Nonlinear mappings preserving at least one eigenvalue

作者：Costara Constantin^*; Repovs Dusan

来源：Studia Mathematica, 2010, 200(1): 79-89.

DOI：10.4064/sm200-1-5

• spectrum
• Lipschitz mapping
• holomorphic mapping
• preserver

We prove that if F is a Lipschitz map from the set of all complex n x 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 F(x) = uxu(-1) or F(x) = ux(t)u(-1) for all x, for some invertible n x n matrix u. We arrive at the same conclusion by supposing F to be of class C(1) on a domain in M(n) containing the null matrix, instead of Lipschitz. We also prove that if F is of class C(1) on a domain containing the null matrix satisfying F(0) = 0 and p(F(x) - F(y)) = p(x - y) for all x and y, where rho(.) denotes the spectral radius, then there exists gamma epsilon C of modulus one such that either gamma(-1) F or gamma(-1) (F) over bar is of the above form, where (F) over bar is the (complex) conjugate of F.

• 出版日期2010