Short paths for symmetric norms in the unitary group

ANTEZANA, JORGE; LAROTONDA, GABRIEL; VARELA, ALEJANDRO

CABA

Seminario; Seminario de Análisis Funcional; 2010

Instituto Argentino de Matemática (CONICET)

For a given symmetrically normed ideal I on an infinite dimensional Hilbert space H, we study the rectifiable distance in the classical Banach-Lie unitary group U = {u a unitary operator in H and (u -1) in I }.We prove that one-parameter subgroups of U are short paths, provided the spectrum of the exponent is bounded by pi, and that any two elements of U can be joined with a short path, thus obtaining a Hopf-Rinow theorem in this infinite dimensional setting, for a wide and relevant class of (non necessarily smooth) metrics. Then we prove that the one-parameter groups are the unique short paths joining given endpoints, provided the symmetric norm considered is strictly convex.