Let H be an infinite-dimensional separable complex Hilbert space and B(H) the algebra of all bounded linear operators on H. For an operator T in B(H), let σg(T) denote the generalized spectrum of T. In this paper, we prove that if φ: B(H) → B(H) is a surjective linear map, then φ preserves the generalized spectrum (i.e. σg(φ(T)) = σg(T) for every T ∈ B(H)) if and only if there is A ∈ B(H) invertible such that either φ(T) = ATA-1 for every T ∈ B(H), or φ(T) = ATtrA-1 for every T ∈ B(H). Also, we prove that γ(φ(T)) = γ(T) for every T ∈ B(H) if and only if there is U ∈ B(H) unitary such that either φ(T) = UTU* for every T ∈ B(H) or φ(T) = UTtrU* for every T ∈ B(H). Here γ(T) is the reduced minimum modulus of T.
@article{urn:eudml:doc:41869,
title = {Linear maps preserving the generalized spectrum.},
journal = {Extracta Mathematicae},
volume = {22},
year = {2007},
pages = {45-54},
zbl = {1160.47033},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:41869}
}
Mbekhta, Mostafa. Linear maps preserving the generalized spectrum.. Extracta Mathematicae, Tome 22 (2007) pp. 45-54. http://gdmltest.u-ga.fr/item/urn:eudml:doc:41869/