  We study an infinite-dimensional
  Ornstein-Uhlenbeck process $(X_t)$ in a given
  Hilbert space $H$. This is driven by a cylindrical symmetric L\'evy process
without a Gaussian component and taking values in a Hilbert space $U$ which
usually contains $H$. We give if and only if conditions under which $X_t$ takes
values in $H$ for some $t>0$ or for all $t>0$. Moreover, we prove
irreducibility for $(X_t)$.
