A Banach space X with a Schauder basis is defined to have the restricted quotient hereditarily indecomposable property if X/Y is hereditarily indecomposable for any infinite-codimensional subspace Y with a successive finite-dimensional decomposition on the basis of X. The following dichotomy theorem is proved: any infinite-dimensional Banach space contains a quotient of a subspace which either has an unconditional basis, or has the restricted quotient hereditarily indecomposable property.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm180-2-2,
author = {Valentin Ferenczi},
title = {A Banach space dichotomy theorem for quotients of subspaces},
journal = {Studia Mathematica},
volume = {178},
year = {2007},
pages = {111-131},
zbl = {1131.46008},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm180-2-2}
}
Valentin Ferenczi. A Banach space dichotomy theorem for quotients of subspaces. Studia Mathematica, Tome 178 (2007) pp. 111-131. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm180-2-2/