Quotient p-Schatten metrics on spheres

• Autores: Andruchow, Esteban; Antúnez, Andrea C.
• Año de publicación: 2017
• Idioma: inglés
• Tipo de recurso: artículo
• Fuente: Revista de la Unión Matemática Argentina. Abr. 2017; 58(1): 21-36; https://inmabb.criba.edu.ar/revuma/revuma.php?p=toc/vol58
• Materia: Sphere; Schatten ideals
• Institución: Universidad Nacional de General Sarmiento

Fil: Andruchow, Esteban. Universidad Nacional de General Sarmiento. Instituto de Ciencias; Argentina.
Fil: Andruchow, Esteban. Consejo Nacional de Investigaciones Científicas y Técnicas. Instituto Argentino de Matemática "Alberto P. Calderón"; Argentina.
Fil: Antunez, Andrea. Universidad Nacional de General Sarmiento. Instituto de Ciencias; Argentina.
Let S(H) be the unit sphere of a Hilbert space H and Up(H) thegroup of unitary operators in H such that u?1 belongs to the p-Schatten idealBp(H). This group acts smoothly and transitively in S(H) and endows it witha natural Finsler metric induced by the p-norm kzkp = tr(zz*)p/21/p. Thismetric is given bykvkx,p = min{kz ? ykp : y ? gx},where z ? Bp(H)ah satisfies that (d?x)1(z) = z · x = v and gx denotes theLie 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 kvkx,p = kz0kp. Inthis paper we show properties of minimal liftings and we treat the problemof finding short curves ? such that ?(0) = x and ??(0) = v with x ? S(H)and v ? TxS(H) given. Also we consider the problem of finding short curveswhich join two given endpoints x, y ? S(H).