Quotient Hereditarily Indecomposable Banach Spaces

http://dx.doi.org/10.4153/CJM-1999-026-4
Canad. J. Math. 51(1999), 566-584

  Published:1999-06-01
 Printed: Jun 1999

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.