Let $l^{p,q}$, $1\le p,\,q\le\infty$, be the mixed-norm sequence space. We investigate the Hausdorff measure of noncompactness of the operator $T_\lambda:l^{r,s}\mapsto l^{u,v}$, defined by the multiplier $T_\lambda(a)=\{\lambda_na_n\}$, $\lambda=\{\lambda_n\}\in l^\infty$, $a=\{a_n\}\in l^{r,s}$, and prove necessary and sufficient conditions for $T_\lambda$ to be a compact.