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Measure of noncompactness of matrix operators on some difference sequence spaces of weighted means. (English) Zbl 1247.47009

Summary: For a sequence $x=\left({x}_{k}\right)$, we denote the difference sequence by ${\Delta }x=\left({x}_{k}-{x}_{k-1}\right)$. Let $u={\left({u}_{k}\right)}_{k=0}^{\infty }$ and $v={\left({v}_{k}\right)}_{k=0}^{\infty }$ be sequences of real numbers such that ${u}_{k}\ne 0$, ${v}_{k}\ne 0$ for all $k\in ℕ$. The difference sequence spaces of weighted means $\lambda \left(u,v,{\Delta }\right)$ are defined as $\lambda \left(u,v,{\Delta }\right)=\left\{x=\left({x}_{k}\right):W\left(x\right)\in \lambda \right\}$, where $\lambda$ is either of $c,{c}_{0},{\ell }_{\infty }$ and the matrix $W=\left({w}_{nk}\right)$ is defined by

${w}_{nk}=\left\{\begin{array}{cc}{u}_{n}\left({v}_{k}-{v}_{k+1}\right)\hfill & \text{if}\phantom{\rule{4.pt}{0ex}}kn·\hfill \end{array}\right\$

In this paper, we establish some identities or estimates for the operator norms and the Hausdorff measures of noncompactness of certain matrix operators on $\lambda \left(u,v,{\Delta }\right)$. Further, we characterize some classes of compact operators on these spaces by using the Hausdorff measure of noncompactness.

##### MSC:
 47B37 Operators on special spaces (weighted shifts, operators on sequence spaces, etc.) 47B07 Operators defined by compactness properties 47H08 Measures of noncompactness and condensing mappings, $K$-set contractions, etc. 40C05 Matrix methods in summability 46B45 Banach sequence spaces