Disjoint invariant subspaces

Abstract

Let .%, denote the (separable) Hubert space of all functions F(e i ) defined on the unit circle with values in the separable (usually infinite dimensional) Hubert space %?', and which are weakly in the Hardy class H 2 . For a closed subspace of H%, "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 ^H^, where ^(e ) is a.e. a unitary operator on %f\ i.e., an inner function. We show that if Sf is infinite dimensional there exists an uncountable family {^*C} of invariant subspaces of .^ of full range such that ^C -^ = (0) if a.