Exponentially Dichotomous Operators and Exponential Dichotomy of Evolution Equations on the Half-Line

To an evolution family on the half-line \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$ \mathcal{U} = (U(t, s))_{t\geq s\geq 0} $$ \end{document} of bounded operators on a Banach space X we associate operators IX and IZ related to the integral equation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$ u(t) = U(t, s)u(s) + \int^{t}_{s} U(t, \xi) f (\xi)d\xi $$ \end{document} and a closed subspace Z of X. We characterize the exponential dichotomy of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$ \mathcal{U} $$ \end{document} by the exponential dichotomy and the quasi-exponential dichotomy of the operators X we associate operators IX and IZ, respectively.