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2007 | 182 | 2 | 141-164
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Subspaces with a common complement in a Banach space

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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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  • Department of Financial and Management Engineering, University of the Aegean, 31, Fostini St., 82100 Chios, Greece
  • Department of Mathematics, National Technical University of Athens, Iroon Polytexneiou 9, 15780 Zografou, Greece
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bwmeta1.element.bwnjournal-article-doi-10_4064-sm182-2-4
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