Counterexamples Concerning Observation Operators for C₀ -Semigroups

This paper concerns systems of the form $\dot{x}(t) = Ax(t)$, $y(t) = Cx(t)$, where $A$ generates a $C_0$-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 $C_0$-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 $A_e$ on a Hilbert space such that $(sI -A_e)$ is uniformly left-invertible, but its semigroup does not have this property.