Perturbation and Spectral Discontinuity in Banach Algebras

TL;DR

This paper demonstrates spectral discontinuity phenomena in Banach algebras, extending previous examples and showing how inessential ideals can induce spectrum function discontinuities through perturbations.

Contribution

It generalizes spectral discontinuity results from Hilbert spaces to arbitrary Banach algebras and constructs examples with non-commutative inessential ideals causing spectrum discontinuities.

Findings

Spectral discontinuity can be induced by perturbations in inessential ideals.

Extension of spectral discontinuity examples from Hilbert spaces to Banach algebras.

Construction of Banach algebras with spectrum discontinuities at specific elements.

Abstract

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 $z$.