and \(\beta \). Some restrictions are, however, usually imposed on \(\alpha \) and \(\beta \) in the literature (see e.g., [3, 12]).

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Acknowledgements

The authors thank the referee for his/her very useful and detailed comments and suggestions, which greatly improved this presentation. Na Liu was supported by the Doctor Scientific Research Fund of Zhengzhou University of Light Industry (2020BSJJ041). Qingxiang Xu was partially supported by the National Natural Science Foundation of China (11971136).

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Authors and Affiliations

  1. College of Mathematics and Information Science, Zhengzhou University of Light Industry, Zhengzhou, 450002, People’s Republic of China

    Na Liu

  2. Department of Mathematics, Shanghai Normal University, Shanghai, 200234, People’s Republic of China

    Qingxiang Xu & Xiaofeng Zhang

Authors
  1. Na Liu
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  2. Qingxiang Xu
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  3. Xiaofeng Zhang
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Correspondence to Qingxiang Xu.

Additional information

Communicated by Ngai-Ching Wong.

The original online version of this article was revised: “The original version of this article, published on 19 January 2023, unfortunately contained an error. Parenthesis should be present in citing all equations. The original article has been corrected.

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Liu, N., Xu, Q. & Zhang, X. The \((n+1)\)-centered operator on a Hilbert \(C^*\)-module. Banach J. Math. Anal. 17, 19 (2023). https://doi.org/10.1007/s43037-022-00240-3

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  • DOI: https://doi.org/10.1007/s43037-022-00240-3

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