A note on the range of the operator \(X\mapsto TX - XT\) defined on \(\mathcal C_{2}(\mathcal H)\) (Q1028875)

Summary: We show how a proof of \textit{J.\,G.\thinspace Stampfli} [Proc.\ Am.\ Math.\ Soc.\ 52, 117--120 (1975; Zbl 0315.47019)] can be extended to prove that the operator \(X\mapsto TX - XT\) 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 \(\sigma (T)\) is finite.