Exact observability, square functions and spectral theory

In the first part of this article we introduce the notion of a backwardforward conditioning (BFC) system that generalises the notion of zero-class admissibiliy introduced in [21]. We can show that unless the spectum contains a halfplane, the BFC property occurs only in siutations where the underlying semigroup extends to a group. In a second part we present a sufficient condition for exact observability in Banach spaces that is designed for infinitedimensional output spaces and general strongly continuous semigroups. To obtain this we make use of certain weighted square function estimates. Specialising to the Hilbert space situation we obtain a result for contraction semigroups without an analyticity condition on the semigroup.