Let Hℋ2 denote the
(separable) Hilbert space of all functions F(ei𝜃) defined on the unit circle
with values in the separable (usually infinite dimensional) Hilbert space
ℋ, and which are weakly in the Hardy class H2. For a closed subspace of
Hℋ2 “invariant” means invariant under the right shift operator. Such an
invariant subspace is said to be of full range if it is of the form 𝒰Hℋ2 , where
𝒰(ei𝜃) is a.e. a unitary operator on ℋ; i.e., an inner function. We show
that if ℋ is infinite dimensional there exists an uncountable family {ℳα}
of invariant subspaces of Hℋ2 of full range such that ℳα∩ℳβ= (0) if
α≠β.
