Critical operators for the degree of the minimal polynomial of derivations restricted to Grassmann spaces (Q886150)

Let \(V\) be a finite dimensional vector space. For a linear operator \(f\) on \(V\), let \(D(f)\) be the derivation associated with \(f\) to the \(m\)th Grassmann space of \(V\). In [Bull. Lond. Math. Soc. 26, No.~2, 140--146 (1994; Zbl 0819.11007)] \textit{J.A. Dias da Silva} and \textit{Y. O. Hamidoune} obtained a lower bound for the degree of the minimal polynomial of \(D(f)\), over an arbitrary field. Over a field of zero characteristic that lower bound is given by \[ \deg(P_{D(f)})\geq m(\deg(P_f)-m)+1. \] Using additive number theory results, results on the elementary divisors of \(D(f)\), and methods of \textit{M. Marcus} and \textit{M. S. Ali} [J. Algebra 22, 12--33 (1972; Zbl 0243.15011)], the author obtains a characterization of equality cases in the former inequality, over a field of zero characteristic, whenever \(m\) does not exceed the number of distinct eigenvalues.