Generalized numerical radius and related inequalities

Abstract

They proved several properties and introduced some inequalities. We continue with the study of this generalized numerical radius and we develop diverse inequalities involving w_N. We also study particular cases with a fixed N(.), for instance the p-Schatten norms. In ["A generalization of the numerical radius". Linear Algebra Appl. 569 (2019)], Abu Omar and Kittaneh defined a new generalization of the numerical radius. That is, given a norm $N(\cdot)$ on $\bh$, the space of bounded linear operators over a Hilbert space H, and A in B(H) w_N(A)=sup_{\theta\in \R}N(Re(e^{i\theta}A)). They proved several properties and introduced some inequalities. We continue with the study of this generalized numerical radius and we develop diverse inequalities involving $w_N$. We also study particular cases when N(.) is the p- Schatten norm with p>1.