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Compactness of matrix operators on some new difference sequence spaces. (English) Zbl 1231.47029

The authors establish some identities or estimates for the Hausdorff measures of noncompactness of certain matrix operators on the difference sequence spaces

${c}_{o}^{\lambda }\left({\Delta }\right)=\left\{\left({x}_{k}\right):\phantom{\rule{4pt}{0ex}}\underset{n\to \infty }{lim}\frac{1}{{\lambda }_{n}}\sum _{k=0}^{n}\left({\lambda }_{k}-{\lambda }_{k-1}\right)\left({x}_{k}-{x}_{k-1}\right)=0\right\}$

and

${\ell }_{\infty }^{\lambda }\left({\Delta }\right)=\left\{\left({x}_{k}\right):\underset{n}{sup}\left|\frac{1}{{\lambda }_{n}}\sum _{k=0}^{n}\left({\lambda }_{k}-{\lambda }_{k-1}\right)\left({x}_{k}-{x}_{k-1}\right)\right|<+\infty \right\}\phantom{\rule{0.166667em}{0ex}},$

where $\lambda =\left({\lambda }_{k}\right)$ is a strictly increasing sequence of positive real numbers tending to infinity; see [M. Mursaleen and 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:
 47B37 Operators on special spaces (weighted shifts, operators on sequence spaces, etc.) 47H08 Measures of noncompactness and condensing mappings, $K$-set contractions, etc. 46B45 Banach sequence spaces 46B50 Compactness in Banach (or normed) spaces 46B15 Summability and bases in normed spaces
##### References:
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