FINSLER GEOMETRY AND ACTIONS OF THE p-SCHATTEN UNITARY GROUPS

Let p bean even positive integer and U-p(H) the Banach-Lie group of unitary operators u which verify that u - 1 belongs to the p-Schatten ideal B-p(H). Let O be a smooth manifold on which U-p(H) acts transitively and smoothly. Then one can endow O with a natural Finsler metric in terms of the p-Schatten norm and the action of U-p(H). Our main result establishes that for any pair of given initial conditions x is an element of O and X is an element of (TO)(x) there exists a curve delta(t) = e(tz) . x in O, with z a skew-hermitian element in the p-Schatten class, such that delta(0) = x and (delta) over dot(0) = X, which remains minimal as long as t parallel to z parallel to(p) <= pi/4. Moreover, delta is unique with these properties. We also show that the metric space (O, d) (where d is the rectifiable distance) is complete. In the process we establish minimality results in the groups U-p(H) and a convexity property for the rectifiable distance. As an example of these spaces, we treat the case of the unitary orbit O = {uAu* : u is an element of U-p(H)} of a self-adjoint operator A is an element of B(H).