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 ${\omega }_N(X)=\underset{\theta \in \mathbb{R}}{sup}N(\mathrm{Re}(e^{i\theta }X))$. 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.