On the equations Ax = q and SX − XT = Q

AbstractIn this paper, we first study solutions of the functional equation Ax = q, where A is the generator of a certain semigroup, using mean-ergodic theory. Then we apply the result to study the solvability of the operator equation SX − XT = Q, where S and −T are generators of some semigroups of operators. We give a characterization of the solvability using weak operator limit of an integral involving the Tensor Product Semigroup of the two semigroups generated, respectively, by S and −T