Strong stabilizability of linear “contractive” systems on Hilbert space, that is those systems denoted by $(A,B)$ and described by the equation $\dot x = Ax + Bu$, in which A generates a semigroup of Hilbert space contraction operators, is studied. Necessary and sufficient conditions are given, which are shown to depend on controllability of the system $(A,B)$, and that of $(A^ * ,B)$ also. Our technique is based on some rather simple properties of invariant and reducing subspaces of Hilbert space operators, and on a canonical decomposition of contraction semigroups due to B. Sz-Nagy and C. Foias.