On a Singular Sylvester Equation with Unbounded Self-Adjoint A and B

Abstract

In this paper we solve the Sylvester equation AX- XB= C, where A and B are closed densely defined self-adjoint operators, and C is a linear operator. We obtain sufficient conditions for the existence of infinitely many solutions and we manage to classify them. These results generalize the previously known results regarding singular Sylvester equations. Finally, we illustrate our results on an operator equation that appears in quantum mechanics, called the position-momentum operator equation.