Decompositions of endomorphisms into a sum of roots of the unity and nilpotent endomorphisms of fixed nilpotence (Q6178769)

Let \(V\) be an \(n\)-dimensional vector space over a field \(\mathbb{F}\). An endomorphism \(t\) is a root of unity if some positive power of \(t\) is the identity. Let \(k\geq 2\). The problem considered in this paper is to investigate conditions under which \(f\in \mathrm{End}(V)\) can be written in the form \(f=t+m\) (say (*)), where \(t\) is a root of unity and \(m^{k}=0\). The following are some examples: (1) If the trace of each elementary divisor of \(f \) is a sum of distinct roots of unity then the decomposition (*) can be achieved with \(k=2\) over some extension of \(\mathbb{F}\); (2) If \( \mathbb{F}\) has characteristic \(p>0\) and the coefficients of the characteristic polynomial for \(f\) lie in a finite field, then (*) holds whenever \(f\) has rank at least \(k\). Finally we have a result in a more general setting: (3) Replace \(\mathbb{F}\) by a division ring \( \Delta \) and assume that \(V\) is a left \(n\)-dimensional vector space over \( \Delta \). Suppose \(k=2\) and that \(f\) is a nilpotent endomorphism \(f\) of \(V\). Then (*) if and only if the rank of \(f\) is at least \(n/2\).