A new approach to numerical radius of quadratic operators (Q781730)

The authors give an elementary approach for computation of the numerical radius and norms of the real and imaginary parts of quadratic operators. Generally, this method is based on proving the maximum of a desired function is equal to one of its suitable upper bounds, via successively establishing equality of this maximum with some of its intermediate upper bounds. This method is illustrated in Theorem~2, Lemma~2 and Proposition~1 and it appears that it can be used in other similar situations. Section~2 is focused on reviewing some classical known results on the numerical radius and norms of the real and imaginary parts of quadratic operators. In [Linear Multilinear Algebra 37, No. 1--3, 13--24 (1994; Zbl 0814.15017)] \textit{C. R. Johnson} et al.\ computed the numerical radius of 2-by-2 matrices from the boundary of their numerical ranges. In Section 3, using a new method, the authors obtain the numerical radius of a quadratic operator in some important special cases. Finally in Section 4, utilizing again this method, the authors obtain the norms of the real and imaginary parts of a quadratic operator in terms of its norm.