Inequalities for the norm and numerical radius for Hilbert $C^{*}$-module operators

In this paper, we introduce some inequalities between the operator norm and the numerical radius of adjointable operators on Hilbert $C^{*}$-module spaces. Moreover, we establish some new refinements of numerical radius inequalities for Hilbert space operators. More precisely, we prove that if $T \in B(H)$ and
$$ \min \Big( \frac{\Vert T+ T^* \Vert^ 2 }{2}, \frac{\Vert T- T^* \Vert^ 2 }{2}\Big) \leq \max \Big(\inf_{ \Vert x \Vert=1}{\Vert Tx \Vert^2}, \inf_{ \Vert x \Vert=1}\Vert T^*x \Vert^2\Big), $$
then
\begin{equation*} \Vert T \Vert \leq \sqrt{ 2} \omega(T); \end{equation*}
this is a considerable improvement of the classical inequality
\begin{equation*} \Vert T \Vert \leq 2\omega(T). \end{equation*}