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The characterization of compact operators on spaces of strongly summable and bounded sequences. (English) Zbl 1213.47019
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 $c$.

47B07Operators defined by compactness properties
46B45Banach sequence spaces
47B37Operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
54E45Compact (locally compact) metric spaces
65J05General theory of numerical methods in abstract spaces
47H08Measures of noncompactness and condensing mappings, $K$-set contractions, etc.
Full Text: DOI
[1] Başar, F.; Malkowsky, E.; Altay, B.: Matrix transformations on the matrix domains of triangles in the spaces of strongly C1-summable and bounded sequences, Publ. math. Debrecen 73, No. 1 -- 2, 193-213 (2008) · Zbl 1164.46003
[2] Djolović, I.; Malkowsky, E.: The Hausdorff measure of noncompactness of operators on the matrix domains of triangles in the spaces of strongly C1 summable and bounded sequences, Appl. math. Comput. 216, 1122-1130 (2010) · Zbl 1244.47046
[3] Djolović, I.; Malkowsky, E.: A note on compact operators on matrix domains, J. math. Anal. appl. 340, 291-303 (2008) · Zbl 1147.47002 · doi:10.1016/j.jmaa.2007.08.021
[4] Gohberg, I. T.; Goldenstein, L. S.; Markus, A. S.: Investigations of some properties of bounded linear operators with their q-norms, Učen. zap. Kishinevsk. univ. 29, 29-36 (1957)
[5] Jarrah, A. M.; Malkowsky, E.: Ordinary, absolute and strong summability and matrix transformations, Filomat17, 59-78 (2003) · Zbl 1274.40001
[6] Maddox, I. J.: On kuttner’s theorem, J. lond. Math. soc. 43, 285-290 (1968) · Zbl 0155.38802 · doi:10.1112/jlms/s1-43.1.285
[7] Malkowsky, E.; Rakočević, V.: An introduction into the theory of sequence spaces and measures of noncompactness, Zbornik radova 9, No. 17, 143-234 (2000) · Zbl 0996.46006
[8] Wilansky, A.: Summability through functional analysis, (1984) · Zbl 0531.40008