We extend an example of B. Aupetit, which illustrates spectral discontinuity for operators on an infinite-dimensional separable Hilbert space, to a general spectral discontinuity result in abstract Banach algebras. This can then be used to show that given any Banach algebra, Y, one may adjoin to Y a non-commutative inessential ideal, I, so that in the resulting algebra, A, the following holds: To each x ∈ Y whose spectrum separates the plane there corresponds a perturbation of x, of the form z = x + a where a ∈ I, such that the spectrum function on A is discontinuous at z.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm203-3-3,
author = {Rudi Brits},
title = {Perturbation and spectral discontinuity in Banach algebras},
journal = {Studia Mathematica},
volume = {204},
year = {2011},
pages = {253-263},
zbl = {1254.46049},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm203-3-3}
}
Rudi Brits. Perturbation and spectral discontinuity in Banach algebras. Studia Mathematica, Tome 204 (2011) pp. 253-263. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm203-3-3/