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This paper will study the weak and strong stabilizability problem of linear control systems on Hilbert space. It will be shown that sufficient conditions for stabilizability can be obtained by using a decomposition theorem in the structure theory of Hilbert space operators. The basic idea is "trivializing" the unitary "part" of a semigroup of bonded linear operators by means of a suitable "feedback" perturbation operator Controllability will not be involved in this process. However, it will be seen that further sufficient conditions as well as necessary conditions will be obtained with the aid of controllability. Extensions as well as limitatations of the familiar finite-dimensional results will also be discussed.