In the literature, is there any paper or research investigating [the invariant subspace problem](https://en.wikipedia.org/wiki/Invariant_subspace_problem) with consideration of differential operators acting on an appropriate Sobolev space?In particular is there any research investigating the "Invariant Subspace Conjecture" via derivational operator arising from vector fields which generate a minimal dynamical system? Namely those vector fields without proper compact invariant sets or vector fields whose all trajectories are dense.


A vector field $X$ on a manifold $M$ defines a derivation on the space of smooth functions. What is a relevant Hilbert space of functions invariant under the $X$-derivation? (A kind of infinite order Sobolov space.) Let's denote this Hilbert space by $H^{\infty}(M)$. Is there any research on the invariant subspaces of the derivations on the infinite order Sobolev space?
Of course there is an obvious $1$-dimensional invariant subspace of constant functions, but I guess that this trivial space can be ignored with some Hilbert space techniques.


As a particular case:


Is there a smooth vector field $X$ on the torus $\mathbb{T}^2$ which is tangent to a Kronecker foliation $dx+\theta dy=0$ for some irrational number $\theta$, with an invariant subspace not degenerated to the trivial $1$-dimensional subspace?



A possible dynamical approach to the "Invariant Subspace Problem"
The following papers are related to your question:


1. A. Atzmon, A model for operators with cyclic adjoint, *Integral
equations and Operator theory*, 10(1987), 153-163.
2. A. Atzmon, Nuclear Frechet spaces of entire functions with transitive
differentiation, *J. Analyse Math.* 60(1993), 1-19.


A.Atzmon


