Short paths for symmetric norms in the unitary group

TL;DR

This paper investigates shortest paths in the unitary group with symmetric norms, establishing conditions for their existence and uniqueness, and extending classical geometric results to infinite-dimensional settings.

Contribution

It proves the existence and uniqueness of shortest paths in the unitary group under symmetric norms, extending the Hopf-Rinow theorem to infinite dimensions.

Findings

One-parameter subgroups are shortest paths when spectrum is bounded by π.

Any two elements can be connected by a shortest path under certain conditions.

Uniqueness of shortest paths is guaranteed with strictly convex symmetric norms.

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