Locally quasi-nilpotent elementary operators

TL;DR

This paper investigates the structure of locally quasi-nilpotent elementary operators on dense algebras, establishing bounds on the local dimension of associated vector spaces and characterizing operators of length 3.

Contribution

It provides new bounds on the local dimension of certain operator spaces and characterizes elementary operators of length 3 in this context.

Findings

The local dimension of V(φ) is at most n(n-1)/2 when certain conditions hold.

If the local dimension reaches the maximum, a specific operator representation exists.

Complete characterization of length 3 locally quasi-nilpotent elementary operators.

Abstract

Let $A$ be a unital dense algebra of linear mappings on a complex vector space $X$. Let $\phi=\sum_{i=1}^n M_{a_i,b_i}$ be a locally quasi-nilpotent elementary operator of length $n$ on $A$. We show that, if $\{a_1,\ldots,a_n\}$ is locally linearly independent, then the local dimension of $V(\phi)=\spa\{b_ia_j: 1 \leq i,j \leq n\}$ is at most $\frac{n(n-1)}{2}$. If $\lDim V(\phi)=\frac{n(n-1)}{2} $, then there exists a representation of $\phi$ as $\phi=\sum_{i=1}^n M_{u_i,v_i}$ with $v_iu_j=0$ for $i\geq j$. Moreover, we give a complete characterization of locally quasi-nilpotent elementary operators of length 3.