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 Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: The following dichotomy is proved. Every Banach space either contains a subspace isomorphic to $\ell ^1$, 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 $L^p(\lambda ),\ 1<p<\infty$, 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 $I: L^{\infty }(\lambda )\to L^1(\lambda )$ admits a factorization through a H.I. space. The same result holds for every strictly singular operator $T: \ell ^p\to \ell ^q,\ 1<p,q<\infty$. 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.
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Additional Information
S. A. Argyros
Affiliation:
Department of Mathematics, University of Athens, Athens, Greece
MR Author ID:
26995
Email:
sargyros@atlas.uoa.gr
V. Felouzis
Affiliation:
Department of Mathematics, University of Athens, Athens, Greece
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
