Let A be a unital dense algebra of linear mappings on a complex vector
space X. Let ϕ=∑i=1nMai,bi be a locally quasi-nilpotent
elementary operator of length n on A. We show that, if {a1,…,an}
is locally linearly independent, then the local dimension of
V(\phi)=\spa\{b_ia_j: 1 \leq i,j \leq n\} is at most 2n(n−1). If
\lDim V(\phi)=\frac{n(n-1)}{2} , then there exists a representation of ϕ
as ϕ=∑i=1nMui,vi with viuj=0 for i≥j. Moreover, we
give a complete characterization of locally quasi-nilpotent elementary
operators of length 3.Comment: 15