Chan and Shapiro showed that each (non-trivial) translation operator f(z){mapping} Tλf(z+λ) acting on the Fréchet space of entire functions endowed with the topology of locally uniform convergence supports a universal function of exponential type zero. We show the existence of d-universal functions of exponential type zero for arbitrary finite tuples of pairwise distinct translation operators. We also show that every separable infinite-dimensional Fréchet space supports an arbitrarily large finite and commuting disjoint mixing collection of operators. When this space is a Banach space, it supports an arbitrarily large finite disjoint mixing collection of C 0-semigroups. We also provide an easy proof of the result of Salas that every infinite-dimensional Banach space supports arbitrarily large tuples of dual d-hypercyclic operators, and construct an example of a mixing Hilbert space operator T so that (T, T 2) is not d-mixing. © 2012 Elsevier Inc

The first and the third authors are supported in part by MICINN and FEDER, Project MTM2010-14909. The third author is also supported by Generalitat Valenciana, Project PROMETEO/2008/101. We would like to thank the referee whose careful reading and suggestions produced a great improvement in the presentation of the article.