Numerical Radius Inequalities for Several Operators

  • Omar Hirzallah
  • Fuad Kittaneh

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.
Published
2014-01-17
How to Cite
Hirzallah, O., & Kittaneh, F. (2014). Numerical Radius Inequalities for Several Operators. MATHEMATICA SCANDINAVICA, 114(1), 110-119. https://doi.org/10.7146/math.scand.a-16641
Section
Articles