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The Hausdorff measure of noncompactness of operators on the matrix domains of triangles in the spaces of strongly $C_{1}$ summable and bounded sequences. (English) Zbl 1244.47046
Let $\omega $ be the space of all complex sequences $x=\left( x_{k}\right) _{k=1}^{\infty }$, $A=\left( a_{nk}\right) _{n,k=1}^{\infty }$ be an infinite matrix of complex numbers and $X$ a subset of $\omega$. The set $$X_{A}=\left\{ x\in \omega :Ax=\sum_{k=1}^\infty a_{nk} {x_{k}}\in X\right\} $$ is called the matrix domain of $A$ in $X$. Given any sets $X$ and $Y$ in $\omega$, let $\left( X,Y\right) $ denote the class of all matrices $A$ such that $X\subset Y_{A}$. Let $T=\left( t_{nk}\right) _{n,k=1}^{\infty }$ be a triangle, that is, $ t_{nk}=0$ for $k>n$ and $t_{nn}\neq 0$ $\left( n=1,2, \dots\right) $, let $e\in \omega $ be the sequence with $e_{k}=1$ for all $k$ and $1\leq p<\infty $. The sets of strongly $C_{1}$-summable and bounded sequences $$ w_{0}^{p}=\left\{ x\in \omega : \lim_{n \to \infty}\left( \frac{1}{n}\sum_{k=1}^n \left\vert x_{k}\right\vert ^{p}\right) =0\right\} , $$ $$ w^{p}=\left\{ x\in \omega :x-\xi \cdot e\in w_{0}^{p}\text{ for some complex number }\xi \right\} $$ and $$ w_{\infty}^{p}=\left\{ x\in \omega :\underset{n}\to{\sup }\left( \frac{1}{n} \sum_{k=1}^n \left\vert x_{k}\right\vert ^{p}\right) <\infty \right\} $$ were defined and studied by {\it I. J. Maddox} [“On Kuttner’s theorem”, J. Lond. Math. Soc. 43, 285--290 (1968; Zbl 0155.38802)]. In the paper under review, the authors apply the Hausdorff measure of noncompactness to characterize the classes of compact operators given by infinite matrices $A\in \left( X,Y\right) $, where $X$ is one of spaces $ \left( w_{0}^{p}\right) _{T},\left( w^{p}\right) _{T}$ or $\left( w_{\infty }^{p}\right) _{T}$ and $Y$ is the space $c_{0}$ of null sequences or the space $c$ of convergent sequences. Moreover, they give sufficient conditions for the compactness of operators $A\in \left( X,Y\right) $ when $X$ is again one of spaces $\left( w_{0}^{p}\right) _{T},\left( w^{p}\right) _{T}$ or $\left( w_{\infty }^{p}\right) _{T}$ and the final $Y$ is the space $l_{\infty }$ of bounded sequences.

47H08Measures of noncompactness and condensing mappings, $K$-set contractions, etc.
40C05Matrix methods in summability
40J05Summability in abstract structures
Full Text: DOI
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