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arXiv · 2203.01473

Powers of posinormal Hilbert-space operators

Paul S. Bourdon·C. S. Kubrusly·Derek Thompson
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Abstract

A bounded linear operator $A$ on a Hilbert space $\mathcal{H}$ is posinormal if there exists a positive operator $P$ such that $AA^{*} = A^{*}PA$. We show that if $A$ is posinormal with closed range, then $A^n$ is posinormal and has closed range for all integers $n\ge 1$. Because the collection of posinormal operators includes all hyponormal operators, we obtain as a corollary that powers of closed-range hyponormal operators continue to have closed range. We also present a simple example of a closed-range operator $T: \mathcal{H}\to \mathcal{H}$ such that $T^2$ does not have closed range.

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Paul S. Bourdon, C. S. Kubrusly, Derek Thompson. 2022-03-03. Powers of posinormal Hilbert-space operators. https://arxiv.org/abs/2203.01473

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