Numerical radius inequalities for Hilbert space operators
Volume 168 / 2005
Studia Mathematica 168 (2005), 73-80
MSC: 47A12, 47A30, 47A63, 47B47.
DOI: 10.4064/sm168-1-5
Abstract
It is shown that if $A$ is a bounded linear operator on a complex Hilbert space, then $$ {1 \over 4}\| {A^* A + AA^* } \| \le ( {w(A )} )^2 \le {1 \over 2}\| {A^* A + AA^* }\| , $$ where $w(\cdot )$ and $\| \cdot \| $ are the numerical radius and the usual operator norm, respectively. These inequalities lead to a considerable improvement of the well known inequalities $$ {1 \over 2}\| A \| \le w( A ) \le \| A \| . $$ Numerical radius inequalities for products and commutators of operators are also obtained.
