Short paths for symmetric norms in the unitary group

Abstract

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 $$U_I=uisaunitaryoperatorinH,u-1\in I.$$ We prove that one-parameter subgroups of U_I are short paths, provided the spectrum of the exponent is bounded by $\pi$, and that any two elements of U_I 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.

This preprint has been replaced by arXiv:1107.2439v1, which deals with the far more general case of symmetric Lagrangians in the unitary group

References in corpus (1)

• Optimal paths for symmetric actions in the unitary group