Let $X$ be a Banach space. The adjoint semigroup $\{T(t)^\prime:t\ge 0\}$ consisting of all adjoint operators $T(t)^\prime$ on the dual space $X^\prime$ is, in general, not strongly continuous where $\{T(t):t\ge 0\}$ is a strongly continuous semigroup on $X$.
My question: Suppose $\{T(z):z\in \Delta\}$ is a analytic semigroup on $X$. Is $\{T(z)^\prime:z\in \Delta\}$ (consisting of all adjoint operators $T(z)^\prime$ on the dual space $X^\prime$) an analytic semigroup? If so, what would be its generator?