Interpolating hereditarily indecomposable Banach spaces

Argyros, S. A.; Felouzis, V.

doi:10.1090/S0894-0347-00-00325-8

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
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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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