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Subspaces with a common complement in a Banach space

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Drivaliaris D., Yannakakis N.

Studia Mathematica

2007

tom 182

nr 2

141-164

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.

Ograniczanie wyników

1 Studia Mathematica

1 Drivaliaris D.

1 Yannakakis N.

1 2007

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Subspaces with a common complement in a Banach space