Metric geometry in infinite dimensional Stiefel manifolds

Abstract

Let J 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 in I. In this paper we study metric properties of the J-Stiefel manifold associated to v, namely SG = { v_0 in I , : , v- v_0 in J, , j(v_0*v_0,v*v)=0 }, where j( , ) is the Fredholm index of a pair of projections. Let UJ(H) be the Banach-Lie group of unitary operators which are perturbations of the identity by elements in J. Then SG coincides with the orbit of v under the action of UJ(H) on I given by (u,w) v_0=uv_0w*, u,w in UJ(H) and v_0 in SG. We endow SG with a quotient Finsler metric by means of the Banach quotient norm of the Lie algebra of UJ(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 coincide with the quotient distance of UJ(H) by the isotropy group. Hence this metric defines the quotient topology in SG. The other results concern with minimal curves in J-Stiefel manifolds when the ideal J 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 join with a curve of minimal length.