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

Abstract

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*} ω_A(TS \pm ST) \leq 2\sqrt{2}\min\Big\{f_A(T,S), f_A(S,T) \Big\}, \end{align*} where $$f_A(X,Y)=\Vert Y{\Vert }_A\sqrt{{\omega }_A^2(X)-\frac{\left|{\Vert \frac{X+X^{{♯}_A}}{2}\Vert }_A^2-{\Vert \frac{X-X^{{♯}_A}}{2i}\Vert }_A^2\right|}{2}}.$$ Here ${\omega }_A(\cdot )$ and $\Vert \cdot {\Vert }_A$ are the $A$-numerical radius and the $A$-operator seminorm of semi-Hilbert space operators, respectively and $X^{{♯}_A}$ denotes a distinguished $A$-adjoint operator of $X$.

References in corpus (1)

• Refinement of seminorm and numerical radius inequalities of semi-Hilbertian space operators