The Caratheodory-Fejer extension theorem

A construction of Carathéodory and Fejér 1] produces a function which is bounded and analytic in the unit disk with specified initial coefficients. An operator generalization of the construction is now obtained for application to the invariant subspace problem. A formal proof 2] of the existence of invariant subspaces is given by the theory of square summable power series 3] in its vector formulation 4]. But the justification of the formal argument requires a determination of extreme points of a convex set 5]. A solution is now given to an extension problem for convex decompositions which arises in connection with the Carathéodory-Fejér theorem. A necessary condition for an extreme point is obtained as an application. The condition is conjectured to be sufficient.