Subspaces with a common complement in a Banach space

Dimosthenis Drivaliaris, and Nikos Yannakakis. "Subspaces with a common complement in a Banach space." Studia Mathematica 182.2 (2007): 141-164. <http://eudml.org/doc/285064>.

@article{DimosthenisDrivaliaris2007,
abstract = {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.},
author = {Dimosthenis Drivaliaris, Nikos Yannakakis},
journal = {Studia Mathematica},
keywords = {common complement; algebraic complement; pair of subspaces; relative position; equivalently positioned subspaces; completely asymptotic subspaces; geometry of Banach spaces},
language = {eng},
number = {2},
pages = {141-164},
title = {Subspaces with a common complement in a Banach space},
url = {http://eudml.org/doc/285064},
volume = {182},
year = {2007},
}

TY - JOUR
AU - Dimosthenis Drivaliaris
AU - Nikos Yannakakis
TI - Subspaces with a common complement in a Banach space
JO - Studia Mathematica
PY - 2007
VL - 182
IS - 2
SP - 141
EP - 164
AB - 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.
LA - eng
KW - common complement; algebraic complement; pair of subspaces; relative position; equivalently positioned subspaces; completely asymptotic subspaces; geometry of Banach spaces
UR - http://eudml.org/doc/285064
ER -