# Tauberian Theorem for 1-parameter groups of operators

The Wiener Tauberian Theorem gives condition on an $$f\in L^1(\mathbb{R})$$ such that the "induced 1-parameter family" $$\{T_b(f)\}_{b\in \mathbb{R}}$$ has a dense span in $$L^1(\mathbb{R})$$; where $$T_b$$ is the Group of transation operators on $$L^1(\mathbb{R})$$ defined by $$f\mapsto T_b(\mathbb{R})$$.

Are there other "similar results" stating that: Given a group (under composition) $$(T_t)_{t\in \mathbb{R}}$$ of (continuous linear) operators on $$L^1(\mathbb{R})$$ then: $$\mbox{If f\in L^1(\mathbb{R}) satisfies ___ then \overline{\operatorname{span}(\{T_t(f)\}_{t\in\mathbb{R}}}=L^1(\mathbb{R}).}$$

Note: I would be interested (though less so) in results where either:

• $$L^1(\mathbb{R})$$ is replaced by $$C(\mathbb{R})$$ with the compact-convergence topology.
• $$(T_t)_{t}$$ forms a semi-group,
• If $$\mathbb{R}$$ is replaced by a locally-compact topological group $$G$$ and the Lebesgue measure by the Haar measure on $$G$$

Related Post: Here.

(What I'm not looking for): I just note that conditions on $$f$$ which I would not be interested in, are those relating to hyper/super-cyclicity. This is because, such conditions need $$f$$ to be "infinitely complicated" (see this paper) while I'm looking for "tractable/simple $$f$$" (as in Wiener's case).

• OK, I convert my answer to an "insightful comment" (first time I go that way :). For the combinatorial side of one-parameter deformations, you can have a look here. For the question of existence of a cyclic vector, there is a bunch of examples in § 4.2 for which you can test the density of the orbits. Do not hesitate to interact (I can provide missing proofs and details on request, of course). Jun 12 '21 at 14:53
• Thanks Gérard, I really do appreciate the comment and I currently looking through your paper. I do feel abit silly, but I don't quite see where density in $L^1(\mathbb{R})$ comes in. (P.s.: Let me just briefly mention that I didn't downvote your earlier answer...so no hard feelings :) ) Jun 12 '21 at 14:57
• OK, I understand. The paper does not offer already elaborated statement about density, but, rather, there are a lot of situations and, I think, a setting to explore some type of one-parameter groups (acting on holomorphic or continuous functions). So, ok, do not loose time if it does not fit directly your research thread. Jun 12 '21 at 15:12
• @DuchampGérardH.E. Still thanks a lot, I'll continue reading it and I'll let you know if I have any questions. Thanks again :) Jun 12 '21 at 15:14
• In the direction of replacing $\mathbf{R}$ by other locally compact groups, I believe that Wiener's Tauberian theorem works perfectly well for locally compact abelian groups. For non abelian groups, I do not know what is the current state of knowledge. It was apparently open in 1976 whether this holds for groups with polynomial growth, see zbmath.org/?q=an%3A0338.43005. See also Yulia Kuznetsova's answer here for other examples and references: mathoverflow.net/q/84943 Jun 14 '21 at 10:03