Weak admissibility does not imply admissibility for analytic semigroups

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 ∈ U the vector Bu defines an admissible control operator. The other conjecture says that B is an admissible control operator if a certain resolvent estimate is satisfied. The examples given in this paper show that even for analytic semigroups the conjectures do not hold. In the last section we construct a semigroup example showing that the first estimate in the Hille-Yosida theorem is not sufficient to conclude boundedness of the semigroup.