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.

Automated summary

This article characterizes linear fractional composition operators with universal translates on different Hilbert spaces, providing strong characterizations of minimal invariant subspaces and eigenvectors, offering an alternative approach to the invariant subspace problem.