On the equations \(Ax=q\) and \(SX-XT=Q\)
Let A be the generator of a certain semigroup, the authors prove the existence of a unique solution x of \(Ax=q\). About this, three equivalent characterizing conditions are demonstrated. This result is applied to study the solvability of the operator euqation \[ (*)\quad SX-XT=Q, \] where S and -T are generators of some semigroups of operators. Three equivalent statements are given for the existence of a solution of (*). These results generalize existence theorems of several authors.
scientific article; zbMATH DE number 4056370