Some improvements of numerical radius inequalities of operators and operator matrices

TL;DR

This paper presents new upper bounds for the numerical radius of operator products and matrices, generalizing existing inequalities and improving their tightness using non-negative functions and operator matrices.

Contribution

It introduces improved bounds for the numerical radius of operator products and matrices, extending previous inequalities with novel generalizations and tighter estimates.

Findings

Improved upper bounds for the numerical radius of operator products

Generalized inequalities for $n\times n$ operator matrices using non-negative functions

New bounds for the $B$-numerical radius of operator matrices

Abstract

We obtain upper bounds for the numerical radius of a product of Hilbert space operators which improve on the existing upper bounds. We generalize the numerical radius inequalities of $n\times n$ operator matrices by using non-negative continuous functions on $[0,\infty)$. We also obtain some upper and lower bounds for the $B$-numerical radius of operator matrices, where $B$ is the diagonal operator matrix whose each diagonal entry is a positive operator $A.$ We show that these bounds generalize and improve on the existing bounds.