Proper improvement of well-known numerical radius inequalities and their applications

Abstract

New inequalities for the numerical radius of bounded linear operators defined on a complex Hilbert space $\mathcal{H}$ are given. In particular, it is established that if $T$ is a bounded linear operator on a Hilbert space $\mathcal{H}$ then \[ w^2(T)\leq \min_{0\leq α\leq 1} \left \| αT^*T +(1-α)TT^* \right \|,\] where $w(T)$ is the numerical radius of $T.$ The inequalities obtained here are non-trivial improvement of the well-known numerical radius inequalities. As an application we estimate bounds for the zeros of a complex monic polynomial.

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• Numerical radius inequalities for tensor product of operators
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• New semi-norm of semi-Hilbertian space operators and its application