In the literature, is there any paper or research investigating the 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?