Let $A$ be a bounded linear operator defined on a complex Hilbert space and let $|A|=(A^*A)^{ 1/2}$ be the positive square root of $A$. Among other refinements of the well known numerical radius inequality $w^2(A)\leq \frac12 \|A^*A+AA^*\|$, we show that \begin{ eqnarray*} w^2(A)&\leq&\frac{ 1}{ 4} w^2 \left(|A|+i|A^*|\right)+\frac{ 1}{ 8}\left\||A|^2+|A^*|^2\right \|+\frac{ 1}{ 4}w\left(|A||A^*|\right) &\leq& \frac12 \|A^*A+AA^*\|. \end{ eqnarray*} Also, we develop inequalities involving numerical radius and spectral radius for the sum of the product operators, from which we derive the following inequalities $$ w^p(A) \leq \frac{ 1}{ \sqrt{ 2} } w(|A|^p+i|A^*|^p )\leq \|A\|^p$$ for all $p\geq 1.$ Further, we derive new bounds for the zeros of complex polynomials.
Categories: math.FA