INVARIANT SUBSPACES OF THE DIRICHLET SHIFT AND PSEUDOCONTINUATIONS

In this paper we study extremal functions for invariant subspaces M of the Dirichlet shift, i.e., solutions phi of the extremal problem SUP{\f(n)(0)\/\\f\\D: f is-an-element-of M, f not-equal 0}. Here n is the smallest nonnegative integer such that the sup is positive. It is known that such a function phi generates M. We show that the derivative (zphi)' has a pseudocontinuation to the exterior disc. This pseudocontinuation is an analytic continuation exactly near those points of the unit circle where phi is bounded away from zero. We also show that the radial limit of the absolute value of an extremal function exists at every point of the unit circle. Some of our results are valid for all functions that are orthogonal to a nonzero invariant subspace.