Operator on the space of rapidly decreasing functions with all non-zero vectors hypercyclic

We construct a continuous linear operator T on the Schwartz space S(Rd) of rapidly decreasing functions such that each non-zero orbit of T is dense. The construction is inspired by the work of C. Read on similar operators on the space ℓ1. The construction, due to the structure of a Fréchet space, can be made significantly simpler than the original construction of Read.