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.
@article{bwmeta1.element.bwnjournal-article-smv134i2p99bwm,
author = {Rajendra Bhatia and Peter \v Semrl and A. Sourour},
title = {Maps on matrices that preserve the spectral radius distance},
journal = {Studia Mathematica},
volume = {133},
year = {1999},
pages = {99-110},
zbl = {0927.15006},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv134i2p99bwm}
}
Bhatia, Rajendra; Šemrl, Peter; Sourour, A. Maps on matrices that preserve the spectral radius distance. Studia Mathematica, Tome 133 (1999) pp. 99-110. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv134i2p99bwm/
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