IAM 02674
INSTITUTO ARGENTINO DE MATEMATICA ALBERTO CALDERON
Unidad Ejecutora - UE
artículos
Título:
Finsler geometry and actions of the p-Schatten unitary groups
Autor/es:
ESTEBAN ANDRUCHOW, GABRIEL LAROTONDA, LÁZARO RECHT
Revista:
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY
Editorial:
AMER MATHEMATICAL SOC
Referencias:
Lugar: Providence RI; Año: 2010 vol. 362 p. 319 - 344
ISSN:
0002-9947
Resumen:
Abstract. Let p be an even positive integer and Up(H) the Banach-Lie group
of unitary operators u which verify that u �� 1 belongs to the p-Schatten idealLet p be an even positive integer and Up(H) the Banach-Lie group
of unitary operators u which verify that u �� 1 belongs to the p-Schatten idealu which verify that u �� 1 belongs to the p-Schatten ideal
Bp(H). Let O be a smooth manifold on which Up(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 Up(H). Our main result establishes
that for any pair of given initial conditionsp(H). Let O be a smooth manifold on which Up(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 Up(H). Our main result establishes
that for any pair of given initial conditionsO with a natural Finsler metric in terms of
the p-Schatten norm and the action of Up(H). Our main result establishes
that for any pair of given initial conditionsp-Schatten norm and the action of Up(H). Our main result establishes
that for any pair of given initial conditions
x 2 O and X 2 (TO)x2 O and X 2 (TO)x
there exists a curve (t) = etz x in O, with z a skew-hermitian element in the(t) = etz x in O, with z a skew-hermitian element in the
p-Schatten class, such that-Schatten class, such that
(0) = x and _(0) = X;(0) = x and _(0) = X;
which remains minimal as long as tkzkp =4. Moreover, is unique with
these properties. We also show that the metric space (O; d) (where d is the
recti able distance) is complete. In the process we establish minimality results
in the groups Up(H), and a convexity property for the recti able distance. As
an example of these spaces, we treat the case of the unitary orbittkzkp =4. Moreover, is unique with
these properties. We also show that the metric space (O; d) (where d is the
recti able distance) is complete. In the process we establish minimality results
in the groups Up(H), and a convexity property for the recti able distance. As
an example of these spaces, we treat the case of the unitary orbitO; d) (where d is the
recti able distance) is complete. In the process we establish minimality results
in the groups Up(H), and a convexity property for the recti able distance. As
an example of these spaces, we treat the case of the unitary orbitUp(H), and a convexity property for the recti able distance. As
an example of these spaces, we treat the case of the unitary orbit
O = fuAu : u 2 Up(H)g= fuAu : u 2 Up(H)g
of a self-adjoint operator A 2 B(H).A 2 B(H).
