On the Hautus test for exponentially stable C₀-groups

For finite-dimensional systems the Hautus test is a well-known and easy checkable condition for observability. Russell and Weiss [SIAM J. Control Optim, 32 (1994), pp. 1--32] suggested an infinite-dimensional version of the Hautus test, which is necessary for exact observability and sufficient for approximate observability of exponentially stable systems. In this paper it is shown that this Hautus test is sufficient for exact observability of certain exponentially stable systems generated by a $C_0$-group, and it is proved that the Hautus test is in general not sufficient for approximate observability of strongly stable systems even if the system is modeled by a contraction semigroup and the observation operator is bounded.