Analytic continuation of inner function-operators
Domingo Antonio Herrero · Pacific Journal of Mathematics · 1972
In his "Lectures on Invariant Subspaces", H. Helson has divided the study of the (closed) invariant subspaces of a unilateral shift of countable multiplicity N (regarded as the multiplication by z on Hi, the H z Hardy class of analytic functions in the unit disc D = {z: \z\ < 1} with values in a complex separable Hubert space K of dimension N) into two main sections: "Full-range subspaces" and "Analytic Range Functions".An invariant subspace ^& is a full-range subspace if it can be written as ^£ = UH 2 K , where U is an INNER FUNC-TION-OPERATOR, i.e, U(z) is a bounded analytic function on D with values in the set of all bounded linear operators in K whose nontangential strong limits U(e ix ) (these limits are well-defined a.e.) are unitary operators in K (a.e).Helson's book contains a study of the analytic properties of an inner function operator in the interior and on the boundary of D. In this article the properties of the analytic continuation of these functions outside D are studied; the results also include some information about the cyclic vectors of a CΌo-contraction in a Hubert space.