Short paths for symmetric norms in the unitary group

Details Short paths for symmetric norms in the unitary group Short paths for symmetric norms in the unitary group

2010-01-01

arxiv.org/abs/1001.0263v2

Mathematics

Metric Geometry

This preprint has been replaced by arXiv:1107.2439v1, which deals with the far more general case of symmetric Lagrangians in t

Scientific paper

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 $$ U_I={u is a unitary operator in H, u-1\in I}. $$ 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.

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