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Zbl 1106.47029
Applications of measure of noncompactness in operators on the spaces $s_{\alpha}$, $s_{\alpha}^{0}$, $s_{\alpha}^{(c)}$, $\ell_{\alpha}^{p}$.
(English)
[J] J. Math. Anal. Appl. 323, No. 1, 131-145 (2006). ISSN 0022-247X

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.
[Poom Kumam (Bangkok)]
MSC 2000:
*47B37 Operators on sequence spaces, etc.
47H09 Mappings defined by "shrinking" properties
47A53 (Semi-)Fredholm operators; index theories
46B45 Banach sequence spaces

Keywords: measure of noncompactness; matrix transformation; sequence space

Citations: Zbl 0018.07101; JFM 63.0352.01

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