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Let $\{X_i\}_{i \in \mathbb{N}} $ be a sequence of $n$-dimensional linear subspaces of the separable Hilbert space $H$ and let $\{\phi_i\}_{i \in I}$ be a sequence of continuous injective linear maps from $H$ to itself such that:

  • $\phi_i(X_i)=X_{i+1}$,
  • $X_i\cap X_j =\{0\}$ if $i\neq j$.

Is there a general criteria under which $ X := \cup_{i \in \mathbb{N}} X_i, $ is dense in $H$?

Note 1: I don't want to assume that $\phi_i =\phi^i$ where $\phi$ is a hypercyclic operator and $X_0$ contains a hypercylic vector. Since, typically, those objects can rarely be made explicit and rely on existence type results.

Vague Intuition I expect some type of Wiener-Tauberian-like theorem where one looses the ability to take the span but somehow it is generated by iterations of $\phi$...

WLOG (if it helps), we may assume that $H=L^2(\mathbb{R})$.

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  • $\begingroup$ But then if $n=1$ you want $X_0$ to contain a supercyclic vector $\endgroup$
    – erz
    Commented Apr 18, 2020 at 15:45
  • $\begingroup$ Sure, but if n is large then the condition strictly relaxes $\endgroup$
    – ABIM
    Commented Apr 18, 2020 at 16:04
  • $\begingroup$ May be of interest: arxiv.org/abs/1205.3575 $\endgroup$
    – ABIM
    Commented Apr 18, 2020 at 16:07
  • $\begingroup$ Yes, but this doesn't give me an explicit criterion I can check to verify if $X_0$ has this property also.. $\endgroup$
    – ABIM
    Commented Apr 18, 2020 at 16:09

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