  We derive converse Lyapunov theorems for input-to-state stability (ISS) of
linear infinite-dimensional analytic systems. We show that input-to-state
stability of a linear system does not imply existence of a coercive quadratic
ISS Lyapunov function, even if the input operator is bounded. If, however, the
semigroup is similar to a contraction semigroup on a Hilbert space, then a
quadratic ISS Lyapunov function always exists for any input operator that is
bounded, or more generally, $p$-admissible with $p<2$. The constructions are
semi-explicit and, in the case of self-adjoint generators, coincide with the
canonical Lyapunov function being the norm squared. Finally, we construct a
family of non-coercive ISS Lyapunov functions for analytic ISS systems under
weaker assumptions on $B$.
