No abstract

No abstract

a b s t r a c tWe give a sufficient condition for a smooth not necessarily injective function ψ : R → R ensuring that the composition operator C ψ :range. This generalizes in a special case the result of Kenessey and Wengenroth which gave a description of composition operators C ψ with closed range for smooth injective symbols ψ : R → R d .

In this paper we investigate the dynamical properties of weighted translation operators acting on the Schwartz space S(R) of rapidly decreasing functions, i.e., operators of the form T w : S(R) → S(R), f (•) → w(•) f (•+1). We characterize when those operators are hypercyclic, weakly mixing, mixing and chaotic. Several examples illustrate our results and show which of those classes are different.

We construct a continuous linear operator acting on the space of smooth functions on the real line without non-trivial invariant subspaces. This is a first example of such an operator acting on a Fréchet space without a continuous norm. The construction is based on the ideas due to C. Read who constructed a continuous operator without non-trivial invariant subspaces on the Banach space ℓ1.