General necessary condition for exact observability

Suppose A generates an exponentially stable strongly continuous semigroup on the Hilbert space X,Y is another Hilbert space, and C:D(A)→Y is an admissible observation operator for this semigroup. (This means that to any initial state in X we can associate an output function in L²([0,∞), Y).) This paper gives a necessary condition for the exact observability of the system defined by A and C. This condition, called (E), is related to the Hautus Lemma from finite dimensional systems theory. It is an estimate in terms of the operators A and C alone (in particular, it makes no reference to the semigroup). This paper shows that (E) implies approximate observability and, if A is bounded, it implies exact observability. This paper conjectures that (E) is in fact equivalent to exact observability. The paper also shows that for diagonal semigroups, (E) takes on a very simple form, and applies the results to sequences of complex exponential functions.