Generalized numerical radius and related inequalities

TL;DR

This paper extends the study of the generalized numerical radius w_N for bounded linear operators, developing new inequalities and analyzing specific cases like p-Schatten norms, building on prior foundational work.

Contribution

It introduces new inequalities involving the generalized numerical radius w_N and explores particular cases such as p-Schatten norms, advancing the theoretical understanding of this operator measure.

Findings

Developed diverse inequalities involving w_N

Analyzed special cases with p-Schatten norms

Extended previous properties of the generalized numerical radius

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.