Universal composition operators

A Hilbert space operator U is called \textit{universal} (in the sense of Rota) if every Hilbert space operator is similar to a multiple of U restricted to one of its invariant subspaces. It follows that the \textit{invariant subspace problem} for Hilbert spaces is equivalent to the statement that all minimal invariant subspaces for U are one dimensional. In this article we characterize all linear fractional composition operators Cϕf=f∘ϕ that have universal translates on both the classical Hardy spaces H2(C+) and H2(D) of the half-plane and the unit disk, respectively. The new example here is the composition operator on H2(D) with affine symbol ϕa(z)=az+(1−a) for $0