Exact observability, square functions and spectral theory

Abstract. 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 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.

• Type: Journal
• Year: 2011
• Where: CORR
• Authors: Bernhard Hermann Haak, El-Maati Ouhabaz
• Keywords: Certain Weighted Square, CORR 2011, Education, Infinite-dimensional Output Spaces, Semigroup