Similarity orbits and the range of the generalized derivation $X\to MX-XN$

Abstract:

If $M$ and $N$ are bounded operators on infinite dimensional complex Hilbert spaces $\mathcal {H}$ and $\mathcal {K}$, let $\tau (X) = MX - XN$ for $X$ in $\mathcal {L}(\mathcal {K},\mathcal {H})$. The closure of the range of $\tau$ is characterized when $M$ and $N$ are normal. There is a close connection between the range of $\tau$ and operators $C$ for which $[\begin {array}{*{20}{c}} M \& C \\ 0 \& N \\ \end {array} ]$ is in the closure of the similarity orbit of $[\begin {array}{*{20}{c}} M \& 0 \\ 0 ^ N \\ \end {array} ]$. This latter set is characterized and compared with the closure of the range of $\tau$.