On chaotic $C_0$-semigroups and infinitely regular hypercyclic vectors

A $C_0$-semigroup $\mathcal {T}=(T(t))_{t\geq 0}$ on a Banach space $X$ is called hypercyclic if there exists an element $x\in X$ such that $\{T(t)x; t\geq 0\}$ is dense in $X$. $\mathcal {T}$ is called chaotic if $\mathcal {T}$ is hypercyclic and the set of its periodic vectors is dense in $X$ as well. We show that a spectral condition introduced by Desch, Schappacher and Webb requiring many eigenvectors of the generator which depend analytically on the eigenvalues not only implies the chaoticity of the semigroup but the chaoticity of every $T(t), t>0$. Furthermore, we show that semigroups whose generators have compact resolvent are never chaotic. In a second part we prove the existence of hypercyclic vectors in $D(A^\infty )$ for a hypercyclic semigroup $\mathcal {T}$, where $A$ is its generator.