Admissible observation operators for linear semigroups

Consider a semigroup T on a Banach space X and a (possibly unbounded) operator C densely defined in X, with values in another Banach space. We give some necessary as well as some sufficient conditions for C to be an admissible observation operator for T, i.e., any finite segment of the output function y(t)=C T _t x, t≧0, should be in L ^p and should depend continuously on the initial state x. Our approach is to start from a description of the map which takes initial states into output functions in terms of a functional equation. We also introduce an extension of C which permits a pointwise interpretation of y(t)=C T _t x, even if the trajectory of x is not in the domain of C.