Exact observability, square functions and spectral theory

TL;DR

This paper introduces the concept of backward-forward conditioning systems, explores their spectral properties, and provides new criteria for exact observability in Banach and Hilbert spaces using weighted square function estimates.

Contribution

It generalizes zero-class admissibility to BFC systems and offers a novel sufficient condition for exact observability in infinite-dimensional settings.

Findings

BFC systems occur only when the semigroup extends to a group unless the spectrum contains a half-plane.

A new sufficient condition for exact observability in Banach spaces using weighted square functions.

Results for contraction semigroups in Hilbert spaces without requiring analyticity.

Abstract

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.