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Applications of measure of noncompactness in operators on the spaces ${s}_{\alpha }$, ${s}_{\alpha }^{0}$, ${s}_{\alpha }^{\left(c\right)}$, ${\ell }_{\alpha }^{p}$. (English) Zbl 1106.47029
We characterize some operators and matrix transformations in the sequence spaces ${s}_{\alpha }$, ${s}_{\alpha }^{\left(0\right)}$, ${s}_{\alpha }^{\left(c\right)}$, ${l}_{\alpha }^{p}$. Moreover, using the Hausdorff measure of noncompactness, necessary and sufficient conditions are formulated for a linear operator between the mentioned spaces to be compact. Among other things, some results of L. W. Cohen and N. Dunford [Duke Math. J. 3, 689–701 (1937; Zbl 0018.07101; JFM 63.0352.01)] are recovered.
##### MSC:
 47B37 Operators on special spaces (weighted shifts, operators on sequence spaces, etc.) 47H09 Mappings defined by “shrinking” properties 47A53 (Semi-)Fredholm operators; index theories 46B45 Banach sequence spaces