Compact Groups of Operators on Subproportional Quotients of l 1 m

Mankiewicz, Piotr

Canadian journal of mathematics, Tome 52 (2000) no. 5, pp. 999-1017

Résumé

It is proved that a “typical” $n$ -dimensional quotient ${{X}_{n}}$ of $l_{1}^{m}$ with $n={{m}^{\sigma }},0<\sigma <1$ , has the property $$\text{Average}\int_{G}{||Tx|{{|}_{{{X}_{n}}}}d{{h}_{G}}(T)\ge \frac{c}{\sqrt{n{{\log }^{3}}n}}\left( n-\int_{G}{|trT|d{{h}_{G}}(T)} \right)},$$ for every compact group $G$ of operators acting on ${{X}_{n}}$ , where ${{d}_{G}}(T)$ stands for the normalized Haar measure on $G$ and the average is taken over all extreme points of the unit ball of ${{X}_{n}}$ . Several consequences of this estimate are presented.

Détail

DOI : 10.4153/CJM-2000-042-8

Mots-clés : 46B20, 46B09

Bibliographie

[B] [B] Ball, K., Normed spaces with a weak-Gordon-Lewis property. In: Functional Analysis, Springer Lecture Notes in Math. 1470(1991), 36–47. Google Scholar

[BKPS] [BKPS] Billard, P., Kwapień, S., A. Pełczyński and Ch. Samuel, Biorthogonal systems of random unconditional convergence in Banach spaces. Longhorn Notes, The University of Texas at Austin, 1986. 13–35. Google Scholar

[BP] [BP] Ball, K. and Pajor, A., Convex bodies with few faces. Proc. Amer. Math. Soc. 110(1990), 225–231. Google Scholar

[B1] [B1] Bourgain, J., Real isomorphic complex spaces need not to be complex isomorphic. Proc. Amer. Math. Soc. 96(1986), 221–226. Google Scholar

[B2] [B2] Bourgain, J., On finite dimensional homogenous Banach spaces. In: Geometric Aspects of Functional Analysis, Israel Seminar (eds. Lindenstrauss, J. and Milman, V. D.), Springer Lecture Notes in Math. 1317(1988), 232–239. Google Scholar

[G1] [G1] Gluskin, E. D., The diameter of Minkowski compactum roughly equals to n. (Russian) Funktsional. Anal. i Priloˇzen. 15(1981), 72–73.(English transl.: Funct. Anal. Appl. 15(1981), 57–58). Google Scholar

[G2] [G2] Gluskin, E. D., Finite-dimensional analogues of spaces without a basis. (Russian) Dokl. Acad. Nauk SSSR 261(1981), 1046–1050. Google Scholar

[GG] [GG] Garling, D. J. H. and Gordon, Y., Relations between some constants associated with finite-dimensional Banach spaces. Israel J. Math. 9(1971), 346–361. Google Scholar

[M1] [M1] Mankiewicz, P., Finite dimensional Banach spaces with symmetry constant of order Rn. Studia Math. 79(1984), 193–200. Google Scholar

[M2] [M2] Mankiewicz, P., Subspace mixing properties of operators in Rn with applications to Gluskin spaces. Studia Math. 88(1988), 51–67. Google Scholar

[M3] [M3] Mankiewicz, P., Compact groups of operators on proportional quotients of ln1. Israel J. Math. 109(1999), 75–91. Google Scholar

[MT1] [MT1] Mankiewicz, P., Mankiewicz, P. and Tomczak-Jaegermann, N., A solution of the finite-dimensional homogeneous Banach spaces problem. Israel J. Math. 75(1991), 129–159. Google Scholar

[MT2] [MT2] Mankiewicz, P., Schauder bases in quotients of subspaces of l2(X). Amer. J. Math. 116(1994), 1341–1363. Google Scholar

[MT3] [MT3] Mankiewicz, P., Structural properties of weak cotype 2 spaces. Canad. J. Math. 48(1996), 607–624. Google Scholar

[MT4] [MT4] Mankiewicz, P., Quotients of finite-dimensional Banach spaces; random phenomena. Handbook of geometry of Banach spaces, Elsevier Publ., to appear. Google Scholar

[P] [P] Pisier, G., Volumes of Convex Bodies and Banach Spaces Geometry. Cambridge Univ. Press, 1989. Google Scholar

[S1] [S1] Szarek, S. J., The finite dimensional basis problem with an appendix on nets of Grassmann manifolds. Acta Math. 151(1983), 153–179. Google Scholar

[S2] [S2] Szarek, S. J., On the existence and uniqueness of complex structure and spaces with “few” operators. Trans. Amer.Math. Soc. 293(1986), 339–353. Google Scholar

[T] [T] Tomczak-Jaegermann, N., Banach-Mazur Distances and Finite Dimensional Operator Ideals. Pitman Monographs Surveys Pure Appl. Math. 38, Longman Scientific &amp; Technical, Harlow, and John Wiley, New York, 1989. Google Scholar