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arXiv · 1806.06141

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

Na Liu·Wei Luo·Qingxiang Xu
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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.

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Na Liu, Wei Luo, Qingxiang Xu. 2018-06-15. The polar decomposition for adjointable operators on Hilbert $C^*$-modules and $n$-centered operators. https://arxiv.org/abs/1806.06141

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