Abstract
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.
Highlights
While Enflo constructed an operator on an artificial Banach space, Read was able to build his counterexample on l1
The first author was able to adapt the methods of Read and constructed counterexamples to the invariant subspace problem for many classical Fréchet spaces including the space of holomorphic functions on the unit disc H(D) and the space of rapidly decreasing sequences s
In this paper we construct an operator without non-trivial invariant subspaces on the space of smooth functions on the real line C∞(R) with the usual topology of uniform convergence of functions and their derivatives on compact sets
Summary
The first author was able to adapt the methods of Read and constructed counterexamples to the invariant subspace problem for many classical Fréchet spaces including the space of holomorphic functions on the unit disc H(D) and the space of rapidly decreasing sequences s (see [5], see [6] for an operator without non-trivial invariant subsets on s). In this paper we construct an operator without non-trivial invariant subspaces on the space of smooth functions on the real line C∞(R) with the usual topology of uniform convergence of functions and their derivatives on compact sets This space does not possess a continuous norm and is isomorphic to the countable product of the space of rapidly decreasing sequences sN which plays an important role in the theory of nuclear Fréchet spaces because of the celebrated Komura-Komura theorem
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