Abstract
Suppose that 1<p≦2, 2≦q<∞. The formal identity operatorI:l p→l qfactorizes through any given non-compact operator from ap-smooth Banach space into aq-convex Banach space. It follows that ifX is a 2-convex space andY is an infinite dimensional subspace ofX which is isomorphic to a Hilbert space, thenY contains an isomorphic copy ofl 2 which is complemented inX.
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References
G. Bennett, L. E. Dor, V. Goodman, W. B. Johnson and C. M. Newman,On uncomplemented subspaces of L p, 1 <p < 2, Isr. J. Math.26 (1977), 178–187.
J. Diestel,Sequences and series in Banach spaces, Springer-Verlag, Berlin, 1983.
P. Enflo,Banach spaces which can be given an equivalent uniformly convex norm, Isr. J. Math.13 (1972), 281–288.
R. C. James,Some self-dual properties of normed linear spaces, inSymposium on Infinite Dimensional Topology, Annals of Math. Studies, Vol. 69, 1972, pp. 159–175.
W. B. Johnson,A universal non-compact operator, Colloq. Math.23 (1971), 267–268.
N. J. Kalton,The space Z 2 viewed as a symplectic Banach space, Proceedings of Research Workshop on Banach Space Theory, University of Iowa, 1981.
A. Lebow and M. Schechter,Semigroups of operators and measure of non-compactness, J. Funct. Anal.7 (1971), 1–26.
J. Lindenstrauss,On the modulus of smoothness and divergent series in Banach spaces, Michigan Math. J.10 (1963), 241–252.
J. Lindenstrauss,On subspaces of Banach spaces without quasi-complements, Isr. J. Math.6 (1968), 36–38.
J. Lindenstrauss and L. Tzafriri,Classical Banach Spaces I, Springer-Verlag, Berlin, 1977.
G. Mackey,Note on a theorem of Murray, Bull. Amer. Math. Soc.52 (1946), 322–325.
B. Maurey,Un théorème de prolongement, C. R. Acad. Sci, Série A279 (1974), 329–332.
A. Pełczyński,A note on the paper of I. Singer “Basic sequences and reflexivity of Banach spaces”, Studia Math.21 (1962), 371–374.
A. Pełczyński and H. P. Rosenthal,Localization techniques in L p spaces, Studia Math.52 (1975), 263–289.
A. Pietsch,s-numbers of operators in Banach spaces, Studia Math.51 (1974), 201–223.
G. Pisier,Martingales with values in uniformly convex spaces, Isr. J. Math.20 (1975), 326–350.
G. Pisier,Un théorème sur les opérateurs linéaires entre éspaces de Banach qui se factorisent par un éspace de Hilbert, Ann. Sci. Ecole Norm. Sup.13 (1980), 23–43.
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Dilworth, S.J. Universal non-compact operators between super-reflexive Banach spaces and the existence of a complemented copy of Hilbert space. Israel J. Math. 52, 15–27 (1985). https://doi.org/10.1007/BF02776075
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DOI: https://doi.org/10.1007/BF02776075