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

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 $${\omega }_A(TS)\le \frac{1}{2}{\omega }_A(ST)+\frac{1}{4}(\Vert T{\Vert }_A\Vert S{\Vert }_A+\Vert TS{\Vert }_A),$$ where ${\omega }_A(T)$ and $\Vert T{\Vert }_A$ denote the $A$-numerical radius and the $A$-operator seminorm of an operator $T$.