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Zbl 1213.47019
Başar, Feyzi; Malkowsky, Eberhard
The characterization of compact operators on spaces of strongly summable and bounded sequences.
(English)
[J] Appl. Math. Comput. 217, No. 12, 5199-5207 (2011). ISSN 0096-3003

Let $A=(a_{nk})_{n,k=0}^\infty$ be an infinite matrix of complex numbers, $X$ and $Y$ be subsets of $\omega$. Let $(X,Y)$ denote the class of all matrices $A$ such that $A_n=(a_{nk})_{k=0}^\infty\in X^\beta$ for all $n\in\Bbb N$ and $Ax=(A_nx)_{n=0}^\infty\in Y$ for all $x\in X$, where $X^\beta$ is the $\beta$ dual of $X$ and $A_nx=\sum_{k=0}^\infty a_{nk}x_k$. {\it I. J. Maddox} in [J. Lond. Math. Soc. 43, 285--290 (1968; Zbl 0155.38802)] introduced and studied the following sets of sequences that are strongly summable and bounded with index $p$ ($1\leq p<\infty$) by the Cesàro method of order 1: $$w_0^p= \bigg\{x\in\omega:\ \lim_{n\to \infty} \frac{1}{n}\ \sum_{k=1}^n\,|x_k|^p=0\bigg\},\quad w_\infty^p= \bigg\{x\in\omega:\ \sup_{n\in {\Bbb N}} \frac{1}{n}\ \sum_{k=1}^n\,|x_k|^p<\infty\bigg\}$$ and $$w^p= \bigg\{x\in\omega:\ \lim_{n\to \infty}\ \frac{1}{n}\,\sum_{k=1}^n|x_k-\xi|^p=0 \text{\ for some\ } \xi\in {\Bbb C}\bigg\}.$$ In the paper under review, the authors use the characterizations given in [{\it F. Başar, E. Malkowsky} and {\it B. Altay}, Publ. Math. 73, No.~1--2, 193--213 (2008; Zbl 1164.46003)] of the classes $(w^p_0,c_0)$, $(w^p,c_0)$, $(w^p_\infty,c_0)$, $(w^p_0,c)$, $(w^p,c)$ and $(w^p_\infty,c)$ and the Hausdorff measure of noncompactness to characterize the classes of compact operators from $w^p_0$, $w^p$ and $w^p_\infty$ into $c_0$ and $c$.
[Angela Albanese (Lecce)]
MSC 2000:
*47B07 Operators defined by compactness properties
46B45 Banach sequence spaces
47B37 Operators on sequence spaces, etc.
54E45 Compactness in metric spaces
65J05 General theory of numerical methods in abstract spaces
47H08

Keywords: spaces of strongly bounded and summable sequences; matrix transformations; compact operators; Hausdorff measure of noncompactness

Citations: Zbl 0155.38802; Zbl 1164.46003

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