Multipliers of mixed-norm sequence spaces and measures of noncompactness, II

Abstract

Let $l^{p,q}$, $0<p,q\le\infty$, be the mixed norm sequence space, and $T_{\lambda}\:l^{r,s}\to l^{u,v}$ the operator defined by the multiplier $T_\lambda(a)=\{\lambda_na_n\}$, $\lambda=\{\lambda_n\}\in l^\infty$, $a=\{a_n\}\in l ^{r,s}$. In this paper, we investigate the Hausdorff measure of noncompactness of the operator $T_\lambda$ in the cases when $0\le r,u,s,v\le\infty$, and prove necessary and sufficient conditions for $T_\lambda$ to be compact. The paper is a continuation of [8] where we considered the cases $1\le r,u,s,v\le\infty$.