Copyright © 2009 Vasile Lauric. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We show how a proof of J. Stampfli can be extended to prove that the operator X↦TX−XT defined on the Hilbert-Schmidt class, when T is an M-hyponormal, p-hyponormal, or log-hyponormal operator, has a closed range if and only if σ(T) is finite.