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

Abstract

Let $textit{A}$ be a positive bounded linear operator on a complex Hilbert space $mathcal{H}$ and $B_{A}(mathcal{H})$ be the subspace of all operators which admit $textit{A}$-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 $T,Sin B_{A}(mathcal{H})$ we have $w_{A}(TSpm ST)leq 2sqrt{2}minleft { f_{A}(T,S),f_{A}(S,T) right },$ where $ f_{A}(X,Y)=left | Y right |_{A}sqrt{w_{A}^{2}(X)-frac{left | left |frac{X+X^{sharp}A}{2}right |_{A}^{2} right |}{2}-frac{left | left |frac{X-X^{sharp}A}{2}right |_{A}^{2} right |}{2i}}.$ This covers and improves the well-known inequalities of Fong and Holbrook. Here $w_{A}(.) and left | . right |_{A}$ are the $textit{A}$-numerical radius and the $textit{A}$-operator seminorm of semi-Hilbert space operators, respectively and $X^{sharp A}$ denotes a distinguished $textit{A}$-adjoint operator of $textit{X}$ .