Singular Sylvester equation in Banach spaces and its applications: Fredholm theory approach

We prove that, by assuming the existence of at least one left upper semi-Fredholm operator, then under some natural conditions, the singular operator equation A X − X B = C is solvable if the appropriate matrix equation is solvable. This characterization is convenient because the matrix version of the problem has been closed in [14] and [17] . In addition, we obtain sufficient conditions for A, B and X such that the generalized derivation A X − X B is a compact operator. A connection is established with Frechet derivatives and commutators of idempotents. Applications to Schur coupling and linear time-invariant systems are mentioned.