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Applications of measure of noncompactness in operators on the spaces $s_{\alpha}$, $s_{\alpha}^{0}$, $s_{\alpha}^{(c)}$, $\ell_{\alpha}^{p}$. (English) Zbl 1106.47029
We characterize some operators and matrix transformations in the sequence spaces $s_\alpha$, $s^{(0)}_\alpha$, $s^{(c)}_\alpha$, $l^p_\alpha$. 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 {\it L. W. Cohen} and {\it N. Dunford} [Duke Math. J. 3, 689--701 (1937; Zbl 0018.07101; JFM 63.0352.01)] are recovered.

MSC:
47B37Operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
47H09Mappings defined by “shrinking” properties
47A53(Semi-) Fredholm operators; index theories
46B45Banach sequence spaces
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References:
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