Advanced Refinements of Numerical Radius Inequalities (2011.08443v4)

Abstract: We prove several numerical radius inequalities for linear operators in Hilbert spaces. It is shown, among other inequalities, that if $A$ is a bounded linear operator on a complex Hilbert space, then [\omega \left( A \right)\le \frac{1}{2}\sqrt{\left| {{\left| A \right|}^{2}}+{{\left| {{A}^{*}} \right|}^{2}} \right|+\left| \left| A \right|\left| {{A}^{*}} \right|+\left| {{A}^{*}} \right|\left| A \right| \right|},] where $\omega \left( A \right)$, $\left| A \right|$, and $\left| A \right|$ are the numerical radius, the usual operator norm, and the absolute value of $A$, respectively. This inequality provides a refinement of an earlier numerical radius inequality due to Kittaneh, namely, [\omega \left( A \right)\le \frac{1}{2}\left( \left| A \right|+{{\left| {{A}^{2}} \right|}^{\frac{1}{2}}} \right).] Some related inequalities are also discussed.