$A$-numerical radius inequalities for semi-Hilbertian space operators

Let $A$ be a positive bounded operator on a Hilbert space $\big(\mathcal{H}, \langle \cdot, \cdot\rangle \big)$. The semi-inner product ${\langle x, y\rangle}_A := \langle Ax, y\rangle$, $x, y\in\mathcal{H}$ induces a semi-norm ${\|\cdot\|}_A$ on $\mathcal{H}$. Let ${\|T\|}_A$ and $w_A(T)$ denote the $A$-operator semi-norm and the $A$-numerical radius of an operator $T$ in semi-Hilbertian space $\big(\mathcal{H}, {\|\cdot\|}_A\big)$, respectively. In this paper, we prove the following characterization of $w_A(T)$ \begin{align*} w_A(T) = \displaystyle{\sup_{\alpha^2 + \beta^2 = 1}} {\left\|\alpha \frac{T + T^{\sharp_A}}{2} + \beta \frac{T - T^{\sharp_A}}{2i}\right\|}_A, \end{align*} where $T^{\sharp_A}$ is a distinguished $A$-adjoint operator of $T$. We then apply it to find upper and lower bounds for $w_A(T)$. In particular, we show that \begin{align*} \frac{1}{2}{\|T\|}_A \leq \max\Big\{\sqrt{1 - {|\cos|}^2_AT}, \frac{\sqrt{2}}{2}\Big\}w_A(T)\leq w_A(T), \end{align*} where ${|\cos|}_AT$ denotes the $A$-cosine of angle of $T$. Some upper bounds for the $A$-numerical radius of commutators, anticommutators, and products of semi-Hilbertian space operators are also given.