Universal elements for non-linear operators and their applications

We prove that under certain topological conditions on the set of universal elements of a continuous map T acting on a topological space X, that the direct sum T ⊕Mg is universal, where Mg is multiplication by a generating element of a compact topological group. We use this result to characterize R+-supercyclic operators and to show that whenever T is a supercyclic operator and z1, . . . , zn are pairwise different non-zero complex numbers, then the operator z1T ⊕···⊕znT is cyclic. The latter answers affirmatively a question of Bayart and Matheron.