Inequalities involving the numerical radius

Abstract

For the classical numerical radius f(A) of an n-by-n real or complex matrix A, we consider inequalities of the following two types. in which f and g are polynomials. In the latter case, the event in which f and g are simple powers is of special interest. We present a variety of particular results depending upon the dimension n and the classification of A or the polynomials. A number of natural questions remain open.