Relative demicompactness properties for exponentially founded $C$-semigroups

Let $C$ be an invertible bounded linear operator on a Banach space $X$. In this paper, we use the concept of relative demicompactness in order to investigate some properties for an exponentially bounded $C$-semigroup $(T(t))_{t\geq0}$. More precisely, we prove that the relative demicompactness of $T(t)$ for some positive values of $t$ is equivalent to the relative demicompactness of $C-A$ where $A$ is the infinitesimal generator of $(T(t))_{t\geq0}$. In addition, we study the relative demicompactness of the resolvent. Finally, we present some conditions on exponentially bounded $C$-semigroups in Hilbert space guaranteeing the relative demicompactness of $AC$.