Metric geodesics of isometries in a Hilbert space and the extension problem

Abstract

We consider the problem of finding short smooth curves of isometries in a Hilbert space H. The length of a smooth curve γ(t), t ∈ [0, 1], is measured by means of ∫^1-0 γ^. (t)ǀǀ dt, where ǀǀ ǀǀ denotes the usual norm of operators. The initial value problem is solved: for any isometry Vo and each tangent vector at V0 (which is an operator of the form iXV0 with X* = X) with norm less than or equal to π, there exist curves of the form e^itZ V0, with initial velocity iZV0 = iXV0, which are short along their path. These curves, which we call metric geodesics, need not be unique, and correspond to the so called extension problem considered by M.G. Krein and others: in our context, given asymmetric operator X0|R(V0) : R(V0)→H, find all possible Z* = Z extending X0|R(V0) to all H, with ǀǀZǀǀ= ǀǀX0ǀǀ. We also consider the problem of finding metric geodesics joining two given isometries V0 and V1. It is well known that if there exists a continuous path joining V0 and V1, then both ranges have the same codimension. We show that if this number is finite, then there exist metric geodesics joining V0 and V1.