Improvement of $A$-numerical radius inequalities of semi-Hilbertian space operators

TL;DR

This paper introduces improved bounds for the $A$-numerical radius of operators in semi-Hilbertian spaces and extends these bounds to $2\times 2$ operator matrices, enhancing previous results.

Contribution

It provides new, tighter bounds for the $A$-numerical radius and operator seminorm in semi-Hilbertian spaces, generalizing and improving existing inequalities.

Findings

New bounds for $A$-numerical radius are established.

Bounds for $B$-operator seminorm and $B$-numerical radius are derived.

Results improve upon existing inequalities in the literature.

Abstract

Let $\mathcal{H}$ be a complex Hilbert space and let $A$ be a positive operator on $\mathcal{H}$. We obtain new bounds for the $A$-numerical radius of operators in semi-Hilbertian space $\mathcal{B}_A(\mathcal{H})$ that generalize and improve on the existing ones. Further, we estimate bounds for the $B$-operator seminorm and $B$-numerical radius of $2\times 2$ operator matrices, where $B=\mbox{diag}(A,A)$. The bounds obtained here improve on the existing ones.