Numerical radius inequalities involving commutators of $G_{1}$ operators

Abstract

We prove numerical radius inequalities involving commutators of $G_{1}$ operators and certain analytic functions. Among other inequalities, it is shown that if $A$ and $X$ are bounded linear operators on a complex Hilbert space, then \begin{equation*} w(f(A)X+X\bar{f}(A))\leq {\frac{2}{d_{A}^{2}}}w(X-AXA^{\ast }), \end{equation*} where $A$ is a $G_{1}$ operator with $\sigma (A)\subset \mathbb{D}$ and $f$ is analytic on the unit disk $\mathbb{D}$ such that $\textrm{{Re}}(f)>0$ and $f(0)=1$.