Let X and Y be Banach spaces and ℬ(X) and ℬ(Y) the algebras of all bounded linear operators on X and Y, respectively. We say that A,B ∈ ℬ(X) quasi-commute if there exists a nonzero scalar ω such that AB = ωBA. We characterize bijective linear maps ϕ : ℬ(X) → ℬ(Y) preserving quasi-commutativity. In fact, such a characterization can be proved for much more general algebras. In the finite-dimensional case the same result can be obtained without the bijectivity assumption.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm184-2-7,
author = {Heydar Radjavi and Peter \v Semrl},
title = {Linear maps preserving quasi-commutativity},
journal = {Studia Mathematica},
volume = {187},
year = {2008},
pages = {191-204},
zbl = {1183.15028},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm184-2-7}
}
Heydar Radjavi; Peter Šemrl. Linear maps preserving quasi-commutativity. Studia Mathematica, Tome 187 (2008) pp. 191-204. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm184-2-7/