Proper Improvement of Well-Known Numerical Radius Inequalities and Their Applications

By Pintu Bhunia, Kallol Paul. Results in Mathematics, 2020

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New inequalities for the numerical radius of bounded linear operators defined on a complex Hilbert space H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {H}}$$\end{document} are given. In particular, it is established that if T is a bounded linear operator on a Hilbert space H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {H}}$$\end{document} then w2(T)≤min0≤α≤1αT∗T+(1-α)TT∗,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} w^2(T)\le \min _{0\le \alpha \le 1} \left\| \alpha T^*T +(1-\alpha )TT^* \right\| , \end{aligned}$$\end{document}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.

Subfields: Applied Mathematics, Computational Theory and Mathematics

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