Exact observability, square functions and spectral theory (Q765912)

First, the paper introduces the notion of Backward-Forward Conditioning (BFC) system that occurs only in situations where the underlying semigroup extends to a group (unless the spectrum contains a half-plane) and generalises the notion of zero-class admissibility introduced in [\textit{G. Q. Xu}, \textit{C. Liu} and \textit{S. P. Yung}, Syst. Control Lett. 57, No.~3, 222--227 (2008; Zbl 1130.93019)]. Next, by using certain weighted square function estimates, a sufficient condition for exact and final state observability in Banach spaces is presented, that is designed for infinite-dimensional output spaces and general strongly continuous semigroups. In the case of Hilbert space, a result for contraction semigroups without an analyticity condition on the semigroup is derived.