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It is shown that a large class of weighted shift operators T have the property that for every λ in the interior of the spectrum of T the operator U = T − λId is universal in the sense of Caradus; i.e., every Hilbert space operator has a nonzero multiple similar to the restriction of U to an invariant subspace. As an application, composition operators induced by power mappings on the L2 and Sobolev spaces of the unit interval are shown to have the same property: thus a complete knowledge of their minimal invariant subspaces would imply a solution to the invariant subspace problem for Hilbert space. A new Muntz-like theorem is proved: this is used to show that generalized polynomials are cyclic vectors for these operators in the L2 case but not in the Sobolev case.

Universal shifts and composition operators