Resumen de Quotient p -Schatten metrics on spheres

Esteban Andruchow, Andrea C. Antúnez

• Let S(H) be the unit sphere of a Hilbert space H and Up(H) the group of unitary operators in H such that u−1 belongs to the p-Schatten ideal Bp(H). This group acts smoothly and transitively in S(H) and endows it with a natural Finsler metric induced by the p-norm ∥z∥p=tr((zz)p/2)1/p. This metric is given by ∥v∥x,p=min{∥z−y∥p:y∈gx}, where z∈Bp(H)ah satisfies that (dπx)1(z)=z⋅x=v and gx denotes the Lie algebra of the subgroup of unitaries which fix x. We call z a lifting of v. A lifting z0 is called a minimal lifting if additionally ∥v∥x,p=∥ z0∥p. In this paper we show properties of minimal liftings and we treat the problem of finding short curves α such that α(0)=x and ˙α(0)=v with x∈S(H) and v∈xS(H) given. Also we consider the problem of finding short curves which join two given endpoints x,y∈S(H).