Numerical radius: New Extensions and Inequalities

We firstly define a seminorm on the space of bounded linear operators on a Hilbert space, which generalizes the numerical radius norm. We investigate basic properties of this seminorm and prove inequalities involving it. Further, for a positive element a in a unital C ∗ -algebra 21 we define a semi-norm on 21, which generalizes the a-operator semi-norm and the a-numerical radius.