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

Abstract

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