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 HK2, the H2 Hardy class of analytic functions in the unit disc
D = {z : |z| < 1} with values in a complex separable Hilbert space K of
dimension N) into two main sections: “Full-range subspaces” and “Analytic Range
Functions”.
An invariant subspace ℳ is a full-range subspace if it can be written as
ℳ = UHK2 , where U is an INNER FUNCTION-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(eix) (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 C00-contraction in a Hilbert
space.
