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Abstract. Let A be a unital dense algebra of linear mappings on a complex vector space X . Let φ = ∑ n i=1 M a i ,b i be a locally quasi-nilpotent elementary operator of length n on A . We show that, if {a 1 ,... ,a n } is locally linearly independent, then the local dimension of V (φ ) = span{b i a j : 1 i, j n} is at most

We discuss some necessary and some sufficient conditions for an elementary operator x → n i=1 a i xb i on a Banach algebra A to be spectrally bounded. In the case of length three, we obtain a complete characterisation when A acts irreducibly on a Banach space of dimension greater than three.

In this paper we introduce the notion of evolution rank and give a decomposition of an evolution algebra into its annihilator plus extending evolution subspaces having evolution rank one. This decomposition can be used to prove that in non-degenerate evolution algebras, any family of natural and orthogonal vectors can be extended to a natural basis. Central results are the characterization of those families of orthogonal linearly independent vectors which can be extended to a natural basis.We also consider ideals in perfect evolution algebras and prove that they coincide with the basic ideals.Nilpotent elements of order three can be localized (in a perfect evolution algebra over a field in which every element is a square) by merely looking at the structure matrix: any vanishing principal minor provides one. Conversely, if a perfect evolution algebra over an arbitrary field has a nilpotent element of order three, then its structure matrix has a vanishing principal minor.We finish by considering the adjoint evolution algebra and relating its properties to the corresponding in the initial evolution algebra.