Disproof of an admissibility conjecture of Weiss

Two conjectures on admissible control operators by George Weiss are disproved in this paper. One conjecture says that an operator $B$ defined on an infinite-dimensional Hilbert space $U$ is an admissible control operator if for every element $u \in U$ the vector $Bu$ defines an admissible control operator. The other conjecture says that $B$ is an admissible control operator if a certain resolvent condition is satisfied. The examples given in this paper show that even for analytic semigroups the conjectures do not hold. In the last section we show that this example leads to a semigroup example showing that the first estimate in the Hille-Yosida Theorem is not sufficient to conclude boundedness of the semigroup.