Commutators and hyponormal operators on a Hilbert space

Document Type : Research Article

Authors

1 Kazan National Research Technological University, Kazan, Russia

2 Department of Mathematics and Mechanics, Kazan Federal University, Kazan, Russia

Abstract

Let $\mathcal{H}$ be an infinite-dimensional Hilbert space over the field $\mathbb{C}$, $\mathcal{B}(\mathcal{H})$ be the $\ast$-algebra of all linear bounded operators on $\mathcal{H}$, let $|X|=\sqrt{X^*X}$ for $X\in \mathcal{B}(\mathcal{H})$. An operator $A\in \mathcal{B}(\mathcal{H})$ is a commutator if $A=[S, T]=ST-TS$ for some $S, T\in \mathcal{B}(\mathcal{H})$. Let $X, Y \in \mathcal{B}(\mathcal{H})$ and $X\geq 0$. If the operator $XY$ is a non-commutator, then $X^pYX^{1-p}$ is a non-commutator for every $0<p<1$. Let $A \in \mathcal{B}(\mathcal{H})$ be $p$-hyponormal for some $0<p\leq 1$. If $|A^*|^r$ is a non-commutator for some $r>0$ then $|A|^q$ is a non-commutator 
for every $q>0$. Let $\mathcal{H}$ be separable and $A \in \mathcal{B}(\mathcal{H})$ be a non-commutator. If $A$ is hyponormal (or cohyponormal) then $A$ is normal. We also present results in the case of a finite-dimensional Hilbert space.

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References

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Journal of the Iranian Mathematical Society
Volume 4, Issue 1
January 2023
Pages 67-78

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