Commutants and derivation ranges

summary: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