Interpolating hereditarily indecomposable Banach spaces
Authors:
S. A. Argyros and V. Felouzis
Journal:
J. Amer. Math. Soc. 13 (2000), 243-294
MSC (2000):
Primary 46B20, 46B70; Secondary 46B03, 52A07, 03E05
DOI:
https://doi.org/10.1090/S0894-0347-00-00325-8
Published electronically:
January 31, 2000
MathSciNet review:
1750954
Full-text PDF
Abstract | References | Similar Articles | Additional Information
The following dichotomy is proved.
Every Banach space either contains a subspace isomorphic to , or it has an infinite-dimensional closed subspace which is a quotient of a Hereditarily Indecomposable (H.I.) separable Banach space.
In the particular case of , it is shown that the space itself is a quotient of a H.I. space. The factorization of certain classes of operators, acting between Banach spaces, through H.I. spaces is also investigated. Among others it is shown that the identity operator
admits a factorization through a H.I. space. The same result holds for every strictly singular operator
.
Interpolation methods and the geometric concept of thin convex sets together with the techniques concerning the construction of Hereditarily Indecomposable spaces are used to obtain the above mentioned results.
- [AA] D. Alspach and S. A. Argyros. Complexity of weakly null sequences, Dissertationes Mathematicae CCCXXI (1992). MR 93j:46014
- [AD1] S. A. Argyros and I. Deliyanni. Banach spaces of the type of Tsirelson, preprint 1992.
- [AD2]
S. A. Argyros and I. Deliyanni. Examples of asymptotic
Banach spaces, Trans. AMS 349 (1997), 973-995. MR 97f:46021
- [ADKM] S. A. Argyros, I. Deliyanni, D. Kutzarova and A. Manoussakis, Modified mixed Tsirelson spaces, J. of Func. Anal. 159 (1998), 43-109. CMP 99:04
- [AG] S. A. Argyros and I. Gasparis. Unconditional structures of weakly null sequences (Preprint).
- [AMT] S. A. Argyros, S. Merkourakis and A. Tsarpalias. Convex unconditionality and summability of weakly null sequences, Isr. J. Math. 107 (1998), 157-193. MR 99m:46021
- [AO] G. Androulakis and E. Odell. Distorting mixed Tsirelson spaces (Preprint).
- [B]
S. F. Bellenot. Tsirelson superspaces and
, Journ. of Funct. Analysis. 69 (1986), No2, 207-228. MR 88f:46033
- [Bo1] J. Bourgain. Convergent sequences of continuous functions, Bull. Soc. Math. Belg. Ser. B 32 (1980), 235-249. MR 84e:46018
- [Bo2] J. Bourgain. La propriété de Radon-Nicodym, Math. Univ. Pierre et Marie Curie 36 (1979).
- [D] J. Diestel. Sequences and series in Banach spaces, Graduate texts in Math. 92, Springer-Verlag, 1984. MR 85i:46020
- [DFJP] W. J. Davis, T. Figiel, W. B. Johnson and A. Pelczynski. Factoring Weakly Compact Operators, Journ. of Funct. Analysis 17 (1974), 311-327. MR 50:8010
- [F1] V. Ferenczi. Quotient Hereditarily Idecomposable spaces (Preprint).
- [F2] V. Ferenczi. A uniformly convex hereditarily indecomposable Banach space, Israel J. Math. 102 (1997), 199-225. MR 98m:46013
- [J] R. C. James. Bases and reflexivity of Banach spaces, Ann. of Math. 52 (1950), 518-527. MR 12:616b
- [G] W. T. Gowers. A new dichotomy for Banach spaces, GAFA 6 (1996), 1083-1093. MR 97m:46017
- [G1]
W. T. Gowers. A Banach space not containing
,
or a reflexive subspace, Trans. Amer. Math. Soc. 344 (1994), 407-420. MR 94j:46024
- [G2] W. T. Gowers. A remark about the Scalar-Plus-Compact Problem, Convex Geometric Analysis 34 (1998), 111-115. MR 99m:46015
- [GM] W. T. Gowers and B. Maurey. The unconditional basic sequence problem, Journal of AMS 6 (1993), 851-874. MR 94k:46021
- [Gr] A. Grothendieck. Critères de compacité dans les espaces fonctionelles généraux, Amer. J. Math. 74 (1952), 168-186. MR 13:857e
- [H]
P. Habala. Banach spaces all of whose subspaces fail the Gordon-Lewis property, Math. Ann. 310 (1998), N
2, 197-212. CMP 98:07
- [Ke] A. Kechris. Classical Descriptive Set Theory, Springer-Verlag, Berlin (1994). MR 96e:03057
- [KM] K. Kuratowski and A. Mostowski. Set Theory, Amsterdam (1968). MR 37:5100
- [KW] N. Kalton and A. Wilanski. Tauberian operators on Banach spaces, Proc. A.M.S. 57 (1976), 251-255. MR 57:13555
- [L] H. E. Lacey. The Isometric Theory of Classical Banach Spaces, Springer-Verlag, Berlin, 1974. MR 58:12308
- [Le]
D. H. Leung. On
-saturated spaces, Illinois J. Math. 39 (1995), 15-29. MR 96h:46024
- [LT] J. Lindenstrauss and L. Tzafriri. Classical Banach spaces I, Springer Verlag 92, 1977. MR 58:17766
- [MR] B. Maurey and H. Rosenthal. Normalized weakly null sequences with no unconditional subsequence , Studia Math. 61 (1977), 77-98. MR 55:11010
- [MS] V. Milman and G. Schechtman. Asymptotic Theory of Finite Dimensional Normed Spaces, Lecture Notes in Math. 1200, Springer-Verlag. MR 87m:46038
- [N]
R. Neidinger. Factoring Operators through hereditarily-
spaces, Lecture Notes in Math. 1166 (1985).
- [N1] R. Neidinger. Properties of Tauberian Operators, Dissertation. University of Texas at Austin, 1984.
- [NR] R. Neidinger and H. P. Rosenthal. Norm-attainment of linear functional on subspaces and characterizations of Tauberian operators, Pac. J. Math. 118 (1985), 215-228. MR 86f:46013
- [OS] E. Odell and T. Schlumprecht. The distortion problem, Acta Math. 173 (1994), 259-281. MR 96a:46031
- [OS1] E. Odell and T. Schlumprecht. A Banach space block finitely universal for monotone bases, Trans. Amer. Math. Soc. (to appear). CMP 98:16
- [S] T. Schlumprecht. An arbitrarily distortable Banach space, Israel J. Math 76 (1991), 81-95. MR 93h:46023
- [To]
N. Tomczak-Jaegermann. Banach spaces of type
have arbitrarily distortable subspaces, GAFA 6 (1996), 1075-1082. MR 98g:46020
- [T] A. Tsarpalias. A note on Ramsey property, Proc. A.M.S. 127 (1999), 583-587. MR 99c:04005
- [Ts]
B. S. Tsirelson. Not every Banach space contains
or
, Funct. Anal. Appl. 8 (1974), 138-141.
Retrieve articles in Journal of the American Mathematical Society with MSC (2000): 46B20, 46B70, 46B03, 52A07, 03E05
Retrieve articles in all journals with MSC (2000): 46B20, 46B70, 46B03, 52A07, 03E05
Additional Information
S. A. Argyros
Affiliation:
Department of Mathematics, University of Athens, Athens, Greece
Email:
sargyros@atlas.uoa.gr
V. Felouzis
Affiliation:
Department of Mathematics, University of Athens, Athens, Greece
DOI:
https://doi.org/10.1090/S0894-0347-00-00325-8
Keywords:
Interpolation methods,
hereditarily indecomposable spaces,
thin convex sets,
Schreier families,
summability methods
Received by editor(s):
April 14, 1998
Received by editor(s) in revised form:
June 8, 1999
Published electronically:
January 31, 2000
Article copyright:
© Copyright 2000
American Mathematical Society