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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 \(n\in\mathbb N\) 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 \(c\).

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.
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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
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