Spectral Characterization of Algebraic Elements
Thomas Ransford, Michael White · Bulletin of the London Mathematical Society · 2001
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 ( ‖ x ‖ γ ) as x → 0. We prove that the converse is also true, provided that A is semisimple.