We show that every spectrally bounded linear map Φ from a Banach algebra onto a standard operator algebra acting on a complex Banach space is square-zero preserving. This result is used to show that if Φ₂ is spectrally bounded, then Φ is a homomorphism multiplied by a nonzero complex number. As another application to the Hilbert space case, a classification theorem is obtained which states that every spectrally bounded linear bijection Φ from ℬ(H) onto ℬ(K), where H and K are infinite-dimensional complex Hilbert spaces, is either an isomorphism or an anti-isomorphism multiplied by a nonzero complex number. If Φ is not injective, then Φ vanishes at all compact operators.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm150-3-4,
author = {Jianlian Cui and Jinchuan Hou},
title = {The spectrally bounded linear maps on operator algebras},
journal = {Studia Mathematica},
volume = {151},
year = {2002},
pages = {261-271},
zbl = {1008.47040},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm150-3-4}
}
Jianlian Cui; Jinchuan Hou. The spectrally bounded linear maps on operator algebras. Studia Mathematica, Tome 151 (2002) pp. 261-271. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm150-3-4/