Quotient Hereditarily Indecomposable Banach Spaces

A Banach space $X$ is said to be 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 $[\text{GM...

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Bibliographic Details
Published in:Canadian journal of mathematics 1999-06, Vol.51 (3), p.566-584
Main Author: Ferenczi, V.
Format: Article
Language:English
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Summary:A Banach space $X$ is said to be 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 $[\text{GM}]$ . 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.
ISSN:0008-414X
1496-4279
DOI:10.4153/CJM-1999-026-4