Numerical radius of bounded operators with lp-norm

Abstract

We study numerical radius of bounded operators on direct sum of a family of  Hilbert spaces with respect to the $\ell^p$-norm, where $1 \leq p \leq \infty.$  We propose  a new method which enable 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 $\ell^p$-norm, where  $1 \leq p \leq 2$.  We also provide an  example to show that some known results on numerical radius are not true for  space is the space of bounded operators on $\ell^p$-sum of Hilbert spaces provided that   $2< p < \infty$. We also present some applications of our results.