Abstract

It is known that if a is an algebraic element of a Banach algebra A, then its spectrum σ(a) is finite, and there exists γ > 0 such that the Hausdorff distance to spectra of nearby elements satisfies  

Δ(σ(a+x),σ(a))=0(xγ)asx0.
We prove that the converse is also true, provided that A is semisimple.

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