General Numerical Radius for Products of Sectorial Matrices

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