Exact controllability of C0-groups with one-dimensional input operators

Assume that A generates a C-0-group T(t) on the complex, separable Hilbert space H, and that b epsilon D(A*)' is an admissible control operator for the semigroup T(t). It is known that exact controllability of the system defined by A and b implies that every element of the spectrum of A is an eigenvalue. We develop equivalent conditions for exact controllability of the system defined by A and b. These conditions are given in terms of the eigenvalues and eigenvectors of A and the control operator b. Also necessary conditions are included. A necessary condition is that one over the eigenvalues is a sequence in l(1+epsilon), epsilon > 0. If additionally, A is a diagonal operator, then we prove that the conjecture of Russell and Weiss [12] holds.