S-nets in the space of tempered distributions and generated operators

In this paper we study new concepts introduced by D. Carfì in the sphere of the theory of tempered distributions. In particular we establish some their algebraic properties. The new concepts are: 1) nets of class S; 2) operators generated by nets of class S; 3) nets of class LS. These new concepts represent the foundation of a generalization of linear algebra in the space of tempered distributions, that permits a more deeply study of some problems faced by linear algebra, as the theory of system, theory of decision, the optimal control, the quantum mechanics and so on.