Quotient hereditarily indecomposable Banach spaces

Abstract

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 XGM constructed by W. T. Gowers and B. Maurey in [GM]. Then we provide an example of a reflexive hereditarily indecomposable space X̂ whose dual is not hereditarily indecomposable; so X̂ is not quotient hereditarily indecomposable. We also show that every operator on X̂* is a strictly singular perturbation of an homothetic map.