  In the first part of this article we introduce the notion of a
backward-forward conditioning (BFC) system that generalises the notion of
zero-class admissibiliy introduced in [Xu,Liu,Yung]. 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 infinite-dimensional 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.
