Universality of weighted composition operators on L2([0, 1]) and Sobolev spaces

It is shown that a class of composition operators Cφ has the property that for every λ in the interior of the spectrum of Cφ the operator U = Cφ − λId 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 a generalization, weighted composition operators on the L2 and Sobolev spaces of the unit interval are shown to have the same property and thus a complete knowledge of their minimal invariant subspaces would imply a solution to the invariant subspace problem for Hilbert space. Moreover, a generalization of sufficient conditions for an operator to be universal is obtained. Cyclicity and non-cyclicity results for a certain class of weights and composition functions are also proved.