Interpolating hereditarily indecomposable Banach spaces

Argyros, S. A.; Felouzis, V.

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

Interpolating hereditarily indecomposable Banach spaces
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by S. A. Argyros and V. Felouzis

J. Amer. Math. Soc. 13 (2000), 243-294

DOI: https://doi.org/10.1090/S0894-0347-00-00325-8

Published electronically: January 31, 2000

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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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