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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 ${t}_{nn}\ne 0$ $\left(n=1,2,\cdots \right)$, let $e\in \omega$ be the sequence with ${e}_{k}=1$ for all $k$ and $1\le p<\infty$. The sets of strongly ${C}_{1}$-summable and bounded sequences

${w}_{0}^{p}=\left\{x\in \omega :\underset{n\to \infty }{lim}\left(\frac{1}{n}\sum _{k=1}^{n}{\left|{x}_{k}\right|}^{p}\right)=0\right\},$
${w}^{p}=\left\{x\in \omega :x-\xi ·e\in {w}_{0}^{p}\phantom{\rule{4.pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\text{some}\phantom{\rule{4.pt}{0ex}}\text{complex}\phantom{\rule{4.pt}{0ex}}\text{number}\phantom{\rule{4.pt}{0ex}}\xi \right\}$

and

${w}_{\infty }^{p}=\left\{x\in \omega :\underset{n}{sup}\left(\frac{1}{n}\sum _{k=1}^{n}{\left|{x}_{k}\right|}^{p}\right)<\infty \right\}$

were defined and studied by 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.

##### MSC:
 47H08 Measures of noncompactness and condensing mappings, $K$-set contractions, etc. 40C05 Matrix methods in summability 40J05 Summability in abstract structures