Constant characteristic functions and homogeneous operators

To a completely non unitary (cnu) contraction T on a separable Hilbert space H, Nagy and Foias (cf. [8]) associate a contraction valued holomorphic function ΘT on the open unit disk ID such that ΘT (0) is a pure contraction. This is called the characterstic function of the operator T. Conversely, given any such holomorphic function Θ on ID, there is a completely nonunitary contraction TΘ whos