Maps on matrices that preserve the spectral radius distance
Volume 134 / 1999
Studia Mathematica 134 (1999), 99-110 DOI: 10.4064/sm-134-2-99-110
Let ϕ be a surjective map on the space of n×n complex matrices such that r(ϕ(A)-ϕ(B))=r(A-B) for all A,B, where r(X) is the spectral radius of X. We show that ϕ must be a composition of five types of maps: translation, multiplication by a scalar of modulus one, complex conjugation, taking transpose and (simultaneous) similarity. In particular, ϕ is real linear up to a translation.