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Zbl 1231.47029
Mursaleen, M.; Noman, Abdullah K.
Compactness of matrix operators on some new difference sequence spaces.
(English)
[J] Linear Algebra Appl. 436, No. 1, 41-52 (2012). ISSN 0024-3795

The authors establish some identities or estimates for the Hausdorff measures of noncompactness of certain matrix operators on the difference sequence spaces $$c_{o}^{\lambda }(\Delta) = \Big\{ ( x_{k}):\ \lim_{n\rightarrow \infty }\frac{1}{\lambda _{n}}\sum_{k=0}^{n} ( \lambda _{k}-\lambda _{k-1})(x_{k}-x_{k-1}) =0\Big\}$$ and $$\ell_\infty^{\lambda }( \Delta) = \Big\{( x_{k}): \sup_{n}\left|\frac{1}{\lambda _{n}}\sum_{k=0}^{n}(\lambda _{k}-\lambda _{k-1})(x_{k}-x_{k-1})\right|<+\infty\Big\}\,,$$ where $\lambda =\left( \lambda _{k}\right)$ is a strictly increasing sequence of positive real numbers tending to infinity; see [{\it M. Mursaleen} and {\it A. K. Noman}, Math. Comput. Modelling 52, No.~ 3--4, 603--617 (2010; Zbl 1201.40003)]. Furthermore, they characterize some classes of compact operators on these spaces.
MSC 2000:
*47B37 Operators on sequence spaces, etc.
47H08
46B45 Banach sequence spaces
46B50 Compactness in normed spaces
46B15 Summability and bases in normed spaces

Keywords: BK space; matrix transformation; compact operator; Hausdorff measure of noncompactness

Citations: Zbl 1201.40003

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