Let $\Cal H$ be a separable infinite dimensional complex Hilbert space, and let $\Cal L(\Cal H)$ denote the algebra of all bounded linear operators on $\Cal H$ into itself. Let $A=(A_{1},A_{2},\dots ,A_{n})$, $B=(B_{1},B_{2},\dots ,B_{n})$ be $n$-tuples of operators in $\Cal L(\Cal H)$; we define the elementary operators $\Delta_{A,B}\:\Cal L(\Cal H)\mapsto\Cal L(\Cal H)$ by $\Delta_{A,B}(X)=\sum_{i=1}^nA_iXB_i-X.$ In this paper, we characterize the class of pairs of operators $A,B\in\Cal L(\Cal H)$ satisfying Putnam-Fuglede’s property, i.e, the class of pairs of operators $A,B\in\Cal L(\Cal H)$ such that $\sum_{i=1}^nB_iTA_i=T$ implies $\sum_{i=1}^nA_i^*TB_i^*=T$ for all $T\in\Cal C_1(\Cal H)$ (trace class operators). The main result is the equivalence between this property and the fact that the ultraweak closure of the range of the elementary operator $\Delta_{A,B}$ is closed under taking adjoints. This leads us to give a new characterization of the orthogonality (in the sense of Birkhoff) of the range of an elementary operator and its kernel in $C_1$ classes.
In this paper we obtain some results concerning the set ${\mathcal M} = \cup \bigl \lbrace \overline{R(\delta _A)}\cap \lbrace A\rbrace ^{\prime }\: A\in {\mathcal L(H)}\bigr \rbrace $, where $\overline{R(\delta _A)}$ is the closure in the norm topology of the range of the inner derivation $\delta _A$ defined by $\delta _A (X) = AX - XA.$ Here $\mathcal H$ stands for a Hilbert space and we prove that every compact operator in $\overline{R(\delta _A)}^w\cap \lbrace A^*\rbrace ^{\prime }$ is quasinilpotent if $A$ is dominant, where $\overline{R(\delta _A)}^w$ is the closure of the range of $\delta _A$ in the weak topology.