Abstract
In the case of Banach algebras we improve the classical result of A. M. Ostrowski concerning the behavior of the spectrum. More precisely, if a is an element of a Banach algebra such that p( a) = 0 for some polynomial p, there exists a constant C such that ∥ a − x∥ ⩽ 1 implies Δ( Sp a, Sp x) ⩽ C∥a − x∥ 1 m , where m denotes the maximum of the multiplicities of the roots of p, ∥ ∥ denotes the complete norm in the algebra, and Δ denotes the Hausdorff distance for compact sets.
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