$k$-reflexivity defect of the image of a generalized derivation

Let $\mathcal{X}$ be a finite-dimensional complex vector space and let k be a positive integer. An explicit formula for the k-reflexivity defect of the image of a generalized derivation on $L(\mathcal{X})$, the space of all linear transformations on $\mathcal{X}$, is given. Using latter, we also study the k-reflexivity defect of the image of an elementary operator of the form $\Delta(T)=ATB-T$ $(T \in L(\mathcal{X}))$.