Remarks on the existence and uniqueness of solutions to the infinite dimensional sylvester equations

In various control problems for infinite-dimensional systems (see, e.g., [1], [2], [3], [4], [7]) one can found usefulness of Sylvester type operator equations. The aim of the paper is to popularize the implemented semigroup framework (e.g. [8], [5], [6]) by showing how it helps to deal with the infinite-dimensional algebraic Sylvester equation both in differential (DSE) and in the algebraic (ASE) form. The main emphasis is put on some basic results concerning the existence and uniqueness of solutions. The collected results are presented from the point of view of potential applications in the systems and control theory.