# In a shadow of the RH: Cyclic vectors of Hardy spaces on the Hilbert multidisc

@article{Nikolski2012InAS, title={In a shadow of the RH: Cyclic vectors of Hardy spaces on the Hilbert multidisc}, author={Nikolai K. Nikolski}, journal={Annales de l'Institut Fourier}, year={2012}, volume={62}, pages={1601-1626} }

— Completeness of a dilation system (φ(nx))n>1 on the standard Lebesgue space L2(0, 1) is considered for 2-periodic functions φ. We show that the problem is equivalent to an open question on cyclic vectors of the Hardy space H(D2 ) on the Hilbert multidisc D2 . Several simple sufficient conditions are exhibited, which include however practically all previously known results (Wintner; Kozlov; Neuwirth, Ginsberg, and Newman; Hedenmalm, Lindquist, and Seip). For instance, each of the following…

## 38 Citations

The periodic dilation completeness problem: cyclic vectors in the Hardy space over the infinite‐dimensional polydisk

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The classical completeness problem raised by Beurling and independently by Wintner asks for which ψ∈L2(0,1) , the dilation system {ψ(kx):k=1,2,…} is complete in L2(0,1) , where ψ is identified with…

Orthogonality, density and shift-invariance in the Hardy space approach to the RH

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A Hardy space approach to the Nyman-Beurling and Báez-Duarte criterion for the Riemann Hypothesis (RH) was introduced in [16]. It states that the RH holds true if and only if the linear manifold N…

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The semi-group of weighted composition operators (Wn)n≥1 where Wnf(z) = (1 + z + . . .+ z )f(z) on the classical Hardy-Hilbert space H of the open unit disk is related to the Riemann Hypothesis (RH)…

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A correction to “In a shadow of the RH: Cyclic vectors of Hardy spaces on the Hilbert multidisc”

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- 2018

— This note corrects some inaccuracies remarked in the paper mentioned in the title. It contains also a few references to recent developments on the dilations f(nx) completeness problem and points…

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This paper is concerned with polynomially generated submodules of the Hardy module H2(D∞ 2 ). Since the polynomial ring P∞ in infinitely many variables is not Noetherian, some standard tricks for…

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The completeness, minimality, and basis property in L2[0, π] and Lp[0, π], p ≠ 2, are considered for systems of dilated functions un(x) = S(nx), n ∈ N, where S is the trigonometric polynomial S(x) =…

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This paper concerns a long-standing problem raised by Beurling and Wintner on completeness of the dilation system {φ(kx) : k = 1, 2, · · · } generated by the odd periodic extension on R of any φ ∈…

Dilation theory and analytic model theory for doubly commuting sequences of $C_{.0}$-contractions

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Sz.-Nagy and Foias proved that each $C_{\cdot0}$-contraction has a dilation to a Hardy shift and thus established an elegant analytic functional model for contractions of class $C_{\cdot0}$. This has…

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