In this paper we establish some properties concerning the class of operators $A\in {\cal L(H)}$ that satisfy $\overline{ { \cal R }(\delta_{A})}\cap\{A\}'=\{0\}$, where $\overline{ { \cal R }(\delta_{A})}$ is the norm closure of the range of the inner derivation $\delta_{A},$ defined on ${\cal L(H)}$ by $\delta_{A}(X)=AX-XA$. Here ${\cal H}$ stands for a Hilbert space; as a consequence, we show that the set $\{ A \in { \cal L(H)}\;\;/\;\;\overline{ { \cal R }(\delta_{A})}\cap\{A\}'=\{0\} \}$ is norm-dense. We also describe some classes of operators $A,\;B$ for which we have $\overline{ { \cal R }(\delta_{A,B})}\cap\ker(\delta_{A^{\ast},B^{\ast}})=\{0\}$ ($\ker(\delta_{A^{\ast},B^{\ast}})$ is the kernel of the generalized derivation $\delta_{A^{\ast},B^{\ast}}$ defined on ${\cal L(H)}$ by $\delta_{A^{\ast},B^{\ast}}(X)=A^{\ast}X-XB^{\ast}$).