Abstract. It is shown that matrices over algebraically closed fields that are farthest apart in the commuting graph must be non-derogatory. Rank-one matrices and diagonalizable matrices are also characterized in terms of the commuting graph.
Abstract. A local version of the Kleinecke-Shirokov theorem is proved. The results easily extend to bounded linear derivations on Banach algebras. In case of algebraic elements, an improved bound on the nilindex of a commutator is obtained as a consequence.Mathematics subject classification (2000): 47B47, 47B48, 46H05.
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