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  • 1 Departamento de Matemática – Av, Universidade Estadual de Maringá – UEM, Colombo 5790, Maringá, PR, Brazil, 87020-900
  • 2 Departamento de Matemática, Universidade Estadual de Maringá – Unicamp, Caixa Postal, 6065, Campinas, SP, Brazil, 13083-859
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Abstract

New measures of noncompactness for bounded sets and linear operators, in the setting of abstract measures and generalized limits, are constructed. A quantitative version of a classical criterion for compactness of bounded sets in Banach spaces by R. S. Phillips is provided. Properties of those measures are established and it is shown that they are equivalent to the classical measures of noncompactness. Applications to summable families of Banach spaces, interpolations of operators and some consequences are also given.

  • [1] Akhmerov, R. R., Kamenskii, M. I., Potapov, A. S., Rodkina, A. E., Sadovskii, B. N. 1992 Measures of Noncompactness and Condensing Operators Birkhäuser Verlag Basel.

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  • [2] Toledano, J. M. Ayerbe, Domínguez Benavides, T., López Acedo, G. 1997 Measures of Noncompactness in Metric Fixed Point Theory Birkhäuser Verlag Basel.

    • Search Google Scholar
    • Export Citation
  • [3] Banás, J., Goebel, K. 1980 Measures of Noncompactness in Banach Spaces Lecture Notes in Pure and Applied Mathemathics 60 Marcel Dekker New York.

    • Search Google Scholar
    • Export Citation
  • [4] Banás, J., Martinon, A. 1992 Measures of noncompactness in Banach sequence spaces Math. Slovaca 42 497503.

  • [5] Brandani da Silva, E., Fernandez, D. L. 2006 Hausdorff measures of noncompactness and interpolation spaces Serdica Math. J. 32 179184.

    • Search Google Scholar
    • Export Citation
  • [6] Brooks, J. K., Dinculeanu, N. 1979 Conditional expectation and weak and strong compactness in spaces of Bochner integrable functions J. Mult. Analysis 9 420427 .

    • Crossref
    • Search Google Scholar
    • Export Citation
  • [7] Dunford , Schwarz, J. 1967 Linear Operators I Interscience Publ. Inc. New York.

  • [8] Edmunds, D. E., Evans, W. D. 1990 Spectral Theory and Differential Operators Clarendon Press Oxford.

  • [9] Fernández-Cabrera, M. L. 2002 Compact operators between real interpolation spaces Math. Ineq. & Appl. 5 283289.

  • [10] McShane, E. J., Botts, T. A. 1959 Real Analysis D. Van Nostrand Co., Inc. Princeton, NJ.

  • [11] Phillips, R. S. 1940 On linear transformations Trans. Amer. Math. Soc. 48 516541.

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