Numerical Radius Inequalities for Nonlinear Operators in Hilbert Spaces

In this paper, the numerical radius of nonlinear operators in Hilbert spaces is studied. First, the relationship between the spectral radius and the numerical radius of nonlinear operators is given. Then, the famous inequality 12‖T‖≤w(T)≤‖T‖\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{1}{2}\Vert T\Vert \le w(T)\le \Vert T\Vert $$\end{document} and inclusion σ(A-1B)⊆W(B)¯W(A)¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma (A^{-1}B)\subseteq \frac{\overline{W(B)}}{\overline{W(A)}}$$\end{document} of bounded linear operators are generalized to the case of certain nonlinear operators, where w(·)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w(\cdot )$$\end{document}, ‖·‖\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert \cdot \Vert $$\end{document} and σ(·)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma (\cdot )$$\end{document} are the numerical radius, the usual operator norm and the spectrum, respectively. Finally, some numerical radius inequalities for nonlinear operator matrices are obtained.