Functional calculus, variational methods and Liapunov's theorem

Given the generator −A of a holomorphic semigroup on a Hilbert space H, we show that A is associated with a closed form if and only if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $A+w\in BIP(H)$\end{document} for some \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $w\in\Bbb{R}$\end{document}. Under this condition we also show that Liapunov's classical theorem is true, in the linear as well as the semilinear case.