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In a shadow of the RH: cyclic vectors of Hardy spaces on the Hilbert multidisc. (À l’ombre de l’HR: Vecteurs cycliques de l’espace de Hardy du multidisque hilbertien.) (English. French summary) Zbl 1267.30108

Ann. Inst. Fourier 62, No. 5, 1601-1626 (2012); correction ibid. 68, No. 2, 563-567 (2018).
Given the family of isometries \(T_n(f)(z)=f(z^n),\, n\in\mathbb{N}\), on the Hardy space \(H^2_0=\{f\in H^2(D):f(0)=0\}\), the \((T_n)\)-cyclic vectors are the functions \(f\in H^2_0\) such that \(\text{span}_{H^2_0}\{T_n(f):n\in\mathbb{N}\}=H^2_0\). Here, \(D\) denotes the open unit disc in \(\mathbb{C}\) and \(\text{span}_{H^2_0}\) means the closed linear hull in \(H^2_0\).
In this paper the author obtains several conditions on a function \(f\in H^2_0\) which implies its \((T_n)\)-cyclicity. These conditions are given in terms of the Bohr transform of \(f\), and, in some special cases, in terms of the Taylor coefficients of \(f\). The results proved in this paper include several previous known results due to different authors.
One of the motivations of the study of these problems is the relation between the Riemann Hypothesis on the zeros of the \(\zeta\)-function, the invariant subspaces of the semigroup \((T_n)\) and the \((T_n)\)-cyclic vectors (the paper contains an interesting abridged story about these relations). We recall that if \(\{p_j\}\) denotes the increasing sequence of prime numbers, \(p_j\geq 2\), \(n=p_1^{\alpha_1(n)}\cdots p_k^{\alpha_k(n)}\cdots\) and \(D^\infty_2=\{w\in l^2: \|w\|_{l^{\infty}}<1\}\), then the Bohr transform \(U\) from \(H^2_0\) to the Hardy space \(H^2(D^\infty_2)\) is defined by \[ U\left(\sum_{n=1}^\infty \hat f(n) z^n\right) =\sum_{n=1}^\infty \hat f(n) \zeta_1^{\alpha_1(n)}\cdots \zeta_k^{\alpha_k(n)}. \] An important property of \(U\) is the fact that \(f\) is \((T_n)\)-cyclic if and only if \(Uf\) is cyclic with respect to multiplications by independent variables (shift operators), that is, if \(Uf\cdot H^\infty(D^\infty_2)\) is dense in \(H^2(D^\infty_2)\).
Beurling proved that if \(f\) is \((T_n)\)-cyclic, then \(Uf\) does not vanish on \(D^\infty_2\). In this paper the author shows that this necessary condition is also sufficient if we assume that the Fourier spectrum of \(f\) is finitely generated in \(\mathbb{N} \) and the Taylor coefficients of \(f\) satisfy \(\hat{f}(n)=O(n^{-\varepsilon})\) for some \(\varepsilon>0\). It is also proved in the paper that if \(Uf\) depends only on \(N\) variables, then \(f\) is \((T_n)\)-cyclic if and only if \(Uf\) is cyclic in \(H^2(D^N)\).
These results provide several sufficient conditions, some of which were already known. For instance, \(f\) is \((T_n)\)-cyclic if one of the following conditions is satisfied:
1) \(Uf\in H^2(D^\infty_2)\) and \(1/Uf\in H^\infty(D^\infty_2)\) (proved in [H. Hedenmalm, P. Lindqvist and K. Seip, Duke Math. J. 86, No. 1, 1–37 (1997; Zbl 0887.46008)]).
2) There exist \(\epsilon>0\) and \(\delta>0\) such that \((Uf)^{1+\varepsilon}\) and \((Uf)^{-\delta}\) are in \(H^2(D^\infty_2)\).
3) \(f\neq 0\) and \(\text{Re}(Uf(\zeta))\geq 0\).
4) \(Uf\in \text{Hol}((1+\varepsilon)D^N)\) and \(Uf(\zeta)\neq 0\) for \(\zeta\in D^N\) (proved in [J. H. Neuwirth, J. Ginsberg and D. J. Newman, J. Funct. Anal. 5, 194–203 (1970; Zbl 0189.12902)] for polynomials).
5) \(Uf=Uf_1\cdot Uf_2\cdot Uf_3\cdot Uf_4\), where the functions \(f_i\) satisfies conditions \(i)\) above, \(i=1,2,3,4\).
The paper concludes with some examples where the cyclicity properties of a function \(f\in H^2_0\) are given in terms of its Taylor coefficients.

MSC:

30H10 Hardy spaces
47A16 Cyclic vectors, hypercyclic and chaotic operators
42C30 Completeness of sets of functions in nontrigonometric harmonic analysis
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References:

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