Improved inequalities related to the $A$-numerical radius for commutators of operators

Let $A$ be a positive bounded linear operator on a complex Hilbert space $\mathcal{H}$ and $\mathcal{B}_{A}(\mathcal{H})$ be the subspace of all operators which admit $A$-adjoints operators. In this paper, we establish some inequalities involving the commutator and the anticommutator of operators in semi-Hilbert spaces, i.e. spaces generated by positive semidefinite sesquilinear forms. Mainly, among other inequalities, we prove that for $T, S\in\mathcal{B}_{A}(\mathcal{H})$ we have \begin{align*} \omega_A(TS \pm ST) \leq 2\sqrt{2}\min\Big\{f_A(T,S), f_A(S,T) \Big\}, \end{align*} where $$f_A(X,Y)=\|Y\|_A\sqrt{\omega_A^2(X)-\frac{\left|\,\left\|\frac{X+X^{\sharp_A}}{2}\right\|_A^2-\left\|\frac{X-X^{\sharp_A}}{2i}\right\|_A^2\right|}{2}}.$$ Here $\omega_A(\cdot)$ and $\|\cdot\|_A$ are the $A$-numerical radius and the $A$-operator seminorm of semi-Hilbert space operators, respectively and $X^{\sharp_A}$ denotes a distinguished $A$-adjoint operator of $X$.