Critical polynomials related to generalized derivations

The relations between the degree of the minimal polynomial of linear operators in tensor products and the cardinality of their spectrum have been used with success in additive number theory. In a previous paper it was proved that if V is a vector space and T is linear operator with minimal polynomial of degree n, then, is a lower bound for the degree of the minimal polynomial of the k-derivation of T on mV. In this article we study the structure of the linear operators T, whose minimal polynomial of the k-derivation has degree precisely equal to the bound above.