The polar decomposition for adjointable operators on Hilbert $C^*$-modules and $n$-centered operators

Na Liu; Wei Luo; Qingxiang Xu

The polar decomposition for adjointable operators on Hilbert $C^*$-modules and $n$-centered operators

Operator Algebras 2018-07-16 v4 Functional Analysis

Authors: Na Liu , Wei Luo , Qingxiang Xu

Abstract

Let $n$ be any natural number. The $n$ -centered operator is introduced for adjointable operators on Hilbert $C^*$ -modules. Based on the characterizations of the polar decomposition for the product of two adjointable operators, $n$ -centered operators, centered operators as well as binormal operators are clarified, and some results known for the Hilbert space operators are improved. It is proved that for an adjointable operator $T$ , if $T$ is Moore-Penrose invertible and is $n$ -centered, then its Moore-Penrose inverse is also $n$ -centered. A Hilbert space operator $T$ is constructed such that $T$ is $n$ -centered, whereas it fails to be $(n+1)$ -centered.

Keywords

hilbert modules operator theory on hilbert spaces operator theory

Cite

@article{arxiv.1806.06141,
  title  = {The polar decomposition for adjointable operators on Hilbert $C^*$-modules and $n$-centered operators},
  author = {Na Liu and Wei Luo and Qingxiang Xu},
  journal= {arXiv preprint arXiv:1806.06141},
  year   = {2018}
}

Comments

The first version has been expanded and re-organized. The series will be divided into several parts. The first part [arXiv:1807.01598v2] is accepted for publication in Advances in Operator Theory. This is the second part. Some citation labels of Ref.[15] have been changed

The polar decomposition for adjointable operators on Hilbert $C^*$-modules and $n$-centered operators

Abstract

Keywords

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