Başar, Feyzi; Malkowsky, Eberhard The characterization of compact operators on spaces of strongly summable and bounded sequences. (English) Zbl 1213.47019 Appl. Math. Comput. 217, No. 12, 5199-5207 (2011). 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 \(n\in\mathbb N\) and \(Ax=(A_nx)_{n=0}^\infty\in Y\) for all \(x\in X\), where \(X^\beta\) is the \(\beta\) dual of \(X\) and \(A_nx=\sum_{k=0}^\infty a_{nk}x_k\). 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:\[ w_0^p= \bigg\{x\in\omega:\;\lim_{n\to \infty} \frac{1}{n}\;\sum_{k=1}^n\,|x_k|^p=0\bigg\},\quad w_\infty^p= \bigg\{x\in\omega:\;\sup_{n\in {\mathbb N}} \frac{1}{n}\;\sum_{k=1}^n\,|x_k|^p<\infty\bigg\} \]and \[ w^p= \bigg\{x\in\omega:\;\lim_{n\to \infty}\;\frac{1}{n}\,\sum_{k=1}^n|x_k-\xi|^p=0 \text{\;for some\;} \xi\in {\mathbb C}\bigg\}. \]In the paper under review, the authors use the characterizations given in [F. Başar, E. Malkowsky and 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 \(c\). Reviewer: Angela Albanese (Lecce) Cited in 1 ReviewCited in 17 Documents MSC: 47B07 Linear operators defined by compactness properties 46B45 Banach sequence spaces 47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.) 54E45 Compact (locally compact) metric spaces 65J05 General theory of numerical analysis in abstract spaces 47H08 Measures of noncompactness and condensing mappings, \(K\)-set contractions, etc. Keywords:spaces of strongly bounded and summable sequences; matrix transformations; compact operators; Hausdorff measure of noncompactness Citations:Zbl 0155.38802; Zbl 1164.46003 PDFBibTeX XMLCite \textit{F. Başar} and \textit{E. Malkowsky}, Appl. Math. Comput. 217, No. 12, 5199--5207 (2011; Zbl 1213.47019) Full Text: DOI References: [1] Başar, F.; Malkowsky, E.; Altay, B., Matrix transformations on the matrix domains of triangles in the spaces of strongly \(C_1\)-summable and bounded sequences, Publ. Math. Debrecen, 73, 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 \(C_1\) 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 [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), Russian [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 [7] Malkowsky, E.; Rakočević, V., An introduction into the theory of sequence spaces and measures of noncompactness, Zbornik radova, 9, 17, 143-234 (2000), Matematički institut SANU, Belgrade · Zbl 0996.46006 [8] Wilansky, A., Summability Through Functional Analysis (1984), North-Holland Mathematics Studies 85: North-Holland Mathematics Studies 85 Amsterdam · Zbl 0531.40008 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.