The generalized Schwarz inequality for semi-Hilbertian space operators and Some $A$-numerical radius inequalities

Abstract

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 $f(t)g(t)=t$ $(t\ge 0)$, 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 $T^{\ast }A$ is a subset in the range of $A$, such that $A$ commutes with $T$, and for all vectors $x,y\in \mathcal{H}$, where ${\left|T\right|}_A={\left(A{T}^{{♯}_A}T\right)}^{1/2}$ such that $T^{{♯}_A}=A^{†}{T}^{\ast }A$, where $A^{†}$ is the Moore-Penrose inverse of $A$. Based on that, some inequalities for the $A$-numerical radius are introduced.

17 pages