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Generalized measures of noncompactness of sets and operators in Banach spaces. (English) Zbl 1234.47040

The paper begins with an axiomatic definition of measure of noncompactness which is a variant of the definition given by J. Banaś and K. G. Goebel [Measures of noncompactness in Banach spaces. Lecture Notes in Pure and Applied Mathematics 60. New York, Basel: Marcel Dekker (1980; Zbl 0441.47056)]. After a discussion of the relationship of this definition with general measures of noncompactness, and with the definition in the Hausdorff class, the authors recall notions on generalized limits and use them in their study of the Phillips measure of noncompactness [R. S. Phillips, Trans. Am. Math. Soc. 48, 516–541 (1940; Zbl 0025.34202)]. It is shown that the measures considered by the authors are equivalent to the Hausdorff measure. This is followed by discussions on measures of noncompactness in spaces of summable families; an application to bounded mappings is given. The final sections of the paper deal with generalized measures of noncompactness of linear operators on Banach spaces and estimates for Phillips-like measures.

MSC:

47H08 Measures of noncompactness and condensing mappings, \(K\)-set contractions, etc.
46B45 Banach sequence spaces
46B50 Compactness in Banach (or normed) spaces
46B70 Interpolation between normed linear spaces
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References:

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