# Short paths for symmetric norms in the unitary group - Mathematics > Metric Geometry

Abstract: For a given symmetrically normed ideal I on an infinite dimensional Hilbertspace H, we study the rectifiable distance in the classical Banach-Lie unitarygroup $$U I={u is a unitary operator in H, u-1\in I}.$$ We prove thatone-parameter subgroups of U I are short paths, provided the spectrum of theexponent is bounded by $\pi$, and that any two elements of U I can be joinedwith a short path, thus obtaining a Hopf-Rinow theorem in this infinitedimensional setting, for a wide and relevant class of non necessarily smoothmetrics. Then we prove that the one-parameter groups are the unique short pathsjoining given endpoints, provided the symmetric norm considered is strictlyconvex.

Author: Jorge Antezana, Gabriel Larotonda, Alejandro Varela

Source: https://arxiv.org/