Perturbation and spectral discontinuity in Banach algebras

Rudi Brits¹

¹ Department of Mathematics University of Johannesburg Johannesburg, South Africa

Studia Mathematica, Tome 203 (2011) no. 3, pp. 253-263

Résumé

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 $x\in Y$ whose spectrum separates the plane there corresponds a perturbation of $x$, of the form $z=x+a$ where $a\in I$, such that the spectrum function on $A$ is discontinuous at $z$.

Détail

DOI : 10.4064/sm203-3-3

Keywords: extend example aupetit which illustrates spectral discontinuity operators infinite dimensional separable hilbert space general spectral discontinuity result abstract banach algebras given banach algebra may adjoin non commutative inessential ideal resulting algebra following holds each whose spectrum separates plane there corresponds perturbation form where spectrum function discontinuous