Powers of posinormal Hilbert-space operators

Abstract

A bounded linear operator $A$ on a Hilbert space $\mathcal{H}$ is posinormal if there exists a positive operator $P$ such that $A{A}^{\ast }=A^{\ast }PA$. 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.

4 pages. After posting v1, the authors discovered v1's Main Theorem is already in the literature, expressed in different language and having different ancestry. V2 provides details