Multiplicative perturbations of local C-semigroups

Abstract

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 \(\overline {D(A)}\) into R(C) such that C B=B C on \(\overline {D(A)}\), 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 C B=B C on \(\overline {D(A)}\). 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.