Metric geometry in infinite dimensional Stiefel manifolds

Abstract

Let I be a separable Banach ideal in the space of bounded operators acting in a Hilbert space H and I the set of partial isometries in H . Fix v ∈ I . In this paper we study metric properties of the I -Stiefel manifold associated to v, namely S t I ( v ) = { v 0 ∈ I : v − v 0 ∈ I , j ( v 0 ∗ v 0 , v ∗ v ) = 0 } , where j ( , ) is the Fredholm index of a pair of projections. Let U I ( H ) be the Banach–Lie group of unitary operators which are perturbations of the identity by elements in I . Then S t I ( v ) coincides with the orbit of v under the action of U I ( H ) × U I ( H ) on I given by ( u , w ) ⋅ v 0 = u v 0 w ∗ , u , w ∈ U I ( H ) and v 0 ∈ S t I ( v ) . We endow S t I ( v ) with a quotient Finsler metric by means of the Banach quotient norm of the Lie algebra of U I ( H ) × U I ( H ) by the Lie algebra of the isotropy group. We give a characterization of the rectifiable distance induced by this metric. In fact, we show that the rectifiable distance coincides with the quotient distance of U I ( H ) × U I ( H ) by the isotropy group. Hence this metric defines the quotient topology in S t I ( v ) . The other results concern with minimal curves in I -Stiefel manifolds when the ideal I is fixed as the compact operators in H . The initial value problem is solved when the partial isometry v has finite rank. In addition, we use a length-reducing map into the Grassmannian to find some special partial isometries that can be joined with a curve of minimal length.