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, for every operators $T$ and $S$ which admit $A$-adjoints, it is proved 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 $T$ respectively.