Subspaces with a common complement in a Banach space

Details Subspaces with a common complement in a Banach space Subspaces with a common complement in a Banach space

2008-05-30

arxiv.org/abs/0805.4707v1

Studia Mathematica 182 (2) (2007), 141-164

Mathematics

Functional Analysis

Scientific paper

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.

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