Numerical radius inequalities and estimation of zeros of polynomials

Abstract

Let $A$ be a bounded linear operator defined on a complex Hilbert space and let $|A|=(A^{\ast }A{)}^{1/2}$ be the positive square root of $A$. Among other refinements of the well known numerical radius inequality $w^2(A)\le \frac{1}{2}\Vert A^{\ast }A+A{A}^{\ast }\Vert$, 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 $$w^p(A)\le \frac{1}{\sqrt{2}}w(|A{|}^p+i|A^{\ast }{|}^p)\le \Vert A{\Vert }^p$$ for all $p\ge 1.$ Further, we derive new bounds for the zeros of complex polynomials.

16 pages

References in corpus (3)

• Furtherance of Numerical radius inequalities of Hilbert space operators
• Improved inequalities for the numerical radius via Cartesian decomposition
• Annular bounds for the zeros of a polynomial from companion matrix