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# Wydawnictwa / Czasopisma IMPAN / Studia Mathematica / Wszystkie zeszyty

## Subspaces with a common complement in a Banach space

### Tom 182 / 2007

Studia Mathematica 182 (2007), 141-164 MSC: Primary 46B20; Secondary 46C05, 47A05. DOI: 10.4064/sm182-2-4

#### Streszczenie

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.

#### Autorzy

• Dimosthenis DrivaliarisDepartment of Financial
and Management Engineering
University of the Aegean
31, Fostini St.
82100 Chios, Greece
e-mail
• Nikos YannakakisDepartment of Mathematics
National Technical University of Athens
Iroon Polytexneiou 9
15780 Zografou, Greece
e-mail

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