An operator on a Fréchet space with no common invariant subspace with its inverse (Q801521)

The main result of this paper is the construction of a Fréchet-Montel space F of formal Laurent series \(\sum^{\infty}_{-\infty}a_ nz^ n\) in which \((z_ n)^{\infty}_{-\infty}\) is a Schauder basis, such that the operators of formal multiplication by z and \(z^{-1}\) are continuous on F and have no common invariant subspace. This answers in the negative the problem of existence of hyperinvariant subspaces for operators on general Fréchet spaces [the problem for Banach spaces remains open]. The strong dual of the above Fréchet space F also yields an example of an abelian complete locally convex algebra, with unit, which does not have any nontrivial closed ideal.