Counterexample to a conjecture of Elsner on the spectral variation of matrices Original Research Article

In 1985, Elsner proved that the Hausdorff distance Δ between the spectra of two n×n matrices A and B satisfiesimagewhere short parallel·short parallel denotes the operator norm with respect to the Euclidean norm on Cn. He further conjectured that the same inequality holds for all operator norms. We disprove this conjecture, and also the weaker conjecture where (short parallelAshort parallel+short parallelBshort parallel) is replaced by 2max(short parallelAshort parallel,short parallelBshort parallel).