Advanced Search

Query:

Fill in the form and click »Search«...

Format:

Display: entries per page entries

Zbl 1244.47046
Djolović, Ivana; Malkowsky, Eberhard
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)
[J] Appl. Math. Comput. 216, No. 4, 1122-1130 (2010). ISSN 0096-3003

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)]. \par 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.
[Giulio Trombetta (Arcavaceta di Rende)]

MSC 2000:: *47H08
40C05 Matrix methods in summability
40J05 Summability in abstract structures

Keywords: strongly bounded and summable sequences; matrix transformations; compact operators; Cesàro method

Citations: Zbl 0155.38802

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Advanced Search

Zentralblatt MATH Berlin [Germany]