We study the problem of the existence of a common algebraic complement for a pair of closed subspaces of a Banach space. We prove the following two characterizations: (1) The pairs of subspaces of a Banach space with a common complement coincide with those pairs which are isomorphic to a pair of graphs of bounded linear operators between two other Banach spaces. (2) The pairs of subspaces of a Banach space X with a common complement coincide with those pairs for which there exists an involution S on X exchanging the two subspaces, such that I + S is bounded from below on their union. Moreover, we show that, in a separable Hilbert space, the only pairs of subspaces with a common complement are those which are either equivalently positioned or not completely asymptotic to one another. We also obtain characterizations for the existence of a common complement for subspaces with closed sum.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm182-2-4,
author = {Dimosthenis Drivaliaris and Nikos Yannakakis},
title = {Subspaces with a common complement in a Banach space},
journal = {Studia Mathematica},
volume = {178},
year = {2007},
pages = {141-164},
zbl = {1134.46008},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm182-2-4}
}
Dimosthenis Drivaliaris; Nikos Yannakakis. Subspaces with a common complement in a Banach space. Studia Mathematica, Tome 178 (2007) pp. 141-164. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm182-2-4/