Numerical Radius Inequalities for Several Operators

Abstract

Let $A$, $B$, $X$, and $A_{1},\dots,A_{2n}$ be bounded linear operators on a complex Hilbert space. It is shown that \[ w\Bigl(\sum_{k=1}^{2n-1}A_{k+1}^{\ast}XA_{k}+A_{1}^{\ast}XA_{2n}\Bigr) \leq 2\Bigl( \sum_{k=1}^{n}\Vert A_{2k-1}\Vert^{2}\Bigr)^{1/2}\Bigl(\sum_{k=1}^{n}\left\Vert A_{2k}\right\Vert^{2}\Bigr)^{1/2}w(X) \] and \[ w(AB\pm BA)\leq 2\sqrt{2}\,\Vert B\Vert \sqrt{w^{2}(A)-\frac{\vert \Vert {\operatorname{Re} A}\Vert^{2}-\Vert {\operatorname{Im} A}\Vert^{2}\vert}{2}}, \] where $w(\cdot)$ and $\left\Vert \cdot \right\Vert$ are the numerical radius and the usual operator norm, respectively. These inequalities generalize and refine some earlier results of Fong and Holbrook. Some applications of our results are given.