Nonlinear mappings preserving at least one eigenvalue

• Autorzy: Constantin Costara , Dušan Repovš
• Języki publikacji: EN

Abstrakty

EN

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 $F(x) = uxu^{-1}$ or $F(x) = ux^{t}u^{-1}$ for all x, for some invertible n × n matrix u. We arrive at the same conclusion by supposing F to be of class 𝓒¹ on a domain in ℳₙ containing the null matrix, instead of Lipschitz. We also prove that if F is of class 𝓒¹ 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 $γ^{-1}F$ or $γ^{-1}F̅$ is of the above form, where F̅ is the (complex) conjugate of F.

Kategorie tematyczne

• 47B49: Transformers, preservers (operators on spaces of operators)
• 15A18: Eigenvalues, singular values, and eigenvectors

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2010
• Tom: 200
• Numer: 1
• Strony: 79-89

Daty

wydano

2010

Twórcy

autor

Constantin Costara

• Faculty of Mathematics and Informatics, Ovidius University, Mamaia Blvd. 124, Constanţa, Romania

autor

Dušan Repovš

• Faculty of Mathematics and Physics, and Faculty of Education, University of Ljubljana, P.O. Box 2964, Ljubljana 1001, Slovenia

Identyfikatory

DOI

10.4064/sm200-1-5