Numerical radius inequalities for products and sums of semi-Hilbertian space operators

TL;DR

This paper establishes new inequalities for the $A$-numerical radius of products and sums of operators in semi-Hilbert spaces, enhancing understanding of operator behavior in these generalized spaces.

Contribution

It introduces novel inequalities for the $A$-numerical radius of operator products and sums in semi-Hilbert spaces, expanding the theoretical framework.

Findings

Derived an inequality for $oldsymbol{ ext{$A$-numerical radius}}$ of operator products.

Established bounds involving $A$-operator seminorms and $A$-numerical radius.

Provided theoretical tools for analyzing operators in semi-Hilbert spaces.

Abstract

New inequalities for the $A$-numerical radius of the products and sums of operators acting on a semi-Hilbert space, i.e. a space generated by a positive semidefinite operator $A$, are established. In particular, it is proved for operators $T$ and $S,$ having $A$-adjoint, that $$ \omega_A(TS) \leq \frac{1}{2}\omega_A(ST)+\frac{1}{4}\Big(\|T\|_A\|S\|_A+\|TS\|_A\Big),$$ where $\omega_A(T)$ and $\|T\|_A$ denote the $A$-numerical radius and the $A$-operator seminorm of an operator $T$.