No abstract

No abstract

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 −λ Id is universal in the sense of Caradus; i.e., every Hilbert space operator has a non-zero multiple similar to the restriction of U to an invariant subspace. As an application, composition operators induced by power mappings on the L 2 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 Müntz-like theorem is proved: this is used to show that generalized polynomials are cyclic vectors for these operators in the L 2 case but not in the Sobolev case.