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}
}
  • N. Nikolski
  • Published 2012
  • Mathematics
  • Annales de l'Institut Fourier
— 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… 
The periodic dilation completeness problem: cyclic vectors in the Hardy space over the infinite‐dimensional polydisk
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  • Mathematics
    Journal of the London Mathematical Society
  • 2020
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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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)
A correction to “In a shadow of the RH: Cyclic vectors of Hardy spaces on the Hilbert multidisc”
— 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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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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References

SHOWING 1-10 OF 28 REFERENCES
A Hilbert space of Dirichlet series and systems of dilated functions in $L^2(0,1)$
For a function $\varphi$ in $L^2(0,1)$, extended to the whole real line as an odd periodic function of period 2, we ask when the collection of dilates $\varphi(nx)$, $n=1,2,3,\ldots$, constitutes a
The validity of Beurling theorems in polydiscs
Abstract : Let Z be the set of integers. We denote by m,n etc. the elements of Z sub 2. Let U sub 2 be the open unit disc and T the boundary of U in the complex plane C slashed. Let Z sub 2, U sub 2
A CLOSURE PROBLEM RELATED TO THE RIEMANN ZETA-FUNCTION.
  • A. Beurling
  • Mathematics
    Proceedings of the National Academy of Sciences of the United States of America
  • 1955
TLDR
This note will deal with a duality of the indicated kind which may be of some interest due to its simplicity in statement and proof.
Invariant subspaces in the polydisk
This note is a study of unitary equivalence of invariant subspaces of H2 of the polydisk. By definition, this means joint unitary equivalence of the shift operators restricted to the invariant
ON WEAKLY INVERTIBLE FUNCTIONS IN THE UNIT BALL AND POLYDISK AND RELATED PROBLEMS
We will present an approach to deal with a problem of existence of (not) weakly invertible functions in various spaces of analytic functions in the unit ball and polydisk based on estimates for
On the shift semigroup on the Hardy space of Dirichlet series
AbstractWe develop a Wold decomposition for the shift semigroup on the Hardy space $$ \mathcal{H}^2 $$ of square summable Dirichlet series convergent in the half-plane $$ \Re (s) > 1/2 $$. As an
On Nyman, Beurling and Baez-Duarte’s Hilbert space reformulation of the Riemann hypothesis
There has been a surge of interest of late in an old result of Nyman and Beurling giving a Hilbert space formulation of the Riemann hypothesis. Many authors have contributed to this circle of ideas,
A strengthening of the Nyman-Beurling criterion for the Riemann Hypothesis
Let $\rho(x)=x-[x]$, $\chi=\chi_{(0,1)}$. In $L_2(0,\infty)$ consider the subspace $\B$ generated by $\{\rho_a | a \geq 1\}$ where $\rho_a(x):=\rho(\frac{1}{ax})$. By the Nyman-Beurling criterion the
The primes contain arbitrarily long arithmetic progressions
We prove that there are arbitrarily long arithmetic progressions of primes. There are three major ingredients. The first is Szemeredi�s theorem, which asserts that any subset of the integers of
Spaces of Holomorphic Functions in the Unit Ball
Preliminaries.- Bergman Spaces.- The Bloch Space.- Hardy Spaces.- Functions of Bounded Mean Oscillation.- Besov Spaces.- Lipschitz Spaces.
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