Let
$H$
be a separable, infinite-dimensional, complex Hilbert space and let
$A,\,B\,\in \,\mathcal{L}\left( H \right)$
, where
$\mathcal{L}(H)$
is the algebra of all bounded linear operators on
$H$
. Let
${{\delta }_{AB}}\,:\mathcal{L}\left( H \right)\to \mathcal{L}\left( H \right)$
denote the generalized derivation
${{\delta }_{AB}}\left( X \right)\,=\,AX\,-\,XB$
. This note will initiate a study on the class of pairs
$\left( A,\,B \right)$
such that
$\overline{R\left( {{\delta }_{AB}} \right)}\,=\,\overline{R\left( {{\delta }_{A*\,B*}} \right)}$
.