Some inequalities for the numerical radius and spectral norm for operators in Hilbert $C^{\ast}$-modules space

Abstract

In this paper, a fresh approach to investigating the numerical radius of bounded operators on Hilbert $C^*$-modules is presented. By using our approach, we can produce some novel findings and extend certain established theorems for bounded adjointable operators on Hilbert $C^*$-module spaces. Moreover, we find an upper bound for power of the numerical radius of $t^{\alpha}ys^{1-\alpha}$
under assumption $0\leq \alpha\leq 1$. In fact, we prove
$$w_c\left(t^{\alpha}ys^{1-\alpha}\right)\leq{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert y \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}^r{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert \alpha t^{r}+(1-\alpha)s^{r}\right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}$$
for all $0\leq \alpha\leq 1$ and $r \geq 2$.