The Kato square root problem for higher order elliptic operators and systems on $ \Bbb R^n $

We prove the Kato conjecture for elliptic operators and N×N-systems in divergence form of arbitrary order 2m on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $ \Bbb R^n $\end{document}. More precisely, we assume the coefficients to be bounded measurable and the ellipticity is taken in the sense of a Gårding inequality. We identify the domain of their square roots as the natural Sobolev space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $ H^m(\Bbb R^n,\Bbb C^N) $\end{document}. We also make some remarks on the relation between various ellipticity conditions and Gårding inequality.