Vol. 27, No. 3, 1968

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Disjoint invariant subspaces

Malcolm Jay Sherman

Vol. 27 (1968), No. 3, 619–620

Let H2 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 , and which are weakly in the Hardy class H2. For a closed subspace of H2 “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 𝒰H2 , where 𝒰(ei𝜃) is a.e. a unitary operator on ; i.e., an inner function. We show that if is infinite dimensional there exists an uncountable family {ℳα} of invariant subspaces of H2 of full range such that α ∩ℳβ = (0) if αβ.

Mathematical Subject Classification
Primary: 47.35
Received: 15 November 1967
Published: 1 December 1968
Malcolm Jay Sherman