Powers of posinormal Hilbert-space operators

TL;DR

This paper investigates the properties of posinormal operators on Hilbert spaces, showing that powers of such operators with closed range also have closed range, and explores related examples and implications for hyponormal operators.

Contribution

It proves that powers of posinormal operators with closed range remain posinormal and have closed range, extending known properties of hyponormal operators.

Findings

Powers of posinormal operators with closed range also have closed range.

Hyponormal operators are included in posinormal operators, so their powers also have closed range.

An example demonstrates an operator whose square does not have closed range.

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.