In this paper, the mixed norm sequence spaces $\ell^{p,q}$ for $1\leq p,q\leq\infty$ are the subject of our research; we establish conditions for an operator $T_\lambda$ to be compact, where $T_\lambda$ is given by a diagonal matrix. This will be achieved by applying the Hausdorff measure of noncompactness and the theory of BK spaces. This problem was treated and solved in [5,6], but in a different way, without the application of the theory of infinite matrices and BK spaces. Here, we will present a new approach to the problem. Some of our results are known and others are new.