PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 138, Number 1, January 2010, Pages 341–353 S 0002-9939(09)10080-1 Article electronically published on August 28, 2009
GEOMETRY OF I-STIEFEL MANIFOLDS
EDUARDO CHIUMIENTO
(Communicated by Marius Junge)
Abstract. Let I be a separable Banach ideal in the space of bounded oper- ators acting in a Hilbert space H and U(H)I the Banach-Lie group of unitary operators which are perturbations of the identity by elements in I.Inthis paper we study the geometry of the unitary orbits
{ UV : U ∈U(H)I} and ∗ { UV W : U, W ∈U(H)I}, where V is a partial isometry. We give a spatial characterization of these orbits. It turns out that both are included in V + I, and while the first one consists of partial isometries with the same kernel of V , the second is given by partial isometries such that their initial projections and V ∗V have null index as a pair of projections. We prove that they are smooth submanifolds of the affine Banach space V + I and homogeneous reductive spaces of U(H)I and U(H)I ×U(H)I respectively. Then we endow these orbits with two equivalent Finsler metrics, one provided by the ambient norm of the ideal and the other given by the Banach quotient norm of the Lie algebra of U(H)I (or U(H)I × U(H)I) by the Lie algebra of the isotropy group of the natural actions. We show that they are complete metric spaces with the geodesic distance of these metrics.
1. Introduction Let H be an infinite dimensional separable Hilbert space and B(H)thespaceof bounded linear operators acting in H. It will cause no confusion to denote by . the spectral norm and the norm of H. By a Banach ideal we mean a two-sided ∗ ideal I of B(H) equipped with a norm . I satisfying T ≤ T I = T I and AT B I ≤ A T I B whenever A, B ∈B(H). In the sequel, I stands for a separable Banach ideal. Denote by U(H) the group of unitary operators in H and by U(H)I the group of unitaries which are perturbations of the identity by an operator in I, i.e.
U(H)I = { U ∈U(H):U − I ∈ I }.
It is a real Banach-Lie group with the topology defined by the metric (U1,U2) → U1 − U2 I (see [5]). The Lie algebra of U(H)I is given by ∗ Iah = { A ∈ I : A = −A }.
Received by the editors September 18, 2008, and, in revised form, April 22, 2009. 2000 Mathematics Subject Classification. Primary 22E65; Secondary 47B10, 58B20. Key words and phrases. Partial isometry, Banach ideal, Finsler metric.