The Multiplicative Properties Of Numerical Radius

The multiplicative properties of the numerical radius is studied by methods of the polar decomposition.First of all,the multiplicative prop-erties of the numerical radius for normal operator is proved.Secondly,the multiplicative properties of the numerical radius for normal operator is extended to the case of quasi-normal operator.Furthermore,the multi-plicative properties of the numerical radius of the product of two operators is extended to the multiplicative properties of the numerical radius for the product of three operators.