In this work, the mixed Schwarz inequality for semi-Hilbertian space operators is proved. Namely, for every positive Hilbert space operator $A$. If $f$ and $g$ are nonnegative continuous functions on $\left[0,\infty\right)$ satisfying $f(t)g(t) =t$ $(t\ge0)$, then \begin{ align*} \left| { \left\langle { T x,y} \right\rangle_A } \right| \le \left\| { f\left( { \left| T \right|_A x} \right)} \right\|_A \left\| { g\left( { \left| { T^{ \sharp_A } } \right|_A y} \right)} \right\|_A \end{ align*} for every Hilbert space operator $T$ such that the range of $T^* A$ is a subset in the range of $A$, such that $A$ commutes with $T$, and for all vectors $x,y\in \mathscr{ H}$, where $\left| T \right|_A = \left(AT^{ \sharp_A}T\right)^{ 1/2}$ such that $T^{ \sharp_A}=A^\dagger T^*A$, where $A^\dagger$ is the Moore-Penrose inverse of $A$. Based on that, some inequalities for the $A$-numerical radius are introduced.
Categories: math.FA