Every inner operator function \(\vartheta\), defined in \(D=\{| z| <1\}\) and valued in the space of bounded linear operators on a fixed separable Hilbert space E, determines a co-invariant subspace H of the operator of multiplication by z in the E-valued Hardy space \(H^ 2_ E: H=H^ 2_ E\theta \vartheta H^ 2_ E\). [For general references see: H. Helson, Lectures on invariant subspaces (1964; Zbl 0119.113)]. If U is a unitary operator in E and \(t\in \partial D\), let \(\gamma_ tE\) denote the closure of the set of all functions of the form \(\gamma_ t(z)a\), \(a\in E\), lying in H \((\gamma_ t(z)=(1-z\bar t)^{-1}[I- \vartheta (z)U^*]\), I - the identity mapping in E). Let \(T_ U\) be the set of all \(t\in \partial D\) such that \(\gamma_ tE\neq \{0\}\) and let \(G_ U=\{\gamma_ tE:t\in T_ U\}.\)
The main result of the paper is: Let the operator \(I-\vartheta (z)U^*\) have a bounded inverse for every \(z\in D\). If \((1-r)^{- 1}Re\{[I+\vartheta (rt)U^*][I-\vartheta (rt)U^*]^{-1}\}\) is uniformly bounded in r, \(0\leq r<1\), for all \(t\in \partial D\) except for a countable set, then the family \(G_ U\) is orthogonal and complete in H. This generalizes an analogous result of D. N. Clark in the scalar case dim E\(=1\) [J. Anal. Math. 25, 169-191 (1972; Zbl 0252.47010)].