Solving the Operator Equation AX-XB=C with Closed A and B

Abstract

We solve the operator equation AX- XB= C, where A and B are closed operators whose point spectra intersect. We obtain sufficient conditions for the existence of solutions and provide a way of constructing them. As a corollary, we obtain a result that gives us new insight on the matrix equations of the form AX= XB, where A and B share non-zero eigenvalues. Afterwards, we illustrate our results on Sturm–Liouville operators.