UNIVERSAL COMPOSITION OPERATORS

A Hilbert space operator U is called 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 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 C-phi f = f circle phi f that have universal translates on both the classical Hardy spaces H-2(C+) and H-2(D) of the half-plane and the unit disk, respectively. The new example here is the composition operator on H-2 (D) with affine symbol phi a (z) = az + (1-a) for 0 < a < 1. This leads to strong characterizations of minimal invariant subspaces and eigenvectors of C-phi a and offers an alternative approach to the ISP.