2011/12/27 by Marianne Akian, Stéphane Gaubert, Stephane Gaubert +5
Economics, Econometrics and Finance · Mathematics · #47H07 #47H08 #91A25 #Advanced Banach Space Theory #Advanced Harmonic Analysis Research #FOS: Mathematics #Functional Analysis (math.FA) #Italy: Economic History and Contemporary Issues #Mathematical and Theoretical Analysis #Optimization and Control (math.OC) #Spectral Theory (math.SP) #Stochastic processes and financial applications #math.FA #math.OC #math.SP #msc:47H07 #msc:47H08 #msc:91A25
paper · pdf · doi:10.48550/arxiv.1112.5968
24 pages, v2: minor improvements and updated references
openalex publication_date 2011/12/27 · arxiv created 2014/04/10 · arxiv updated 2014/04/14 · openalex created_date 2022/09/05 · openalex updated_date 2026/07/28
Several notions of spectral radius arise in the study of nonlinear order-preserving positively homogeneous self-maps of cones in Banach spaces. We give conditions that guarantee that all these notions lead to the same value. In particular, we give a Collatz-Wielandt type formula, which characterizes the growth rate of the orbits in terms of eigenvectors in the closed cone or super-eigenvectors in the interior of the cone. This characterization holds when the cone is normal and when a quasi-compactness condition, involving an essential spectral radius defined in terms of k-set-contractions, is satisfied. Some fixed point theorems for non-linear maps on cones are derived as intermediate results. We finally apply these results to show that non-linear spectral radii commute with respect to suprema and infima of families of order preserving maps satisfying selection properties.