Let $\mathscr{H}$ be a separable infinite dimensional complex Hilbert space, and let $\mathcal{L}\left(\mathscr{H}\right)$ denote the algebra of all bounded linear operators on $\mathscr{H}$ into itself. Let $A=({A}_{1},{A}_{2},\cdots ,{A}_{n})$, $B=({B}_{1},{B}_{2},\cdots ,{B}_{n})$ be $n$-tuples of operators in $\mathcal{L}\left(\mathscr{H}\right)$; we define the elementary operators ${\Delta}_{A,B}\phantom{\rule{0.222222em}{0ex}}\mathcal{L}\left(\mathscr{H}\right)\mapsto \mathcal{L}\left(\mathscr{H}\right)$ by ${\Delta}_{A,B}\left(X\right)={\sum}_{i=1}^{n}{A}_{i}X{B}_{i}-X.$ In this paper, we characterize the class of pairs of operators $A,B\in \mathcal{L}\left(\mathscr{H}\right)$ satisfying Putnam-Fuglede’s property, i.e, the class of pairs of operators $A,B\in \mathcal{L}\left(\mathscr{H}\right)$ such that ${\sum}_{i=1}^{n}{B}_{i}T{A}_{i}=T$ implies ${\sum}_{i=1}^{n}{A}_{i}^{*}T{B}_{i}^{*}=T$ for all $T\in {\mathcal{C}}_{1}\left(\mathscr{H}\right)$ (trace class operators). The main result is the equivalence between this property and the fact that...