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# Quotient Hereditarily Indecomposable Banach Spaces

Published:1999-06-01
Printed: Jun 1999
• V. Ferenczi
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## Abstract

A Banach space $X$ is said to be {\it quotient hereditarily indecomposable\/} if no infinite dimensional quotient of a subspace of $X$ is decomposable. We provide an example of a quotient hereditarily indecomposable space, namely the space $X_{\GM}$ constructed by W.~T.~Gowers and B.~Maurey in \cite{GM}. Then we provide an example of a reflexive hereditarily indecomposable space $\hat{X}$ whose dual is not hereditarily indecomposable; so $\hat{X}$ is not quotient hereditarily indecomposable. We also show that every operator on $\hat{X}^*$ is a strictly singular perturbation of an homothetic map.
 MSC Classifications: 46B20 - Geometry and structure of normed linear spaces 47B99 - None of the above, but in this section