Annales de l’institut Fourier (AIF)

Nikolski, Nikolai

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ANNALES
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L’INSTITUT FOURIER
Nikolai NIKOLSKI
A correction to “In a shadow of the RH: Cyclic vectors of Hardy spaces on
the Hilbert multidisc”
Tome 68, no 2 (2018), p. 563-567.
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Ann. Inst. Fourier, Grenoble
68, 2 (2018) 563-567
A CORRECTION TO “IN A SHADOW OF THE RH:
CYCLIC VECTORS OF HARDY SPACES ON THE
HILBERT MULTIDISC”
Annales de l’institut Fourier, vol. 62 (2012), n°5, 1601–1626
by Nikolai NIKOLSKI
Abstract. — 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 out some old statements of
V. Ya. Kozlov from early 1950’s still unproved.
Résumé. — Quelques inexactitudes dans l’article mentionné dans le titre sont
corrigées, et une information sur les progrès récents dans le sujet (sur la complétude
des dilatations f(nx)) est ajoutée. On discute également quelques assertions sur la
complétude (de V. Ya. Kozlov, 1950) qu’on n’a pas réussi à démontrer jusqu’à nos
jours.
In this note I correct a few vague points of my presentation and report on
some additional information on the subject of which I was unaware when
writing my paper in 2012.
I would like to thank Boris Mityagin of Ohio State University for indi-
cating me some incomplete and/or mistaken arguments in the paper, as
well as for interesting discussions.
I also indebted to the referee indicating a few misprints.
Keywords: dilations completeness, Hilbert’s multidisc, Bohr transform, cyclic vectors,
Riemann hypothesis.
2010 Mathematics Subject Classification: 32A35, 32A60, 42B30, 42C30, 47A16.
564 Nikolai NIKOLSKI
1. How to define real powers of a function F ∈ H2(D∞2 ),
F (ζ) 6= 0 (ζ ∈ D∞2 ), point 2.1(12)-i) of the paper in
question.
At this point of the paper, I have used the inequality |at− bt| 6 C|a− b|t
(which is true for real positive a and b but is inapplicable to arbitrary
a, b ∈ C) instead of applying property 2.1(2), as follows.
By 2.1(2), a sequence (Fn)n>1, Fn ∈ H2(Dn) with a chain property
Fn+1(ζ, 0) = Fn(ζ) (ζ ∈ Dn) and supn ‖Fn‖2 < ∞, uniquely defines a
function F ∈ H2(D∞2 ) such that F(n) = Fn (F(n) =: F (ζ, 0, . . . ), ζ ∈ Dn
are defined in 2.1(2)). Now, given F ∈ H2(D∞2 ), F (ζ) 6= 0 (ζ ∈ D∞2 )
and F (0) = 1, we have F(n)(ζ) 6= 0 (ζ ∈ Dn), and hence there exists
a holomorphic logF(n), which is uniquely defined by the choice log 1 =
0. Setting F t(n)(ζ) = exp(t logF(n)(ζ)) (ζ ∈ Dn) we obtain a family of
functions satisfying F t(n)F s(n) = F
t+s
(n) for all 0 < t, s < 1. Since |F
t
(n)(ζ)| =
|F(n)(ζ)|t for all ζ ∈ Dn, we have ‖F t(n)‖2/t = ‖F(n)‖t2 6 ‖F‖t2 for every n.
Since F(n)(0) = 1, we also get F t(n+1)(ζ, 0) = F t(n)(ζ) (ζ ∈ Dn), and hence
obtain functions F t ∈ H2(D∞2 ) such that F tF s = F t+s. By the previous
estimates for ‖F t(n)‖2/t, it is clear that F t ∈ H2/t(D∞2 ). 
2. How to prove that polynomials are dense in Hp(D∞2 ),
2 6 p <∞, point 2.1(12)-iii).
In my article, such a proof is also based on the mentioned above inequal-
ity inapplicable to complex a and b. Now, we give an alternative proof.
(a) First, we show that the union of
⋃
n>1 H
p(Dn) is dense in Hp(D∞2 )
(as usual, a function F ∈ Hp(Dn) is also considered in Hp(D∞2 ) as F (ζ) =
F (ζ1, . . . , ζn), ζ ∈ D∞2 ). Indeed, if F ∈ Hp(D∞2 ), we have limn F(n)(ζ) =
F (ζ) for every ζ ∈ D∞2 (it is clear that, as in 2.1(5), this property is true for
all F ∈ H2(D∞2 )) and ‖F(n)‖p 6 ‖F‖p. In a moment, we will verify that the
linear hull L =: Lin(ϕζ : ζ ∈ D∞2 ) of functionals ϕζ(f) =: f(ζ), is norm
dense in the dual space (Hp(D∞2 ))∗, which implies that limn F(n) = F
weakly in Hp(D∞2 ). Then, by a known functional analysis argument we
conclude that there exist convex combinations fk of functions F(n) such
that limk ‖F − fk‖p = 0.
In order to check the above density of L, we mention that Hp(D∞2 ) is
a uniformly convex (and so, reflexive) Banach space and the family (ϕζ :
ζ ∈ D∞2 ) is total (ϕζ(f) = f(ζ) = 0, ∀ζ ∈ D∞2 ⇒ f = 0), hence L is
ANNALES DE L’INSTITUT FOURIER
IN A SHADOW OF THE RH: CYCLIC VECTORS 565
norm dense (Hahn–Banach). The completeness and uniform convexity of
Hp(D∞2 ) follow, as above, from a limiting reasoning with F(n) and the
corresponding properties of Hp(Dn); for example, given F,G ∈ Hp(D∞2 ),
such that ‖F‖p 6 1, ‖G‖p 6 1 and ‖F − G‖p > ε, we get ‖F(n)‖p 6 1,
‖G(n)‖p 6 1 and ‖F(n) −G(n)‖p > ε (for sufficiently large n), which imply∥∥ 1
2 (F(n) + G(n))
∥∥
p
6 1 − 1p (ε/2)
p (following [2] for every Lp(µ) space); it
remains to use limn
∥∥ 1
2 (F +G)(n)
∥∥
p
=
∥∥ 1
2 (F +G)
∥∥
p
. 
(b) Since the polynomials are dense in each Hp(Dn) (see [7], §3.4), the
claimed density in Hp(D∞2 ) follows. 
3. The statements of point (4) of Theorem 3.4 and
Lemma 3.5
Unfortunately, the formula for an invariant subspace E in point (4) of
Theorem 3.4 and Lemma 3.5 are stated unproperly.
In fact, the last sentence of point (4) should say:
Moreover, if E is a (Mζ)-invariant subspace of H2(D∞2 ) generated
by F = Uf , then
E = (I ◦ ζα(n)) ·H2(D∞2 ) ,
where I is the inner part of ϕ.
Indeed, since F = ϕ ◦ ζα(n), and the multiplication by I ◦ ζα(n), h 7→
(I ◦ ζα(n))h, is isometric on H2(D∞2 ), we have E ⊂ (I ◦ ζα(n)) ·H2(D∞2 ). By
Beurling invariant subspace theorem, there exist polynomials pm (in one
variable) such that ‖pmϕ − I‖H2(D) =
∥∥(pm ◦ ζα(n))F − I ◦ ζα(n)∥∥H2(D∞2 )
tends to zero (as m → ∞), which shows that (I ◦ ζα(n)) ∈ E, and hence
(I ◦ ζα(n)) ·H2(D∞2 ) ⊂ E. 
For Lemma 3.5, what is really proved (and what is a correct statement
of the Lemma) is the following:
Z(F ) is the union
⋃
σ(Aσ′ × Dσ) (σ runs over all proper subsets
of {1, 2, . . . , n}), where Aσ′ are the same quality subsets of Tσ
′ as
in Lemma (finite unions of real analytic varieties of real dimension
less than card(σ′)).
This corrected form of the Lemma (a finite union of sets Aσ′×Dσ instead
of just one of them) can be used in the proof of Theorem 3.4 (points (2)
and (3)) just in the same way as it is done in the text. 
TOME 68 (2018), FASCICULE 2
566 Nikolai NIKOLSKI
4. Comments to Example 4.1(4)
The second statement of Example 4.1(4), i.e.
f = z(λ− z)N , 1 < |λ| < N =⇒ f is non-cyclic in H20 ,
is proved for real positive numbers λ only (in fact, the proof is valid for all
λ, 0 6 λ 6 N).
For complex λ, for the moment, we can prove some weaker non-cyclicity
statements (see [6]), for example: (a) if q is a prime such that
√
N + 1 < q 6
N + 1 and |λ| < (Cq−1N )1/q−1, then f is non-cyclic in H20 ; (b) if N = q2− 1
and |λ| 6 q− 1 then f is non-cyclic; (c) given ε > 0, there exists N(ε) such
that N > N(ε), |λ| < N1−ε ⇒ fλ,N is non-cyclic.
Notice that the first (unchanged) statement of the same Example can be
read as follows: if |λ| > N/ log 2 ≈ 1, 45 ·N then f is cyclic.
5. V. Ya. Kozlov’s enigmatic statements
In 1948–1950, V. Ya. Kozlov published a few notes in Doklady AN SSSR
where he claimed several beautiful results on the completeness (cyclicity)
of the dilates systems (h(nx))n>1 in L2odd(−1, 1), [3, 4] (see also D. Bour-
gin [1], of the same period). It is easy to see that this system is com-
plete in L2odd(−1, 1) if and only if (f(zn))n>1 is complete in H20 , where
f =
∑
k>1 ĥ(k)zk. In particular, Kozlov stated the following astonishing
results:
Let θ, 0 < θ 6 1, and h = hθ be an odd and 2-periodic function
defining for 0 < x < 1 by
hθ(x) = χ(0,θ)(x);
then, hθ is cyclic for θ = 1, θ = 12 and θ =
2
3 , and is not cyclic for
θ = 13 (and for θ in a neighborhood of
1
3 ); the same non-cyclicity is
claimed for θ = qp where p > 2 is a prime and q an odd integer such
that tan2 qπ2p <
1
p .
(By the way, the latter non-cyclicity condition is NOT fulfilled for values
q = 1, p = 3 corresponding to the previous claim (for θ = 13 )!)
Until now, no proofs of these claims were published (their author died
in 2007). We can quite easily treat the cases θ = 1 and θ = 12 , when
the Bohr transforms Uf of the functions f corresponding to h1 and h1/2
are, respectively, a reproducing kernel of H2(D∞2 ) at a point or a product
of a reproducing kernel with a linear function, and hence are cyclic ([5,
ANNALES DE L’INSTITUT FOURIER
IN A SHADOW OF THE RH: CYCLIC VECTORS 567
Corollary 3.7]). It would be much interesting to recover other Kozlov’s
claims (maybe, also with the use of the Bohr lift U).
BIBLIOGRAPHY
[1] D. G. Bourgin, “A class of sequences of functions”, Trans. Am. Math. Soc. 60
(1946), p. 478-518.
[2] O. Hanner, “On the uniform convexity of Lp and lp”, Ark. Mat. 3 (1955), no. 19,
p. 239-244.
[3] V. Ya. Kozlov, “On the completeness of systems of functions {ϕ(nx)} in the space
L2(0, 2π)”, Doklady Akad. Nauk SSSR 61 (1948), p. 977-980.
[4] ———, “On the completeness of a system of functions of type {ϕ(nx)} in the space
L2”, Doklady Akad. Nauk SSSR 73 (1950), p. 441-444.
[5] N. Nikolski, “In a shadow of the RH: Cyclic vectors of Hardy spaces on the Hilbert
multidisc”, Ann. Inst. Fourier 62 (2012), no. 2, p. 1601-1626.
[6] ———, “Binomials whose dilations generate H2(D)”, Algebra and Analysis 29
(2017), no. 6, p. 159-177.
[7] W. Rudin, Function theory in polydiscs, Mathematics Lecture Note Series,
W. A. Benjamin, Inc, 1969, 188 pages.
Manuscrit reçu le 22 août 2017,
révisé le 11 décembre 2017,
accepté le 13 décembre 2017.
Nikolai NIKOLSKI
Université de Bordeaux
UFR de Mathématiques et Informatique
351 cours de la Libération
33405 Talence (France)
nikolski@math.u-bordeaux.fr
TOME 68 (2018), FASCICULE 2


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