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

Abstract

Let $A$ be a positive bounded linear operator on a complex Hilbert space $\mathcal{H}$ and $\mathbb{B}_{A}(\mathcal{H})$ 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 $T, S\in\mathbb{B}_{A}(\mathcal{H})$ 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 $$f_A(X,Y)=\ Y\ _A\sqrt{\omega_A^2(X)-\frac{\left \,\left\ \frac{X+X^{\sharp_A}}{2}\right\ _A^2-\left\ \frac{X-X^{\sharp_A}}{2i}\right\ _A^2\right }{2}}.$$ This covers and improves the well-known inequalities of Fong and Holbrook. 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 $X$.