Estimates for the spectrum near algebraic elements

We extend a result of S. Friedland (Linear Algebra Appl. 12 (1982) 81-98) o n the variation of eigenvalues of matrices to show that, if a, b are elemen ts of a Banach algebra, both algebraic of degree at most n, then the Hausdo rff distance between their spectra satisfies triangle (sigma(a), sigma (b))(n) less than or equal to c(n) (2M)(n-)1 para llel to a - b parallel to, where M = max(parallel to a parallel to, parallel to b parallel to) and c(n ) less than or equal to 2/3n + 1/3. The same technique also re-proves a loc al form of this result, obtained earlier by B. Aupetit and J. Zemanek (Line ar Algebra Appl. 52/53 (1983) 39-44), but with improved bounds on the const ants. We further investigate the sharpness of these bounds. (C) 2000 Elsevi er Science Inc. All rights reserved.