We prove that if F is a Lipschitz map from the set of all complex n × n matrices into itself with F(0) = 0 such that given any x and y we know that F(x) - F(y) and x-y have at least one common eigenvalue, then either or for all x, for some invertible n × n matrix u. We arrive at the same conclusion by supposing F to be of class ¹ on a domain in ℳₙ containing the null matrix, instead of Lipschitz. We also prove that if F is of class ¹ on a domain containing the null matrix satisfying F(0) = 0 and ρ(F(x) - F(y)) = ρ(x-y) for all x and y, where ρ(·) denotes the spectral radius, then there exists γ ∈ ℂ of modulus one such that either or is of the above form, where F̅ is the (complex) conjugate of F.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm200-1-5,
author = {Constantin Costara and Du\v san Repov\v s},
title = {Nonlinear mappings preserving at least one eigenvalue},
journal = {Studia Mathematica},
volume = {196},
year = {2010},
pages = {79-89},
zbl = {1216.47069},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm200-1-5}
}
Constantin Costara; Dušan Repovš. Nonlinear mappings preserving at least one eigenvalue. Studia Mathematica, Tome 196 (2010) pp. 79-89. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm200-1-5/