Generalizations of the field of values useful in the study of polynomial functions of a matrix Original Research Article

For a given square matrix A and positive integer k, we consider sets Ω in the complex plane satisfying image short parallelp(A)short parallelgreater-or-equal, slantedmaxzset membership, variantΩp(z)for all polynomials p of degree k or less. The largest such set, referred to as the polynomial numerical hull of degree k, was introduced by O. Nevanlinna [Convergence of Iterations for Linear Equations, Birkhäuser, Basel, 1993] and a number of properties of this set were derived for both matrices and linear operators. We give several equivalent characterizations of the polynomial numerical hull of degree k and we actually compute these sets for several matrices. For k=1, this set is just the field of values of A, and for kgreater-or-equal, slantedm, where m is the degree of the minimal polynomial of A, it is the spectrum of A. For 1