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Applications of measure of noncompactness in operators on the spaces s α , s α 0 , s α (c) , α p . (English) Zbl 1106.47029
We characterize some operators and matrix transformations in the sequence spaces s α , s α (0) , s α (c) , l α p . Moreover, using the Hausdorff measure of noncompactness, necessary and sufficient conditions are formulated for a linear operator between the mentioned spaces to be compact. Among other things, some results of L. W. Cohen and N. Dunford [Duke Math. J. 3, 689–701 (1937; Zbl 0018.07101; JFM 63.0352.01)] are recovered.
MSC:
47B37Operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
47H09Mappings defined by “shrinking” properties
47A53(Semi-)Fredholm operators; index theories
46B45Banach sequence spaces