Admissible observation operators for the right-shift semigroup

We give a characterization of infinite-time admissible observation operator s for the right-shift semigroup on L-2[0, infinity]. Our main result is tha t 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 parallel to C(sI - A)(-1) parallel to less than or equal to m/root Re s for all s in the open right half-plane. We derive this using F efferman's theorem on bounded mean oscillation and Hankel operators. This r esult solves a special case of a more general conjecture which says that th e same equivalence is true for any strongly continuous semigroup acting on a Hilbert space. For normal semigroups the conjecture is known to be true a nd then it is equivalent to the Carleson measure theorem. We derive some re lated results and partial results concerning the case when the signals are not scalar but with values in a Hilbert space.