Vol. 40, No. 2, 1972

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Analytic continuation of inner function-operators

Domingo Antonio Herrero

Vol. 40 (1972), No. 2, 327–340

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 is a full-range subspace if it can be written as = 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.

Mathematical Subject Classification 2000
Primary: 47B35
Secondary: 30A78
Received: 10 November 1970
Revised: 21 June 1971
Published: 1 February 1972
Domingo Antonio Herrero