A perturbation theorem for exponentially bounded \(C\)-semigroups (Q1343413)

Let \((X;||)\) be a Banach space. By \(B(X)\) we denote the set of all bounded linear operators in \(X\). Let \(C\in B(X)\) be injective. A strongly continuous family of bounded operators \(\{S(t)\); \(t\geq 0\}\) is called an exponentially bounded \(C\)-semigroup (hereinafter abbreviated to \(C\)-semigroup) on \(X\), if \(S(0)=C\), \(S(t)S(s) =S(t+s)C\), \(\forall t\), \(s\geq 0\), and \(|S(t) |\leq Me^{at}\), \(\forall t\geq 0\). The aim of this note is to derive a perturbation theorem for \(C\)-semigroups without requiring that \(R(C)\) be dense in \(X\). From the relationship between a \(C\)-semigroup \(S(t)\) on \(X\) and a \(C_0\)-semigroup \(T(t)\) on a subspace \(\Sigma\) of \(X\), and by the aid of the well-known perturbation theory for \(C_0\)-semigroups we get our perturbation theorem for \(C\)-semigroups.