Estimates for the spectrum near algebraic elements Original Research Article

We extend a result of S. Friedland (Linear Algebra Appl. 12 (1982) 81–98) on the variation of eigenvalues of matrices to show that, if a,b are elements of a Banach algebra, both algebraic of degree at most n, then the Hausdorff distance between their spectra satisfiesimagewhere M=max(short parallelashort parallel,short parallelbshort parallel) and cnless-than-or-equals, slant23n+13. The same technique also re-proves a local form of this result, obtained earlier by B. Aupetit and J. Zemánek (Linear Algebra Appl. 52/53 (1983) 39–44), but with improved bounds on the constants. We further investigate the sharpness of these bounds.