Primary 47B47, 47A30, 47B20; Secondary 4

This note studies how certain problems in quantum theory have motivated some recent research in pure Mathematics in matrix and operator theory. The mathematical key is that of a commutator. We introduce the notion of the pair (A,B) of operators having the Fuglede-Putnam's property in the ideal of all compact operators. The characterization of this class leads us to generalize some recent results. We also give some applications of these results.;