Short paths for symmetric norms in the unitary group

2010/01/01 by Antezana, Jorge, Larotonda, Gabriel, Varela, Alejandro
#47B10 #53C22 (Secondary) #53C23 (Primary) 51F25 #FOS: Mathematics #Metric Geometry (math.MG) #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.1001.0263

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∈ I. We prove that one-parameter subgroups of U_I are short paths, provided the spectrum of the exponent is bounded by π, 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.

Related

• Unitary subgroups and orbits of compact self-adjoint operators
• Optimal paths for symmetric actions in the unitary group
• The rectifiable distance in the unitary Fredholm group
• Finsler geometry and actions of the p-Schatten unitary groups
• Lie groups with a bi-invariant distance