An extension of the $a$-numerical radius on $C^*$-algebras

TL;DR

This paper introduces a new semi-norm extending the $a$-numerical radius in $C^*$-algebras, explores its properties, and establishes new bounds for the $a$-numerical radius of algebra elements.

Contribution

It defines a generalized semi-norm on $C^*$-algebras that extends existing concepts and derives novel inequalities and bounds for the $a$-numerical radius.

Findings

Established basic properties of the new semi-norm.

Derived new upper and lower bounds for the $a$-numerical radius.

Discussed related inequalities and results.

Abstract

Let $a$ be a positive element in a unital $C^*$-algebra $\mathfrak{A}$. We define a semi-norm on $\mathfrak{A}$, which generalizes the $a$-operator semi-norm and the $a$-numerical radius. We investigate basic properties of this semi-norm and prove inequalities involving it. Further, we derive new upper and lower bounds for the $a$-numerical radii of elements in $\mathfrak{A}$. Some other related results are also discussed.