A Numerical Radius Inequality for Hilbert–Schmidt Operators

摘要

Abstract In this paper, we prove that for every Hilbert–Schmidt operator T which acts on a Hilbert space r(T)≤(2-2)(‖T‖HS2+ρ(T)2),documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$begin{aligned} r(T)le sqrt{(2-sqrt{2})(Vert TVert _{HS}^2+rho (T)^2)}, end{aligned}$$end{document}where ρ(·)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$rho (cdot )$$end{document}, r(·)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$r(cdot )$$end{document} and ‖·‖HSdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$Vert cdot Vert _{HS}$$end{document} denote spectral radius, numerical radius, and Hilbert–Schmidt norm, respectively.