On the equation \(ax-xb=c\) in complex Banach algebras (Q5959502)

The paper discusses the problem of solvability of the equation \(ax-xb=c\) in a complex Banach algebra \(A\) with unity, assuming that \(a,b\in A\) are quasinormal elements. Furthermore, when \(A\) possesses a continuous involution, it is shown that, for \(c\in A\), a necessary and sufficient condition for the solvability of the above mentioned equation is given by the strong similarity of the matrices \(\Bigl(\begin{smallmatrix} a & 0\\ 0 & b\end{smallmatrix} \Bigr)\) and \(\Bigl( \begin{smallmatrix} a & c\\ 0 & b \end{smallmatrix}\Bigr)\).