Subspaces with a common complement in a Banach space

TL;DR

This paper characterizes pairs of closed subspaces in Banach spaces that share a common algebraic complement, linking them to graph representations and involutions, with special results for Hilbert spaces.

Contribution

It provides new characterizations of subspace pairs with common complements using isomorphisms, involutions, and geometric conditions, extending previous understanding.

Findings

Pairs with common complement are isomorphic to pairs of graphs of bounded operators.

Existence of an involution exchanging subspaces characterizes common complements.

In separable Hilbert spaces, only certain geometrically positioned pairs have common complements.

Abstract

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.