On controllability of diagonal systems with one-dimensional input space

This paper deals with diagonal systems on a Hilbert space with a onedimensional input space and a (possibly unbounded) input operator. A priori it is not assumed that the input operator is admissible. Necessary and sufficient conditions for different notions of controllability such as null controllability and exact controllability are presented. These conditions, which are given in terms of the eigenvalues of the diagonal operator and in terms of the input operator, are linked with the theory of interpolation in Hardy spaces.