Optimal paths for symmetric actions in the unitary group

2011/07/13 by Jorge Antezana, Antezana, Jorge, Gabriel Larotonda +3 · 2 citations
Mathematics · #15A18 #51F25 (Primary) 47L20 #53C22 (Secondary) #Advanced Topics in Algebra #Differential Geometry (math.DG) #FOS: Mathematics #Functional Analysis (math.FA) #Geometric Analysis and Curvature Flows #Mathematical Inequalities and Applications

paper · pdf · doi:10.48550/arxiv.1107.2439

openalex publication_date 2011/07/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given a positive and unitarily invariant Lagrangian L defined in the algebra of Hermitian matrices, and a fixed interval [a,b]⊂\mathbb R, we study the action defined in the Lie group of n× n unitary matrices U(n) by S(α)=∫_a^b L(α(t)) dt , where α:[a,b]\toU(n) is a rectifiable curve. We prove that the one-parameter subgroups of U(n) are the optimal paths, provided the spectrum of the exponent is bounded by π. Moreover, if L is strictly convex, we prove that one-parameter subgroups are the unique optimal curves joining given endpoints. Finally, we also study the connection of these results with unitarily invariant metrics in U(n) as well as angular metrics in the Grassmann manifold

Citations

• The Rotation of Eigenvectors by a Perturbation. III

Cited by

• Lie groups with a bi-invariant distance
• The group of L² isometries on H¹₀

Related

• Minimal curves in U(n) and Gl(n)+ with respect to the spectral and the trace norms
• Short paths for symmetric norms in the unitary group
• Unitary subgroups and orbits of compact self-adjoint operators
• Convexity of sums of eigenvalues of a segment of unitaries
• Minimal length curves in unitary orbits of a Hermitian compact operator