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

ANDRUCHOW, ESTEBAN; RECHT LAZARO; VARELA ALEJANDRO

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY

AMER MATHEMATICAL SOC

Año: 2007 vol. 135 p. 2527 - 2537

0002-9939

We consider the problem of finding short smooth curves of isometries in a Hilbert space H. Thelength of a smooth curve (t), t 2 [0, 1], is measured by means ofR 10 k˙ (t)k dt, where k k denotesthe usual norm of operators. The initial value problem is solved: for any isometry V0 and eachtangent vector at V0 (which is an operator of the form iXV0 with X = X) with norm less than orequal to , there exist curves of the form eitZV0, with initial velocity iZV0 = iXV0, which are shortalong their path. These curves, which we call metric geodesics, need not be unique, and correspondto the so called extension problem considered by M.G. Krein and others: in our context, given asymmetric operatorX0|R(V0) : R(V0) ! H,find all possible Z = Z extending X0|R(V0) to all H, with kZk = kX0k. We also consider theproblem of finding metric geodesics joining two given isometries V0 and V1. It is well known thatif 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.