×

Compact sets of compact operators in absence of \(\ell^{1}\). (English) Zbl 0961.47010

Let \(H\) be a subset of compact operators from a reflexive Banach space \(X\) to another Banach space \(Y\). The (relative) compactness of \(H\) result in F. Galaz-Fontes [Proc. Am. Math. Soc. 126, No. 2, 587-588 (1998; Zbl 0893.46015)] is shown to remain valid even if \(X\) does not contain a copy of \(\ell^1\).
Reviewer: M.Turinici (Iaşi)

MSC:

47B07 Linear operators defined by compactness properties
46B25 Classical Banach spaces in the general theory

Citations:

Zbl 0893.46015
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Philip M. Anselone, Compactness properties of sets of operators and their adjoints, Math. Z. 113 (1970), 233 – 236. · Zbl 0176.11105 · doi:10.1007/BF01110195
[2] N. Bourbaki, Topologie Générale, Tome II: Chapitres 5 à 10, Hermann, Paris, 1974. · Zbl 0337.54001
[3] Joseph Diestel, Sequences and series in Banach spaces, Graduate Texts in Mathematics, vol. 92, Springer-Verlag, New York, 1984. · Zbl 0542.46007
[4] Leonard E. Dor, On sequences spanning a complex \?\(_{1}\) space, Proc. Amer. Math. Soc. 47 (1975), 515 – 516. · Zbl 0296.46014
[5] Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, With the assistance of W. G. Bade and R. G. Bartle. Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers, Ltd., London, 1958. · Zbl 0084.10402
[6] Fernando Galaz-Fontes, Note on compact sets of compact operators on a reflexive and separable Banach space, Proc. Amer. Math. Soc. 126 (1998), no. 2, 587 – 588. · Zbl 0893.46015
[7] Hans Jarchow, Locally convex spaces, B. G. Teubner, Stuttgart, 1981. Mathematische Leitfäden. [Mathematical Textbooks]. · Zbl 0466.46001
[8] Gottfried Köthe, Topological vector spaces. I, Translated from the German by D. J. H. Garling. Die Grundlehren der mathematischen Wissenschaften, Band 159, Springer-Verlag New York Inc., New York, 1969. · Zbl 0179.17001
[9] Gottfried Köthe, Topological vector spaces. II, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], vol. 237, Springer-Verlag, New York-Berlin, 1979. · Zbl 0417.46001
[10] Theodore W. Palmer, Totally bounded sets of precompact linear operators, Proc. Amer. Math. Soc. 20 (1969), 101 – 106. · Zbl 0165.47603
[11] Haskell P. Rosenthal, A characterization of Banach spaces containing \?\textonesuperior , Proc. Nat. Acad. Sci. U.S.A. 71 (1974), 2411 – 2413. · Zbl 0297.46013
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.