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In the first part of this article we introduce the notion of abackward-forward conditioning (BFC) system that generalises the notion ofzero-class admissibiliy introduced in [Xu,Liu,Yung]. We can show that unlessthe spectum contains a halfplane, the BFC property occurs only in siutationswhere the underlying semigroup extends to a group. In a second part we presenta sufficient condition for exact observability in Banach spaces that isdesigned for infinite-dimensional output spaces and general strongly continuoussemigroups. To obtain this we make use of certain weighted square functionestimates. Specialising to the Hilbert space situation we obtain a result forcontraction semigroups without an analyticity condition on the semigroup.

Exact observability, square functions and spectral theory