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 $T$.
Categories: math.FA