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Zbl 1247.47009
Mursaleen, M.; Karakaya, V.; Polat, H.; Simşek, N.
Measure of noncompactness of matrix operators on some difference sequence spaces of weighted means. (English)
[J] Comput. Math. Appl. 62, No. 2, 814-820 (2011). ISSN 0898-1221

Summary: For a sequence $x=(x_{k})$, we denote the difference sequence by $\Delta x=(x_{k} - x_{k - 1})$. Let $u=(u_k)_{k=0}^\infty$ and $v=(v_k)_{k=0}^\infty$ be sequences of real numbers such that $u_{k}\ne 0$, $v_{k}\ne 0$ for all $k\in \bbfN$. The difference sequence spaces of weighted means $\lambda (u,v,\Delta)$ are defined as $\lambda (u,v,\Delta )=\{x=(x_{k}):W(x)\in \lambda\}$, where $\lambda$ is either of $c, c_{0}, \ell _{\infty }$ and the matrix $W=(w_{nk})$ is defined by $$w_{nk}=\cases u_n(v_k-v_{k+1})&\text{if }k<n,\\u_n v_n&\text{if }k=n,\\0&\text{if }k>n.\endcases$$ In this paper, we establish some identities or estimates for the operator norms and the Hausdorff measures of noncompactness of certain matrix operators on $\lambda (u,v,\Delta)$. Further, we characterize some classes of compact operators on these spaces by using the Hausdorff measure of noncompactness.

MSC 2000:: *47B37 Operators on sequence spaces, etc.
47B07 Operators defined by compactness properties
47H08
40C05 Matrix methods in summability
46B45 Banach sequence spaces

Keywords: sequence space; weighted mean; matrix transformation; compact operator; Hausdorff measure of noncompactness

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