Improved Bounds for numerical radius and $a$-numerical radius in $C^{\ast }$-algebras

Abstract

In this article, we derive several significant upper bounds for the numerical radius and $a$-numerical radius of an element in a $C^{\ast }$-algebra by improving inequalities for positive linear functionals. Our findings refine and generalize the existing inequalities. Furthermore, we introduce a new notion to derive improved upper bounds of the numerical radius for an element in a $C^{\ast }$-algebra using the Moore-Penrose inverse.