Invariant subspace problem for classical spaces of functions

We construct continuous linear operators without non-trivial invariant subspaces on several classical non-Banach spaces of analysis: the Schwartz space of rapidly decreasing functions, the space of smooth functions on a compact smooth manifold, the space of holomorphic functions on the unit disc or on an arbitrary polydisc, the space of entire functions on Cd for d⩾2. As these are reflexive spaces, the result gives an analogous statement for the dual spaces, e.g. the space of tempered distributions. The construction works with Köthe sequence spaces and is based on methods developed by Read to construct his famous example on the space l1.