Numerical Radius of Bounded Operators with ℓ^p-Norm

We study the numerical radius of bounded operators on direct sum of a family of Hilbert spaces with respect to the ℓ^p-norm, where 1 ≤ p ≤∞. We propose a new method which enables us to prove validity of many inequalities on numerical radius of bounded operators on Hilbert spaces when the underling space is a direct sum of Hilbert spaces with ℓ^p-norm, where 1 ≤ p ≤ 2. We also provide an example to show that some known results on numerical radius are not true for a space that is the set of bounded operators on ℓ^p-sum of Hilbert spaces where 2 <p < ∞. We also present some applications of our results.