Necessary conditions for the exact observability of systems on Hilbert spaces

In this paper, we introduce the zero class of admissible observation operators, for which the norm of the output map tends to zero as the time tends to zero. We show that for such an observation operator, exact observability implies that the semigroup of the system is bounded from below. In addition, if the semigroup generator has no residual spectrum, then the semigroup can be extended to a C0C0 group.