New approach to some results related to mixed norm sequence spaces

In this paper, the mixed norm sequence spaces $\ell^{p,q}$ for $1\le p,q \le \infty$ 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 I.Jovanovi\'{c}, V.Rako\v{c}evi\'{c}, Multipliers of mixed norm sequence spaces and Measure of Noncompactness, {\it Publications De L'Institut Math\'{e}matique}, 1994, {\bf 56(70)}, 61--68 and I.Jovanovi\'{c}, V.Rako\v{c}evi\'{c}, Multipliers of mixed norm sequence spaces and measure of noncompactness. II, {\it Matemati\v{c}ki vesnik}, 1997, {\bf 49}, 197--206,  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.