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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?

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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

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    $\begingroup$ @A.Atzmon Dear Prof. Atzmon thank you very much for your very excelent answer and your attention to my question. You are mostly well come to MO. $\endgroup$ – Ali Taghavi Apr 10 '18 at 19:35
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Nice try, but I don't think this will help with the ISP.

I think of the "$X$-derivation" as an unbounded skew-adjoint operator $D$ on $L^2(M)$. When you say "Sobolev space" I suppose you mean to take the domain of $D$, equipped with the square norm $\|f\|^2_2 + \|Df\|^2_2$. I'm not sure what you mean about an infinite order space.

Anyway, working on $L^2(M)$ the operator $D$ is $i$ times a self-adjoint operator, so by spectral theory you can go to a multiplication picture in which $D$ becomes multiplication by a purely imaginary function. In this picture you can see that there are lots of invariant subspaces contained in the domain of $D$, i.e., the Sobolev space.

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  • $\begingroup$ Thank you for your answer. The derivation is a bounded opertor from $H^s(M)$ to $H^{s-1}(M)$. Now i search for a legal situation of $s=\infty$. Then a natural question is the decomposition of $H^{\infty}(M)$ to irreducible component.(The component without invariant subspace) $\endgroup$ – Ali Taghavi Mar 29 '18 at 20:44
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    $\begingroup$ That's not going to work, for the same reason given in my answer. When you work in the spectral multiplication picture it's clear that there are lots of subspaces --- e.g., the spectral projection of $iD$ for any bounded interval --- which are invariant under $D^n$ for all $n$. $\endgroup$ – Nik Weaver Mar 29 '18 at 21:43

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