General Numerical Radius for Products of Sectorial Matrices
In this paper, we investigate the generalized numerical radius $ω_N$, associated with a matrix norm $N$ defined by $ω_N(X) = \sup_{θ\in \mathbb{R}} N(\operatorname{Re}(e^{iθ}X))$. We focus on matrices whose numerical ranges are contained in sectors of the complex plane (sectorial matrices) and derive upper bounds for $ω_N(XY)$ and $ω_N(X \circ Y)$ for such matrices $X$ and $Y$. Our results generalize and refine well known numerical radius inequalities. Several known inequalities for $ω(X)$ are recovered as special cases.
math.FA↗