Pseudocomplémentation dans les espaces de Banach

Patric Rauch

ArticleOriginal scientific text

Title

Pseudocomplémentation dans les espaces de Banach

Authors ¹

Affiliations

Équipe d'Analyse, U.A. N° 754 AU C.N.R.S., Université Paris VI, Tour 46, 4ème Étage, 4, Place Jussieu, 75 252 Paris, Cedex 05, France

Abstract

This paper introduces the following definition: a closed subspace Z of a Banach space E is pseudocomplemented in E if for every linear continuous operator u from Z to Z there is a linear continuous extension ū of u from E to E. For instance, every subspace complemented in E is pseudocomplemented in E. First, the pseudocomplemented hilbertian subspaces of

L ¹

are characterized and, in

L^{p}

with p in [1, + ∞[, classes of closed subspaces in which the notions of complementation and pseudocomplementation are equivalent are pointed out. Then, for Banach spaces with the uniform approximation property, Dvoretzky's theorem is strengthened by proving that they contain uniformly pseudocomplemented

ℓ^{2}_n

's. Finally, the study of Banach spaces in which every closed subspace is pseudocomplemented is started.

[B-D-G-J-N] G. Bennnett, L. E. Dor, V. Goodman, W. B. Johnson, and C. M. Newman, On uncomplemented subspaces of $L_{p}$ , 1 < p < 2, Israel. J. Math. 26 (1977), 178-187.
[BOU 1] J. Bourgain, New Classes of $ℒ^{p}$ -Spaces, Lecture Notes in Math. 889, Springer, 1981.
[BOU 2] J. Bourgain, Bounded orthogonal systems and the Λ(p)-set problem, Acta Math. 162 (1989), 227-245.
[D-D-S] W. J. Davis, D. W. Dean and I. Singer, Complemented subspaces and Λ-systems in Banach spaces, Israel J. Math. 6 (1968), 303-309.
[DAY] M. M. Day, On the basis problem in normed spaces, Proc. Amer. Math. Soc. 13 (1962), 655-658.
[DOR] L. E. Dor, On projections in $L_{1}$ , Ann. of Math. 102 (1975), 463-474.
[D-S] L. E. Dor and T. Starbird, Projections of $L_{p}$ onto subspaces spanned by independent random variables, Compositio Math. 39 (2) (1979), 141-175.
[DUR] P. L. Duren, Theory of $H^{p}$ Spaces, Academic Press, New York 1970.
[DVO] A. Dvoretzky, Some results on convex bodies and Banach spaces, in : Proc. Internat. Sympos. Linear Spaces, Jerusalem 1960, Israel Acad. Sci. Humanities, Jerusalem 1961, 123-160.
[FIG] T. Figiel, The exponential estimate for local structure of gaussian subspaces of $L^{p}$ , Bull. Polish Acad. Sci. Math. 36 (3-4) (1988), 133-141.
[F-J-S] T. Figiel, W. B. Johnson and G. Schechtman, Factorization of natural embeddings of $l_{p}^{n}$ into $L_{r}$ , I, Studia Math. 89 (1988), 79-103.
[F-L-M] T. Figiel, J. Lindenstrauss and V. Milman, The dimensions of almost spherical sections of convex bodies, Acta Math. 139 (1977), 53-94.
[HOF] K. Hoffman, Banach Spaces of Analytic Functions, Prentice-Hall, Englewood Cliffs, N.J., 1962.
[LEVY] M. Levy, Prolongement d'un opérateur d'un sous-espace de L¹(μ) dans L¹(ν), Séminaire Maurey-Schwartz, 1979-80, exposé 5, Ecole Polytechnique, Paris.
[L-R] J. Lindenstrauss and H. P. Rosenthal, The $ℒ^{p}$ -spaces, Israel J. Math. 7 (1969), 325-349.
[Ló-R] J. López and K. Ross, Sidon Sets, Dekker, New York 1975.
[L-T 1 et 2] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces, I : Sequence Spaces, II : Function Spaces, Springer, Berlin 1977, 1979.
[L-T 3] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces, Lecture Notes in Math. 338, Springer, 1973.
[L-T 4] J. Lindenstrauss and L. Tzafriri, On the complemented subspace problem, Israel J. Math. 9 (1971), 263-269.
[Mar-P] M. B. Marcus and G. Pisier, Random Fourier Series with Applications to Harmonic Analysis, Ann. of Math. Stud. 101, Princeton Univ. Press, 1981.
[Mau-P] B. Maurey et G. Pisier, Séries de variables aléatoires vectorielles indépendantes et propriétés géométriques des espaces de Banach, Studia Math. 58 (1976), 45-90.
[PEŁ] A. Pełczyński, Projections in certain Banach spaces, ibid. 19 (1960), 209-228.
[PIE] A. Pietsch, Operator Ideals, North-Holland, 1978.
[PIS 1] G. Pisier, Factorization of Linear Operators and Geometry of Banach Spaces, CBMS Regional Conf. Ser. in Math. 60, Amer. Math. Soc., 1986.
[PIS 2] G. Pisier, Probabilistic methods in the geometry of Banach spaces, in: Probability and Analysis, C.I.M.E. Varenna 1985, Lecture Notes in Math. 1206, Springer, 1986, 167-241.
[PIS 3] G. Pisier, Holomorphic semi-groups and the geometry of Banach spaces, Ann. of Math. 115 (1982), 375-392.
[PIS 4] G. Pisier, Bases, suites lacunaires dans les espaces $L^{r}$ d'après Kadec-Pełczyński, Séminaire Maurey-Schwartz, 1972-73, exposé 18, Ecole Polytechnique, Paris.
[PIS 5] G. Pisier, The Volume of Convex Bodies and Banach Space Geometry, Cambridge Univ. Press, 1989.
[PIS 6] G. Pisier, Les inégalités de Khintchine-Kahane d'après C. Borell, Séminaire sur la géométrie des espaces de Banach, 1977-78, n° 7, Ecole Polytechnique, Palaiseau.
[ROS 1] H. P. Rosenthal, Projections onto translation-invariant subspaces of $L^{p} (G)$ , Mem. Amer. Math. Soc. 63 (1966).
[ROS 2] H. P. Rosenthal, On the subspaces of $L^{p} (p > 2)$ spanned by sequences of independent random variables, Israel J. Math. 8 (1970), 273-303.
[SIM] B. Simon, The $P {(ϕ)}_{2}$ Euclidean (Quantum) Field Theory, Princeton Univ. Press, Princeton 1974.
[T-J] N. Tomczak-Jaegermann, Banach-Mazur Distances and Finite-Dimensional Operator Ideals, Longman Scientific & Technical, 1989.

Title

Pseudocomplémentation dans les espaces de Banach

Affiliations

Abstract

Bibliography