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Measure of noncompactness of operators and matrices on the spaces $c$ and $c_{0}$. (English) Zbl 1154.47027
The authors, using the Hausdorff measure of noncompactness, give necessary and sufficient conditions for a linear operator, or a matrix, between the spaces $c$ and $c_0$ to be compact. Some results of {\it L. W.\thinspace Cohen} and {\it N. Dunford} [Duke Math. J. 3, No.4, 689-701 (1937; Zbl 0018.07101; JFM 63.0352.01)] are recovered.

47B37Operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
46B45Banach sequence spaces
47B07Operators defined by compactness properties
47H09Mappings defined by “shrinking” properties
Full Text: DOI
[1] R. R. Akhmerov, M. I. Kamenskiĭ, A. S. Potapov, A. E. Rodkina, and B. N. Sadovskiĭ, Measures of Noncompactness and Condensing Operators, vol. 55 of Operator Theory: Advances and Applications, Birkhäuser, Basel, 1992. · Zbl 0748.47045
[2] J. Banaś and K. Goebel, Measures of Noncompactness in Banach Spaces, vol. 60 of Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, New York, 1980. · Zbl 0441.47056
[3] L. W. Cohen and N. Dunford, “Transformations on sequence spaces,” Duke Mathematical Journal, vol. 3, no. 4, pp. 689-701, 1937. · Zbl 0018.07101 · doi:10.1215/S0012-7094-37-00357-0
[4] I. T. Gohberg, L. S. Goldenstein, and A. S. Markus, “Investigations of some properties of bounded linear operators with their q-norms,” U\vcenie Zapiski, Kishinevskii Gosuniversitet, vol. 29, pp. 29-36, 1957 (Russian).
[5] I. J. Maddox, Elements of Functional Analysis, Cambridge University Press, London, 1970. · Zbl 0193.08601
[6] A. E. Taylor, Introduction to Functional Analysis, John Wiley & Sons, New York; Chapman & Hall, London, 1958. · Zbl 0081.10202