Subspaces with a common complement in a Banach space

Dimosthenis Drivaliaris ¹   ; Nikos Yannakakis ²

¹ Department of Financial and Management Engineering University of the Aegean 31, Fostini St. 82100 Chios, Greece
² Department of Mathematics National Technical University of Athens Iroon Polytexneiou 9 15780 Zografou, Greece

Studia Mathematica, Tome 182 (2007) no. 2, pp. 141-164

Résumé

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.

Détail

DOI : 10.4064/sm182-2-4

Keywords: study problem existence common algebraic complement pair closed subspaces banach space prove following characterizations pairs subspaces banach space common complement coincide those pairs which isomorphic pair graphs bounded linear operators between other banach spaces pairs subspaces banach space common complement coincide those pairs which there exists involution exchanging subspaces bounded below their union moreover separable hilbert space only pairs subspaces common complement those which either equivalently positioned completely asymptotic another obtain characterizations existence common complement subspaces closed sum