This paper concerns systems of the form. (x)over dot(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.