Admissible observation operators for the right-shift semigroup

We give a characterization of infinite-time admissible observation operators for the right-shift semigroup on L²[0, ∞). Our main result is that if A is the generator of this semigroup and C is the observation operator, mapping D(A) into the complex numbers, then C is infinite-time admissible if and only if ∥C(sI-A)^-1∥≤M/√Re s for all s in the open right half-plane. We derive this using Fefferman's theorem on bounded mean oscillation and Hankel operators. This result solves a special case of a more general conjecture which says that the same equivalence is true for any strongly continuous semigroup acting on a Hilbert space. For normal semigroups the conjecture is known to be true and then it is equivalent to the Carleson measure theorem. We derive some related results and partial results concerning the case when the signals are not scalar but with values in a Hilbert space.