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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 $c$.

MSC:
 47B07 Operators defined by compactness properties 46B45 Banach sequence spaces 47B37 Operators on special spaces (weighted shifts, operators on sequence spaces, etc.) 54E45 Compact (locally compact) metric spaces 65J05 General theory of numerical methods in abstract spaces 47H08 Measures of noncompactness and condensing mappings, $K$-set contractions, etc.
Full Text:
References:
 [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