Counterexamples Concerning Observation Operators for <i>C</i>₀ -Semigroups

Abstract

This paper concerns systems of the form $\dot x$(t)=Ax(t), y(t)=Cx(t), where A generates a C0 -semigroup. Two conjectures which were posed in 1991 and 1994 are shown not to hold. The first conjecture (by G. Weiss) states that if the range of C is one-dimensional, then C is admissible if and only if a certain resolvent estimate holds. The second conjecture (by D. Russell and G. Weiss) states that a system is exactly observable if and only if a test similar to the Hautus test for finite-dimensional systems holds. The C0 -semigroup in both counterexamples is analytic and possesses a basis of eigenfunctions. Using the (A,C)-pair from the second counterexample, we construct a generator Ae on a Hilbert space such that (sI-Ae) is uniformly left-invertible, but its semigroup does not have this property.