Further improvements of generalized numerical radius inequalities for Hilbert space operators

Abstract

Several new improvements of the $A$-numerical radius inequalities for operators acting on a semi-Hilbert space, i.e., a space generated by a positive operator $A$, are proved. In particular, among other inequalities, we show that \begin{align*} \frac{1}{4}\|T^{\sharp_A} T+TT^{\sharp_A}\|_A \leq\frac{1}{4}\Big(2ω_A^2(T)+γ(T)\Big) \leq ω_A^2(T), \end{align*} where $$\gamma (T)=\sqrt{{\left(\Vert {\Re }_A(T){\Vert }_A^2-\Vert {\Im }_A(T){\Vert }_A^2\right)}^2+4\Vert {\Re }_A(T){\Im }_A(T){\Vert }_A^2}.$$ Here ${\omega }_A(X)$ and $\Vert X{\Vert }_A$ denote respectively the $A$-numerical radius and the $A$-seminorm of an operator $X$. Also, ${\Re }_A(T):=\frac{T+T^{{♯}_A}}{2}$ and ${\Im }_A(T):=\frac{T-T^{{♯}_A}}{2i}$, where $T^{{♯}_A}$ is a distinguished $A$-adjoint operator of $T$. Further, some new refinements of the triangle inequality related to $\Vert \cdot {\Vert }_A$ are established.