Multiplicative perturbations of local C-semigroups

In this paper, we establish some left and right multiplicative perturbation theorems concerning local C-semigroups when the generator A of a perturbed local C-semigroup S(⋅) may not be densely defined and the perturbation operator B is a bounded linear operator from D(A)¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\overline {D(A)}$\end{document} into R(C) such that CB=BC on D(A)¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\overline {D(A)}$\end{document}, which can be applied to obtain some additive perturbation theorems for local C-semigroups in which B is a bounded linear operator from [D(A)] into R(C) such that CB=BC on D(A)¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\overline {D(A)}$\end{document}. We also show that the perturbations of a (local) C-semigroup S(⋅) are exponentially bounded (resp., norm continuous, locally Lipschitz continuous, or exponentially Lipschitz continuous) if S(⋅) is.