Numerical radius inequalities and estimation of zeros of polynomials

Abstract

Abstract Let A be a bounded linear operator defined on a complex Hilbert space and let | A | = ( A * ⁢ A ) 1 2 {|A|=(A^{*}A)^{\frac{1}{2}}} . Among other refinements of the well-known numerical radius inequality w 2 ⁢ ( A ) ≤ 1 2 ⁢ ∥ A * ⁢ A + A ⁢ A * ∥ {w^{2}(A)\leq\frac{1}{2}\|A^{*}A+AA^{*}\|} , we show that w 2 ⁢ ( A ) ≤ 1 4 ⁢ w 2 ⁢ ( | A ⁢ | + i | ⁢ A * | ) + 1 8 ⁢ ∥ | A | 2 + | A * | 2 ∥ + 1 4 ⁢ w ⁢ ( | A | ⁢ | A * | ) ≤ 1 2 ⁢ ∥ A * ⁢ A + A ⁢ A * ∥ . w^{2}(A)\leq\frac{1}{4}w^{2}(|A|+{\rm i}|A^{*}|)+\frac{1}{8}\||A|^{2}+|A^{*}|^% {2}\|+\frac{1}{4}w(|A||A^{*}|)\leq\frac{1}{2}\|A^{*}A+AA^{*}\|. Also, we develop inequalities involving the numerical radius and the spectral radius for the sum of the product operators, from which we derive the inequalities w p ⁢ ( A ) ≤ 1 2 ⁢ w ⁢ ( | A | p + i ⁢ | A * | p ) ≤ ∥ A ∥ p w^{p}(A)\leq\frac{1}{\sqrt{2}}w(|A|^{p}+{\rm i}|A^{*}|^{p})\leq\|A\|^{p} for all p ≥ 1 {p\geq 1} . Further, we derive new bounds for the zeros of complex polynomials.