I am looking for characterizations of the adjoint operator of the infinitesimal generator of a $C_0$-semigoup in Hilbert spaces. All I could find in the literature is that if $(A, D(A))$ is the infinitesimal generator of a $C_0$ semigroup $S(t)$ in $X$ where $X$ is a reflexive Banach space then $(A^*, D(A^*))$ the adjoint of $(A, D(A))$ is the infinitesimal generator of a $C_0$-semigroup $S^*(t)$ which is the adjoint of $S(t)$, hence the characterizations are provided by Hille-Yosida and Lumer-Phillips theorems.
Are there any other characterizations of the generator $(A, D(A))$ or especially its adjoint $(A^*, D(A^*))$?
Thank you in advance.
