Connected components and isometries in the metric space of all closed linear subspaces of a Hilbert space
by
J.-Ph. Labrousse
University of Nice(France)

ABSTRACT

Let H be a Hilbert space over C and let F (H) be the set of all closed linear subspaces of H.

∀M, N ∈ F(H) set g(M,N ) = ∥P_{M} − P_{N} ∥ ( known as the gap metric ) where

P_{M} , P_{N} denote respectively the orthogonal projections in H on M and on N .

∀M, N ∈ F (H) such that ker(P_{M}+P_{N} − I ) = {0}, Ψ(M, N ), the bisector of M and

N , is the uniquely determined element of F(H) such that (setting Ψ(M, N ) = W ):

(i) P_{M} P_{W} = P_{W} P_{N}

(ii) (P_{M} + P_{N} ) P_{W} = P_{W}(P_{M}+P_{N} ) is positive definite.

A mapping Φ of F(H) into itself is called an isometry if

∀M, N ∈ F (H), g(M, N) = g(Φ(M), Φ(N)).

In this presentation we use the notion of bisector (introduced in [1]) to determine

the arcwise connected components of F(H) and the properties of isometries on that

space. This leads to a number of applications to linear operators and relations.

References

[1] J-Ph Labrousse, Geodesics in the space of linear relations on a Hilbert space,

Proc. of the 18th OT Conference, The Theta Foundation, Bucharest, (2000)

213-234

Date received: May 31, 2008

Copyright © 2008 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cawh-70.