Some inequalities for the numerical radius and spectral norm for operators in Hilbert C^-modules space

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 tαys1-αunder assumption 0≤ α≤ 1. In fact, we prove wctαys1-α≤ yʳα tʳ+(1-α)sʳfor all 0≤ α≤ 1 and r ≥ 2.