Comptes Rendus Mathématique

Curtis R. Harvey

Nikita A. Karpenko

Comptes Rendus
Mathématique
Curtis R. Harvey and Nikita A. Karpenko
Quadratic forms in I n of dimension 2n +2n−1
Volume 362 (2024), p. 1539-1543
Online since: 14 November 2024
https://doi.org/10.5802/crmath.668
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
C EN T R E
MER S ENN E
The Comptes Rendus. Mathématique are a member of the
Mersenne Center for open scientific publishing
www.centre-mersenne.org — e-ISSN : 1778-3569
Comptes Rendus. Mathématique
2024, Vol. 362, p. 1539-1543
https://doi.org/10.5802/crmath.668
Research article / Article de recherche
Algebraic geometry / Géométrie algébrique
Quadratic forms in I n of dimension 2n +2n−1
Formes quadratiques de dimension 2n +2n−1 dans I n
Curtis R. Harvey a and Nikita A. Karpenko ∗,a
a Mathematical & Statistical Sciences, University of Alberta, Edmonton, Canada
URL: www.ualberta.ca/~karpenko
E-mails: rcharvey@ualberta.ca, karpenko@ualberta.ca
Abstract. For n ≥ 3, confirming a weak version of a conjecture of Hoffmann, we show that every anisotropic
quadratic form in I n of dimension 2n + 2n−1 splits over a finite extension of the base field of degree not
divisible by 4. The first new case is n = 4, where we obtain a classification of the corresponding quadratic
forms up to odd degree base field extensions and get this way a strong upper bound on their essential 2-
dimension. As well, we compute the reduced Chow group of the maximal orthogonal grassmannian of the
quadratic form and conclude that its canonical 2-dimension is 2n +2n−2 −2.
Résumé. Pour n ≥ 3, en confirmant une version faible d’une conjecture de Hoffmann, on montre que toute
forme quadratique anisotrope de dimension 2n +2n−1 dans I n se déploie sur une extension finie du corps
de base d’un degré qui n’est pas divisible par 4. Le premier nouveau cas est celui de n = 4, où l’on obtient
une classification des formes quadratiques correspondantes à une extension de degré impair près ce qui
donne une forte borne supérieure pour leur 2-dimension essentielle. De plus, on détermine le groupe de
Chow réduit de la grassmannienne orthogonale maximale de la forme quadratique et on en déduit que sa
dimension 2-canonique est égale à 2n +2n−2 −2.
Keywords. Quadratic forms over fields, projective homogeneous varieties, Chow rings.
Mots-clés. Formes quadratiques sur des corps, variétés projectives homogènes, anneaux de Chow.
2020 Mathematics Subject Classification. 11E04, 14C25.
Funding. This work has been supported by a Discovery Grant from the National Science and Engineering
Research Council of Canada and completed when the second author was a visitor at the Université de
Lorraine.
Manuscript received 10 March 2024, revised 1 July 2024, accepted 10 July 2024.
Let F be a field (of any characteristic) and let I = I (F ) be the Witt group of classes of even-
dimensional non-degenerate quadratic forms over F defined as in [4, Section 8] (and denoted
Iq (F ) there). For n ≥ 2, we write I n = I n(F ) for the subgroup in I (F ) generated by the n-fold
Pfister forms. We refer to [4, 9.B] for other equivalent definitions of I n(F ) (denoted I nq (F ) there).
Any element of I is represented by an anisotropic quadratic form. By the Arason–Pfister
Hauptsatz, the smallest possible dimension of a nonzero anisotropic quadratic form in I n is 2n
(see [4, Theorem 23.7(1)] for the characteristic-free version). The quadratic forms in I n of dimen-
sion 2n are classified: as a consequence of the Arason–Pfister Hauptsatz and [4, Corollary 23.4)],
they are exactly the forms similar to n-fold Pfister forms. In particular, any 2n-dimensional qua-
dratic form in I n splits over a finite base field extension of degree dividing 2.
∗Corresponding author
ISSN (electronic): 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
1540 Curtis R. Harvey and Nikita A. Karpenko
The smallest possible dimension exceeding 2n of an anisotropic quadratic form in I n is
2n + 2n−1. For n = 3 this has been shown in [12] (characteristic ̸= 2) and in [1] (arbitrary
characteristic); for n = 4 in [6] (characteristic ̸= 2) and in [5, Theorem 4.2.11] (characteristic 2); for
arbitrary n and characteristic ̸= 2 a proof has been given in [15, Theorem 5.4] and then extended
to characteristic 2 in [13, Proposition 11.5].
For n = 2, quadratic forms in I n of dimension 2n + 2n−1 are the well-understood Albert
forms. For n ≥ 3, by a conjecture of Hoffmann ([6, Conjecture 2] for characteristic ̸= 2 and [5,
Conjecture 4.3.1] for characteristic 2), quadratic forms in I n of dimension 2n + 2n−1 should be
classified as products of an Albert bilinear form (i.e., a 6-dimensional symmetric bilinear form of
determinant −1) by a Pfister form (of foldness n−2). In particular, such forms should split over a
finite base field extension of degree dividing 2 as well.
However, the two above conjectures are so far proved for n = 3 only: the proof for characteristic
̸= 2 of [12] is extended to characteristic 2 in [5, Proposition 4.1.2]. The main result of the present
note is
Theorem 1. For any n ≥ 3 and in any characteristic, every quadratic form in I n of dimension
2n +2n−1 splits over a finite base field extension of degree not divisible by 4.
Proof. Let X be a connected component of the highest orthogonal grassmannian of a quadratic
form q in I n(F ) of dimension 2n +2n−1. Theorem 1 means that the index i (X ) of the variety X ,
defined as the g.c.d. of the degrees of closed points on X , divides 2. In other terms, taking into
account Springer’s Theorem [4, Corollary 18.5], i (X ) = 2 provided that q is not split.
We write X for X over an algebraic closure of F and we write CH(X ) for the ring given by the
image of the change of field homomorphism CH(X ) → CH(X ) of the Chow rings. Note that the
kernel of the change of field homomorphism is the ideal of the elements of finite order. For this
reason, CH(X ) is sometimes called the reduced Chow group of X .
By [4, Theorem 86.12], the ring CH(X ) is generated by certain homogeneous elements e1, . . . ,el
of codimensions 1, . . . , l := 2n−1 + 2n−2 − 1. It is convenient to define ei := 0 for i > l . For any
i ≥ 1, the element ei is characterized by the property that (−1)i 2ei is the i th Chern class of the
tautological vector bundle on X (see [4, Proposition 87.13]); in particular, 2ei ∈ CH(X ).
Since for any field extension K /F , the anisotropic part of the quadratic form qK over the field
K is either 0, or 2n , or 2n +2n−1, it follows by [4, Corollary 88.6] (see also [4, Corollary 88.7]) that
ei ∈ CH(X ) for all i different from k := 2n−1 −1 and l . By [4, (86.15)], for any i ≥ 1, we have
e2i −2ei−1ei+1 +2ei−2ei+2 −·· ·+ (−1)i−12e1e2i−1 + (−1)i e2i = 0 ∈ CH(X ).
In particular,
2ek el = 2ek+1el−1 −2ek+2el−2 +·· ·±2em−1em+1 ±e2m ∈ CH(X ),
where m := (k + l )/2. Therefore 2e ∈ CH(X ), where e ∈ CH(X ) is the product e1 . . .el of all the
generators. Since e is the class of a 0-cycle of degree 1 (see [4, Corollary 86.10]), the variety X
(over F ) possesses a 0-cycle of degree 2. □
For n = 4, in view of [6, Proposition 4.1] and [5, Proposition 4.3.2], Theorem 1 provides a
classification of the corresponding quadratic forms “up to odd degree extensions” which yields a
strong upper bound on their essential 2-dimension. We provide details right below, starting with
the classification result:
Theorem 2. For a field F (of any characteristic), let q be a quadratic form in I 4(F ) of dimension
24 = 24 +23. Then there exists a finite field extension K /F of odd degree such that qK is isomorphic
to the tensor product of an Albert bilinear form by a Pfister form.
Proof. By Theorem 1, we can find a finite field extension K /F of odd degree and a field extension
L/K of degree dividing 2 such that qL is split. The description of qK then follows from [6,
Proposition 4.1] (for characteristic ̸= 2) and [5, Proposition 4.3.2] (for characteristic 2). □
Curtis R. Harvey and Nikita A. Karpenko 1541
To formulate the result on the essential 2-dimension, let us consider the functor I 424, associat-
ing to every extension field K of a fixed field F the set of isomorphism classes of quadratic forms
in I 4(K ) of dimension 24. The essential 2-dimension of an element in I 424(K ) as well as the essen-
tial 2-dimension ed2 I 424 of the functor I
4
24 are defined as in [3, Section 1].
Corollary 3. One has ed2 I 424 ≤ 7.
Proof. Writing F for the base field and taking q ∈ I 424(K ) for a field extension K /F , we find by
Theorem 2 an odd degree field extension L/K such that qL is isomorphic to the tensor product of
the diagonal Albert bilinear form 〈a1, a2, a3, a4, a5,−a1a2a3a4a5〉 by the Pfister form 〈〈b1,b2]] for
some nonzero a1, . . . , a5,b1 ∈ L and some b2 ∈ L, where in characteristic ̸= 2 the element b2 is also
nonzero. The subfield F (a1, . . . , a5,b1,b2) ⊂ L, whose transcendence degree over F is at most 7, is
then a field of definition of qL . It follows that ed2 q = ed2 qL ≤ 7 and so ed2 I 424 ≤ 7. □
Recall that the essential 2-dimension is a 2-local version of and constitutes a lower bound for
the essential dimension, measuring, informally speaking, how many independent parameters are
required to describe an isomorphism class of the corresponding type of objects; in particular,
ed2 I 424 ≤ ed I 424. For n = 3, since the description of the corresponding quadratic forms does
not involve odd degree extensions, similar to the proof of Corollary 3 arguments show that
ed2 I 312 ≤ ed I 312 ≤ 6. In fact, in characteristic ̸= 2, ed2 I 312 = ed I 312 = 6 by [3, Theorem 7.1]: the lower
bound 6 ≤ ed2 I 312 is obtained by constructing a nontrivial degree 6 cohomological invariant with
coefficients in Z/2Z for I 312.
For n ≥ 4, assuming [6, Conjecture 2], one gets
ed2 I
n
2n+2n−1 ≤ ed I n2n+2n−1 ≤ n +3.
Finally, one has ed2 I n2n = ed I n2n = n +1 for any n. Indeed, as already mentioned, any q ∈ I n2n
is isomorphic to b · 〈〈b1,b2, . . . ,bn]] for some n + 1 parameters b,b1, . . . ,bn , ensuring that n + 1
is an upper bound for ed I n2n . On the other hand, associating in characteristic ̸= 2 to q the
symbol (b,b1, . . . ,bn) in the (n + 1)st Galois cohomology group with coefficients in Z/2Z, one
gets a nontrivial degree n + 1 cohomological invariant showing that n + 1 is a lower bound
for ed2 I n2n (see [11, Theorem 3.4]). The non-triviality of the cohomological invariant is shown
in [2, Section 3]. The characteristic 2 case is treated similarly using cohomological invariants
with values in étale motivic cohomology groups (cf. [14, Section 3] and especially [14, Proof of
Lemma 3.1]); the non-triviality of the cohomological invariant follows from [7].
To conclude, let us return to the case of arbitrary n ≥ 2. Let X be the highest orthogonal
grassmannian of an anisotropic quadratic form q ∈ I n . If dim q = 2n , then i (X ) = 2 and therefore
the ring CH(X ) contains 2CH(X ). By [4, Corollary 88.6], CH(X ) also contains the elements
e1, . . . ,e2n−1−2 – the generators of the ring CH(X ) with exception of the very last one e2n−1−1. Since
i (X ) ̸= 1, we conclude that CH(X ) is exactly the subring in CH(X ) generated by 2CH(X ) and
e1, . . . ,e2n−1−2 (cf. [4, Example 88.10]).
Now let us assume that dim q = 2n +2n−1, where n ≥ 3. Since i (X ) = 2 by Theorem 1, we still
have the inclusion CH(X ) ⊃ 2CH(X ). Besides, it has been shown in the proof of Theorem 1 that
CH(X ) ∋ ei for all i except i = k := 2n−1 −1 and i = l := 2n−1 +2n−2 −1.
Theorem 4. For any n ≥ 3 and any anisotropic quadratic form q in I n of dimension 2n +2n−1, the
ring CH(X ) of its highest grassmannian X is generated by 2CH(X ) and all ei with i ̸∈ {k, l }.
Proof. Since CH(X ) ⊃ 2CH(X ), it suffices to show that the ring Ch(X ) is generated by ei with
i ̸∈ {k, l }, where Ch(X ) := CH(X )/2CH(X ) and Ch(X ) := Im(Ch(X ) → Ch(X )). By [4, Theorem 87.7]
(originally proved in [16]), it suffices to show that neither ek nor el is in Ch(X ).
By [4, Corollary 82.3] once again, the anisotropic part of q over the function field of its quadric
Y has dimension 2n . By [4, Corollary 88.7], we conclude that ek ̸∈ Ch(X ).
1542 Curtis R. Harvey and Nikita A. Karpenko
Finally, let us assume that el ∈ Ch(X ) and seek for a contradiction. By [4, Theorem 90.3]
(originally proved in [16]), the canonical 2-dimension of the variety X equals k and does not
change when the base field is extended to the function field of Y . It follows by [10, Theorem 3.2]
that a shift of the upper Chow motive U (X ) with coefficients Z/2Z is a direct summand of the
motive of Y . On the other hand, by [4, Lemma 82.4], the complete motivic decomposition of the
quadric Y consists only of shifts of the upper motive U (Y ). Moreover, since the variety XF (Y ) has
no 0-cycle of odd degree, the motives U (Y ) and U (X ) are not isomorphic, see [9, Corollary 2.15].
The contradiction obtained proves Theorem 4. □
Regarding the motives of the varieties X and Y from the above proof, each of them decom-
poses in a finite direct sum of indecomposable motives; moreover, by [9, Corollary 2.6] , such a
decomposition is unique in the usual sense. The upper motive U (X ) (resp., U (Y )) is defined as
the summand with nontrivial Ch0 (unique in any decomposition given). By [9, Corollary 2.15],
the motives U (X ) and U (Y ) are isomorphic if and only if each of the two varieties XF (Y ) and YF (X )
possesses a 0-cycle of odd degree.
Let us also recall that canonical dimension cd(X ) of a smooth projective variety X is the
minimum of dimension of the image of a rational self-map X 99K X , c.f. [8]. See also [8,
Definition 1.3] for a definition using the essential dimension of a certain functor related to X .
Canonical 2-dimension, which appeared in the above proof, is its 2-local version also providing a
lower bound for it.
Corollary 5. For any anisotropic q as in Theorem 4, the canonical 2-dimension cd2(X ) of its
highest grassmannian X is equal to k + l = 2n +2n−2 −2.
Proof. As in the proof of Theorem 4, ei ∈ Ch(X ) for i ̸∈ {k, l }. Moreover, it follows from Theorem 4
(and has been shown in its proof explicitly) that neither ek nor el is in Ch(X ). Thus in terms of
the J-invariant in [4, Chapter 88], we have J (q) = {k, l } and so [4, Theorem 90.3] tells us that
cd2(X ) = k + l . □
Acknowledgements
We thank Detlev Hoffmann for providing reference [5]. We also thank Alexander Merkurjev and
Alexander Vishik for useful comments.
Declaration of interests
The authors do not work for, advise, own shares in, or receive funds from any organization
that could benefit from this article, and have declared no affiliations other than their research
organizations.
References
[1] R. Baeza, Quadratic forms over semilocal rings, Springer, 1978, pp. vi+199.
[2] G. Berhuy and G. Favi, “Essential dimension: a functorial point of view (after A. Merkur-
jev)”, Doc. Math. 8 (2003), pp. 279–330.
[3] V. Chernousov and A. Merkurjev, “Essential dimension of spinor and Clifford groups”,
Algebra Number Theory 8 (2014), no. 2, pp. 457–472.
[4] R. Elman, N. A. Karpenko and A. Merkurjev, The algebraic and geometric theory of quadratic
forms, American Mathematical Society, 2008, pp. viii+435.
Curtis R. Harvey and Nikita A. Karpenko 1543
[5] F. Faivre, Liaison des formes de Pfister et corps de fonctions de quadriques en caractéristique
2, PhD thesis, Université de Franche-Compte, Besançon, 2006.
[6] D. W. Hoffmann, “On the dimensions of anisotropic quadratic forms in I 4”, Invent. Math.
131 (1998), no. 1, pp. 185–198.
[7] O. T. Izhboldin, “On the cohomology groups of the field of rational functions”, in Mathe-
matics in St. Petersburg, American Mathematical Society, 1996, pp. 21–44.
[8] N. A. Karpenko, “Canonical dimension”, in Proceedings of the International Congress of
Mathematicians. Volume II , Hindustan Book Agency, 2010, pp. 146–161.
[9] N. A. Karpenko, “Upper motives of algebraic groups and incompressibility of Severi-Brauer
varieties”, J. Reine Angew. Math. 677 (2013), pp. 179–198.
[10] N. A. Karpenko, “Minimal canonical dimensions of quadratic forms”, Doc. Math. 2015
(2015), pp. 367–385.
[11] A. Merkurjev, “Essential dimension: a survey”, Transform. Groups 18 (2013), no. 2, pp. 415–
481.
[12] A. Pfister, “Quadratische Formen in beliebigen Körpern”, Invent. Math. 1 (1966), pp. 116–
132.
[13] E. Primozic, “Motivic Steenrod operations in characteristic p”, Forum Math. Sigma 8
(2020), article no. e52 (25 pages).
[14] B. Totaro, “Essential dimension of the spin groups in characteristic 2”, Comment. Math.
Helv. 94 (2019), no. 1, pp. 1–20.
[15] A. Vishik, “Motives of quadrics with applications to the theory of quadratic forms”, in
Geometric methods in the algebraic theory of quadratic forms, Springer, 2004, pp. 25–101.
[16] A. Vishik, “On the Chow groups of quadratic Grassmannians”, Doc. Math. 10 (2005),
pp. 111–130.


Quadratic forms in $I^n$ of dimension $2^n+2^{n-1}$

Francesco Cellarosi

Zachary Selk

Rough Paths above Weierstrass Functions

Sudip Ranjan Bhuia

Comptes Rendus
Mathématique
Sudip Ranjan Bhuia
Factorization of Hankel operators, range inclusion of Toeplitz and Hankel
operators on the vector-valued Hardy space
Volume 362 (2024), p. 1773-1778
Online since: 25 November 2024
https://doi.org/10.5802/crmath.680
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
C EN T R E
MER S ENN E
The Comptes Rendus. Mathématique are a member of the
Mersenne Center for open scientific publishing
www.centre-mersenne.org — e-ISSN : 1778-3569
Comptes Rendus. Mathématique
2024, Vol. 362, p. 1773-1778
https://doi.org/10.5802/crmath.680
Research article / Article de recherche
Functional analysis, Operator theory / Analyse fonctionnelle, Théorie des
opérateurs
Factorization of Hankel operators, range
inclusion of Toeplitz and Hankel operators on
the vector-valued Hardy space
Factorisation des opérateurs de Hankel, inclusion des
images des opérateurs de Toeplitz et de Hankel sur
l’espace de Hardy à valeurs vectorielles
Sudip Ranjan Bhuia a
a Indian Statistical Institute, Statistics and Mathematics Unit, 8th Mile, Mysore Road,
Bangalore, 560 059, India
E-mail: sudipranjanb@gmail.com
Abstract. Using Douglas theorem on factorization and range inclusion of bounded linear operators, we give
the factorization of Hankel operators, range inclusion of Hankel and Toeplitz operators defined on vector-
valued Hardy spaces.
Résumé. En utilisant le théorème de Douglas sur la factorisation impliquée par l’inclusion des images de
deux opérateurs linéaires bornés, nous donnons la factorisation des opérateurs de Hankel quand il y a
inclusion des images de deux opérateurs de Hankel ou de Toeplitz définis sur des espaces de Hardy à valeur
vectorielle.
Keywords. Toeplitz operators, Hankel operators, Hardy space, inner functions, Douglas Lemma, factoriza-
tion, majorization.
Mots-clés. Opérateurs de Toeplitz, opérateurs de Hankel, espace de Hardy, fonctions intérieures, lemme de
Douglas, factorisation, majorisation.
2020 Mathematics Subject Classification. 47A56, 47A63, 47A68, 47B20, 47B35.
Funding. The research of the author is supported by the NBHM postdoctoral fellowship, Department of
Atomic Energy (DAE), Government of India (File No: 0204/16(21)/2022/R&D-II/11995).
Manuscript received 1 May 2024, revised and accepted 20 August 2024.
1. Introduction and Preliminaries
Let E be a Hilbert space (here all Hilbert spaces are separable and over C) and T be a bounded
linear operator on E (T ∈B(E ) in short). The subspaces R(T ) = {T x : x ∈ E } and N (T ) = {x ∈ E :
T x = 0} denotes the range space and the null space, respectively. A linear operator T ∈ B(E ) is
ISSN (electronic): 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
1774 Sudip Ranjan Bhuia
said to be bounded below if there exists a constant M ≥ 0 such that ∥T x∥ ≥ M ∥x∥ for every x ∈ E ,
T ∈B(E ) is said to normal if T ∗T = T T ∗, hyponormal if T T ∗ ≤ T ∗T , and isometry if T ∗T = I .
LetD be the open unit disc in C, and let E be a Hilbert space. The E -valued Hardy space H 2
E
(D)
(H 2(D) if E = C ) over D is the Hilbert space of all E -valued analytic functions f (z) = ∑∞n=0 an zn ,
an ∈ E , z ∈D such that
∥ f ∥ =
( ∞∑
n=0
∥an∥2
) 1
2
<∞.
Let E∗ be another Hilbert space. Then denote by H∞B(E ,E∗)(D) (H
∞
B(E )(D) if E∗ = E ) the set of
B(E ,E∗)-valued bounded analytic functions on D which is a Banach space with the norm
defined by
∥Θ∥∞ = sup{∥Θ(z)∥ : |z| < 1}.
An operator-valued analytic function Θ ∈ H∞
B(E ,E∗)(D) is said to be inner if Θ(z)
∗Θ(z) = IE a.e.
z ∈T.
1.1. Toeplitz and Hankel Operators
In this section we give definitions of Toeplitz and Hankel operators and their basic properties. Let
E be any Hilbert space. Let L2
E
(T), where T denotes the unit circle in the complex plane, denote
the Hilbert space of all square E -valued Lebesgue integrable functions on T, that is
L2E (T) =
{
f :T−→ E measurable : ∥ f ∥2 =
∫
T
∥ f (z)∥2E dm(z) <∞
}
,
where m represent the normalized Lebesgue measure on T. The Hardy space H 2
E
(D) can also be
identified (via radial limits) with a subspace of E -valued functions in L2
E
(T), which we will also
denote by H 2
E
(D). This subspace consists of functions f for which f̂ (n) = 0 for all n < 0, where
f̂ (n) denotes the n-th Fourier coefficient of f .
Given another Hilbert space E∗, we denote by L∞B(E ,E∗)(T) ( L
∞
B(E )(T) if E∗ = E ) the set of
B(E ,E∗)-valued bounded functions on T.
Let E∗ and E be Hilbert spaces. For Φ ∈ L∞B(E ,E∗)(T), the Laurent operator LΦ : L
2
E
(T) → L2
E∗ (T)
is defined by (LΦ f )(z) = Φ(z) f (z), z ∈ T. In this case, LΦ is bounded and ∥LΦ∥ = ∥Φ∥∞. The
Toeplitz operator TΦ : H 2E (D) → H 2E∗ (D) with (operator-valued) symbolΦ is defined by
TΦ = PH 2
E∗ (D)
LΦ|H 2
E
(D),
where PH 2
E∗ (D)
(in short P ) is the orthogonal projection of L2
E∗ (T) onto H
2
E∗ (D). It is well known
that ∥TΦ∥ = ∥Φ∥∞ (cf. [1, Theorem 1.7, p. 112]).
The Toeplitz operator TΦ = A is characterized by the operator equation
T ∗z ATz = A.
Let E be a Hilbert space, and let Φ ∈ H∞
B(E )(D). The analytic Toeplitz operator TΦ : H
2
E
(D) →
H 2
E
(D) with symbolΦ is defined by
(TΦ f )(z) =Φ(z) f (z) ( f ∈ H 2E (D), z ∈D).
It is known that ∥TΦ∥ = ∥Φ∥∞, and TΦ is an isometry if and only ifΦ is inner [5, Proposition 2.2].
Let A ∈ B(H 2
E
(D)). Then ATz = Tz A if and only if A is an analytic Toeplitz operator that is,
A = TΦ for someΦ ∈ H∞B(E )(D).
Let J be defined on L2
E
(T) by
J f (z) = z f (z), f ∈ L2E (T).
Sudip Ranjan Bhuia 1775
The J maps zH 2
E
(D) onto H 2
E
(D), and J maps H 2
E
(D) onto zH 2
E
(D). This J is a unitary operator with
the following properties:
J∗ = J , J 2 = I , J M∗z = Mz J , JQ = P J , and JP =Q J ,
where Q := I −P the projection from L2
E
(T) to zH 2
E
(D) := L2
E
(T)⊖H 2
E
(D).
The Hankel operator HΦ from H 2E (D) into H
2
E∗ (D) is defined by
HΦh = JQ (Φh) = P J (Φh) , h ∈ H 2E (D).
The Hankel operator HΦ = A is characterized by the operator equation
ATz = T ∗z A. (1.1)
It is easy to verify that H∗Φ = HΦ(z)∗ . The Toeplitz and Hankel operators are related by the following
equation
TΦΨ−TΦTΨ = H∗Φ∗ HΨ equivalently, TΦ̌Ψ̃−TΦ̌TΨ̃ = HΦH∗Ψ,
where qΦ(z) =Φ(z) and Φ̃(z) =Φ(z)∗.
Theorem 1 ([5, p. 69]). Let T ∈B(H ) and T ′ ∈B(H ′) be two contractions.
(1) Let U ∈B(K ) and U ′ ∈B(K ′) be the minimal isometric dilation of T and T ′, respectively.
Then for any operator W ∈ B(H ,H ′) satisfying W T = T ′W, there exists W1 ∈ B(K ,K ′)
such that W1U =U ′W1, ∥W1∥ = ∥W ∥ , W = PH ′ W1|H and W (K ⊖H ) ⊂K ′⊖K ′.
(2) Let U ∈ B(K ) and U ′ ∈ B(K ′) be the co-isometric extension of T and T ′, respectively.
Then for any operator W ∈ B(H ,H ′) satisfying W T = T ′W, there exists W1 ∈ B(K ,K ′)
such that W1U =U ′W1, ∥W1∥ = ∥W ∥ , W = W1|H .
Theorem 2 (Douglas Theorem [2, Theorem 1]). If A,B ∈ B(H ), then the following statements
are equivalent:
(1) R(A) ⊆R(B);
(2) A A∗ ≤λ2BB∗ for some λ≥ 0;
(3) A = BC for some C ∈B(H ).
Moreover, if (1), (2), and (3) are valid, then there exists a unique operator C so that
(a) ∥C∥2 = inf{µ|A A∗ ≤µBB∗};
(b) N (A) =N (C );
(c) R(C ) ⊆R(B∗).
Here C = D∗, where D : R(B∗) → R(A∗) defined by D(B∗ f ) = A∗ f and D = 0 on the orthogonal
complement of R(B∗).
Theorem 3 (Beurling-Lax-Halmos theorem [5, Chapter V, Theorem 3.3]). Any Tz -invariant
subspace of M of H 2
E
(D) is of the form
M =ΘH 2F (D),
where F is a Hilbert space andΘ :D→B(F ,E ) is an inner function.
The main aim of this paper is to give the factorization of Hankel operators using Douglas
theorem on range inclusion, factorization and majorization (cf. Theorem 2). We also studied
the case when the range of the Toeplitz operator is included in the range of the Hankel operator
(cf. Theorem 7). We have extended the results of [4] in the vector-valued Hardy space setup. We
conclude by discussing the hyponormality of Hankel operators.
1776 Sudip Ranjan Bhuia
2. Main results
We begin this section with some observations.
Proposition 4. LetΨ ∈ H∞
B(E )(D) andΦ ∈ L∞B(E )(T). Then
(1) HΦTΨ = HΦΨ.
(2) T ∗ΨHΦ = HΦTΨ̃.
Proof. The proof of (1) follows directly from the definition. Let us prove (2).
T ∗ΨHΦ = (H∗ΦTΨ)∗ = (HΦ̃TΨ)∗ = H∗Φ̃Ψ = HΦΨ̃ = HΦTΨ̃. □
Remarks 5.
(1) For analytic function Φ ∈ H∞
B(E )(D), the subspace N (TΦ) is Tz -invariant, equivalently,
R(T ∗
Φ
) is T ∗z -invariant. By Theorem 3, there exists a Hilbert space F and an inner
functionΘ ∈ H∞
B(F ,E )(D) such that N (TΦ) = TΘH 2F (D).
(2) For any Φ ∈ L∞
B(E )(T), R(H
∗
Φ
) is T ∗z -invariant. By Theorem 3, there exists a Hilbert space
F and an inner function Θ ∈ H∞
B(F ,E )(D) such that N (HΦ) = TΘH 2F (D). If R(HΦ1 ) ⊆
R(HΦ2 ). Then, by Theorem 2, there exists C ∈ B(H 2E (D)) such that HΦ1 = HΦ2C . There-
fore, for any ∆ ∈ H∞
B(E )(D), using Proposition 4, we have
C∗T ∗∆H
∗
Φ2
=C∗H∗Φ2 T∆̃ = H∗Φ1 T∆̃ = T ∗∆H∗Φ1 = T ∗∆C∗H∗Φ2(
C∗T ∗∆ −T ∗∆C∗
)
H∗Φ2 = 0.
Therefore, C∗T ∗∆ −T ∗∆C∗ = 0 on R(H∗Φ2 ) and R(C T∆−T∆C ) ⊆N (HΦ2 ) = TΘH 2F (D).
LetΘ1,Θ2 ∈ H∞B(E )(D) withΘ2 be inner such that R(TΘ1 ) ⊆R(TΘ2 ). Then, by Theorem 2, there
exists C ∈B(H 2
E
(D)) such that TΘ1 = TΘ2C . Next, observe that
TΘ2 TzC = Tz TΘ2C = Tz TΘ1 = TΘ1 Tz = TΘ2C Tz .
Since TΘ2 is isometry, we have C Tz = TzC which implies C = TΘ3 for some Θ3 ∈ H∞B(E )(D). Thus
TΘ1 = TΘ2 TΘ3 .
Again ifΘ1,Θ2 ∈ H∞B(E )(D) withΘ1 is isometry and ∥Θ2∥∞ ≤ 1 such that TΘ1 T ∗Θ1 ≤ TΘ2 T ∗Θ2 , then
TΘ1 = TΘ2 TΘ3 for some Θ3 ∈ H∞B(E )(D) which is inner. This fact is an easy consequence of [5,
Proposition V.5.3].
By using the result in [3], we observe that if Θ1,Θ2 ∈ H∞B(E )(D) with ∥Θi∥∞ ≤ 1 for i = 1,2 such
that TΘ1 T
∗
Θ1
≤ TΘ2 T ∗Θ2 , then TΘ1 = TΘ2 TΘ3 for someΘ3 ∈ H∞B(E )(D) with ∥Θ3∥∞ ≤ 1.
In contrast to the aforementioned observations, we prove the following theorem, which also
generalizes the factorization of contractive analytic functions [5, p. 205]. Although this result may
be known to experts, we provide a different proof using Douglas theorem (cf. Theorem 2).
Theorem 6. Let Θ1,Θ2 ∈ H∞B(E )(D). Then R(TΘ1 ) ⊆ R(TΘ2 ) if and only if Θ1 = Θ2Θ3 for some
Θ3 ∈ H∞B(E )(D).
Proof. Suppose that R(TΘ1 ) ⊆ R(TΘ2 ). Then, by Theorem 2, there exists an operator W :
R(T ∗
Θ2
) → R(T ∗
Θ1
) such that T ∗Θ1 = W T ∗Θ2 . Since TΘi Tz = Tz TΘi , and R(T ∗Θi ) is T
∗
z -invariant for
i = 1,2, we have
W T ∗z T
∗
Θ2
=W T ∗Θ2 T ∗z = T ∗Θ1 T ∗z = T ∗z T ∗Θ1 = T ∗z W T ∗Θ2
W
(
T ∗z
∣∣
R(T ∗
Θ2
)
)
=
(
T ∗z
∣∣
R(T ∗
Θ1
)
)
W.
Therefore, by using (2) of Theorem 1, there exists W1 such that W1T ∗z = T ∗z W1 and W = W1|R(T ∗
Θ2
).
Therefore, we get W ∗1 is an analytic Toeplitz operator TΘ3 . Thus TΘ1 = TΘ2 TΘ3 which implies
Θ1 =Θ2Θ3. The converse trivially follows from the Theorem 2. □
Sudip Ranjan Bhuia 1777
Theorem 7. LetΦ,Φ1,Φ2,Ψ ∈ L∞B(E )(T). Then the following are true:
(1) R(TΦ) ⊆R(HΨ) if and only ifΦ= 0.
(2) Suppose that Φ is not identically zero, then R(H∗Φ) ⊆ R(T ∗Φ) if and only if P is bounded
below onΦH 2
E
(D).
(3) R(HΦ1 ) ⊆R(HΦ2 ) if and only if there existsΩ ∈ H∞B(E ) such that HΦ1 = HΦ2 TΩ = HΦ2Ω.
Proof.
(1). Suppose that R(TΦ) ⊆R(HΨ). Then, for some λ≥ 0
TΦT
∗
Φ ≤λ2HΨH∗Ψ
TΦT
∗
Φ ≤λ2(TΨ̌Ψ̃−TΨ̌TΨ̃).
Thus for every f ∈ H 2
E
(D), we have∥∥TΦ∗ f ∥∥2 +λ2 ∥∥TΨ̃ f ∥∥2 ≤λ2 ∥∥Ψ̃ f ∥∥2∥∥P (Φ∗ f )∥∥2 +λ2 ∥∥P (Ψ̃ f )∥∥2 ≤λ2 ∥∥Ψ̃ f ∥∥2 .
Since {zmη : m ∈N, η ∈ E } is total in H 2
E
(D), we have∥∥P (Φ∗zmη)∥∥2 +λ2 ∥∥P (Ψ̃zmη)∥∥2 ≤λ2 ∥∥Ψ̃(zmη)∥∥2 .
As m →∞, from above expression, we conclude that∥∥Φ∗∥∥2 +λ2 ∥∥Ψ̃∥∥2 ≤λ2 ∥∥Ψ̃∥∥2 ,
which impliesΦ= 0. The converse is evident.
(2). Suppose that R(H∗Φ) ⊆ R(T ∗Φ). Then, by Theorem 2, there exists λ ≥ 0 such that H∗ΦHΦ ≤
λ2T ∗ΦTΦ, which implies
TΦ∗Φ−TΦ∗TΦ ≤λ2T ∗ΦTΦ.
Therefore, for every f ∈ H 2
E
(D), we have∥∥Φ f ∥∥2 −∥∥TΦ f ∥∥2 ≤λ2 ∥∥TΦ f ∥∥2∥∥P (Φ f )∥∥2 ≥ (1+λ2)−1 ∥∥Φ f ∥∥2 .
Which shows that the orthogonal projection operator P is bounded below onΦH 2
E
(D).
Conversely, if P is bounded below on ΦH 2
E
(D), then there exists a constant M ≥ 0 such that∥∥P (Φ f )∥∥2 ≥ M ∥∥Φ f ∥∥2 for every f ∈ H 2
E
(D). Therefore,∥∥P (Φ(ηzn))∥∥2 ≥ M ∥∥Φ(ηzn)∥∥2 .
As n → ∞, we have M ≤ 1. Then choose M = (1+µ2)−1 for some µ ≥ 0, and apply the above
argument in the necessity part to obtain the desired conclusion.
(3). First, we assume that R(HΦ1 ) ⊆R(HΦ2 ). Then, by Theorem 2, we have
HΦ1 H
∗
Φ1
≤λ2HΦ2 H∗Φ2 , HΦ1 = HΦ2C
such that N (C ) = N (HΦ1 ), ∥C∥ ≤ λ2, and R(C ) ⊆ R(H∗Φ2 ). Note that C = W ∗, where W :
R(H∗
Φ2
) →R(H∗
Φ1
) is defined by W (H∗Φ2 f ) = H∗Φ1 f and W = 0 on R(H∗Φ2 )⊥.
Since T ∗z HΦ2 = HΦ2 Tz , we have N (HΦ2 ) is Tz -invariant, equivalently R(H∗Φ2 ) is T ∗z -invariant.
Therefore, by using (1.1), we get
W T ∗z H
∗
Φ2
=W H∗Φ2 Tz = H∗Φ1 Tz = T ∗z H∗Φ1 = T ∗z W H∗Φ2
which implies
W
(
T ∗z
∣∣
R(H∗
Φ2
)
)
=
(
T ∗z
∣∣
R(H∗
Φ1
)
)
W.
1778 Sudip Ranjan Bhuia
Using (2) of Theorem 1, there exists W1 : H 2E (D) → H 2E (D) such that W1T ∗z = T ∗z W1 and W =
W1|R(H∗
Φ2
). Thus W
∗
1 is an analytic Toeplitz operator TΩ and HΦ1 = HΦ2W ∗ = HΦ2 TΩ. The
converse is obvious. □
Corollary 8. Let Φ ∈ L∞
B(E )(T). Then HΦ is hyponormal if and only if HΦ = H∗ΦTΩ for some
Ω ∈ H∞
B(E )(D) with ∥Ω∥∞ ≤ 1.
Proof. Suppose that HΦ is hyponormal that is, HΦH∗Φ ≤ H∗ΦHΦ equivalently, HΦH∗Φ ≤ HΦ̃H∗Φ̃.
Now setΦ1 =Φ andΦ2 = Φ̃. By applying (3) of Theorem 7, we have HΦ = HΦ̃TΩ = H∗ΦTΩ for some
Ω ∈ H∞
B(E )(D) with ∥Ω∥∞ ≤ 1.
Conversely, suppose HΦ = H∗ΦTΩ for someΩ ∈ H∞B(E )(D) with ∥Ω∥∞ ≤ 1. Then ∥TΩ∥ = ∥Ω∥∞ ≤
1. Therefore,
H∗ΦHΦ−HΦH∗Φ = H∗Φ(I −TΩT ∗Ω)HΦ ≥ 0.
Hence HΦ is hyponormal. □
Corollary 9. LetΦ ∈ L∞
B(E )(T). If HΦ is hyponormal, then it is normal.
Proof. Suppose HΦ is hyponormal. By Corollary 8 and (2) of Proposition 4, we observe that
HΦ̃ = H∗Φ = T ∗ΩHΦ = HΦTΩ̃ = (H∗Φ)∗TΩ̃ = H∗Φ̃TΩ̃.
Which shows that if HΦ is hyponormal, then H∗Φ is also hyponormal, and consequently,
normal. □
Declaration of interests
The authors do not work for, advise, own shares in, or receive funds from any organization
that could benefit from this article, and have declared no affiliations other than their research
organizations.
References
[1] H. Bercovici, Operator theory and arithmetic in H∞, American Mathematical Society, 1988,
pp. xii+275.
[2] R. G. Douglas, “On majorization, factorization, and range inclusion of operators on Hilbert
space”, Proc. Am. Math. Soc. 17 (1966), pp. 413–415.
[3] R. B. Leech, “Factorization of analytic functions and operator inequalities”, Integral Equa-
tions Oper. Theory 78 (2014), no. 1, pp. 71–73.
[4] B. A. Lotto, “Range inclusion of Toeplitz and Hankel operators”, J. Oper. Theory 24 (1990),
no. 1, pp. 17–22.
[5] B. Sz.-Nagy, C. Foias, H. Bercovici and L. Kérchy, Harmonic analysis of operators on Hilbert
space, Second edition, Springer, 2010, pp. xiv+474.


Factorization of Hankel operators, range inclusion of Toeplitz and Hankel operators on the vector-valued Hardy space

Sam Edwards

Mikolaj Fraczyk

Minju Lee

Hee Oh

Infinite volume and atoms at the bottom of the spectrum

Patrick Buchfink

Silke Glas

Bernard Haasdonk

Approximation Bounds for Model Reduction on Polynomially Mapped Manifolds

Jin-Hui Fang

Fang-Gang Xue

Comptes Rendus
Mathématique
Jin-Hui Fang and Fang-Gang Xue
On weakly prime-additive numbers with length 4k +3
Volume 362 (2024), p. 275-278
Online since: 2 May 2024
https://doi.org/10.5802/crmath.555
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
C EN T R E
MER S ENN E
The Comptes Rendus. Mathématique are a member of the
Mersenne Center for open scientific publishing
www.centre-mersenne.org — e-ISSN : 1778-3569
Comptes Rendus. Mathématique
2024, Vol. 362, p. 275-278
https://doi.org/10.5802/crmath.555
Research article / Article de recherche
Number theory / Théorie des nombres
On weakly prime-additive numbers with
length 4k +3
Jin-Hui Fang ∗,a and Fang-Gang Xue b
a School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, P.R. China
b Nanjing University of Information Science & Technology, Nanjing 210044, P.R. China
E-mails: fangjinhui1114@163.com, xfg950118@126.com
Abstract. If a positive integer n has at least two distinct prime divisors and can be written as n = pα11 +·· ·+p
αt
t ,
where p1 < ·· · < pt are prime divisors of n and α1, . . . ,αt are positive integers, then we define such n as
weakly prime-additive. Obviously, t ⩾ 3. Following Erdős and Hegyvári’s work, Fang and Chen [J. Number
Theorey 182(2018), 258-270] obtained the following result: for any positive integer m, there exist infinitely
many weakly prime-additive numbers n with m | n and n = pα11 +·· ·+p
α5
5 , where p1, . . . , p5 are distinct prime
divisors of n and α1, . . . ,α5 are positive integers. In this paper, we prove the existence of such n with general
length t , where t ≡ 3 (mod 4) and t > 3. The main result is summarized as follows: for any positive integers
m, t with t ≡ 3 (mod 4) and t > 3, there exist infinitely many weakly prime-additive numbers n with m | n and
n = pα11 +·· ·+p
αt
t , where p1, . . . , pt are distinct prime divisors of n and α1, . . . ,αt are positive integers.
Keywords. weakly prime-additive numbers, Dirichlet’s theorem, the Chinese remainder theorem.
2020 Mathematics Subject Classification. 11A07, 11A41.
Funding. This work is supported by the National Natural Science Foundation of China, Grant No. 12171246
and the Natural Science Foundation of Jiangsu Province, Grant No. BK20211282.
Manuscript received 28 July 2023, revised 14 August 2023, accepted 17 August 2023.
1. Introduction
If a positive integer n has at least two distinct prime divisors and can be written as n = pα11 +·· · + pαtt , where p1 < ·· · < pt are prime divisors of n and α1, . . . ,αt are positive integers, then
we define such n as weakly prime-additive. Obviously, t ⩾ 3. In 1992, Erdős and Hegyvári [2]
firstly considered weakly prime-additive numbers and proved that, for any prime p, there exist
infinitely many weakly prime-additive numbers n such that p|n.
In 2018, Fang and Chen [5] made further research on weakly prime-additive numbers and
obtained the following result:
Theorem 1. For any positive integer m, there exist infinitely many weakly prime-additive num-
bers with t = 3 which are divisible by m if and only if 8 ∤m.
In [5], Fang and Chen also proved that:
∗Corresponding author.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
276 Jin-Hui Fang and Fang-Gang Xue
Theorem 2. For any positive integer m, there exist infinitely many weakly prime-additive num-
bers n with m | n and
n = pα11 +·· ·+pα55 ,
where p1, . . . , p5 are distinct prime divisors of n and α1, . . . ,α5 are positive integers.
Afterwards, Fang [3] proved that, for any positive integers m, t with t ≡ 1 (mod 4) and t > 1,
there exist infinitely many weakly prime-additive numbers n with m | n and n = pα11 + ·· ·+pαtt ,
where p1, . . . , pt are distinct prime divisors of n and α1, . . . ,αt are positive integers. In this paper,
by adding some tricky arguments, we solve the case t ≡ 3 (mod 4), where t > 3, that is:
Theorem 3. For any positive integers m, t with t ≡ 3 (mod 4) and t > 3, there exist infinitely many
weakly prime-additive numbers n with m | n and
n = pα11 +·· ·+pαtt , (1)
where p1, . . . , pt are distinct prime divisors of n and α1, . . . ,αt are positive integers.
Let k, l be integers with k ⩾ 2 and l ⩾ 3. Write
Sk =
{
n : n = ∑
p|n
pk ,ω(n)⩾ 2
}
,
where ω(n) is the number of distinct prime factors of n. In 2005, De Koninck and Luca [1]
obtained many nice results about Sk . As a main result, they considered the nature of Sk and
identified all integers n in S3 withω(n) = 3. Let {Pm}m⩾0 denote the Pell sequence given by P0 = 0,
P1 = 1 and Pm+1 = 2Pm +Pm−1 for m ⩾ 1. In 2022, Fang [4] proved that, a positive integer n can
be expressed as n = 22 + p2 + q2, where p, q are distinct odd prime factors of n, if and only if,
n = 22 +P 22m+1 +P 22m+3 for some positive integer m, where P2m+1, P2m+3 are both primes. One
may refer to [1] and [4] for details.
2. Proof of Theorem 3
The idea is from [5, Theorem 1.3] and [3, Theorem 1].
Proof. Write t = 4k+3 and m = 2am1, where a ⩾ 0 and 2 ∤m1. Then k ⩾ 1. Letφ denote the Euler
totient function. Similar to [3, Theorem 1], we consider the integers
n = 2u +pα11 +pα22 +·· ·+p
α4k+1
4k+1 +p4k+2, (2)
where pi are primes fixed later,
u = (a +2)φ(m1)
4k+2∏
i=1
(pi −1) and αi =
4k+2∏
j=i+1
p j −1
2
for i = 1,2, . . . ,4k +1.
(The method of this proof and the construction are the same as [3, Theorem 1], but we use a tricky
idea during the choices of p1, p2, . . . , p4k+2, which is the key point during the proof.)
By the Chinese remainder theorem and Dirichlet’s theorem there exists a prime p1 >
max{2a+2,m1,5} such that
p1 ≡ 1 (mod 2a+2), p1 ≡−1 (mod m1).
By the Chinese remainder theorem and Dirichlet’s theorem there exists a prime p2 > p1 such that
p2 ≡−1 (mod 2a+2), p2 ≡ 1 (mod m1), p2 ≡ 1 (mod p1).
By the Chinese remainder theorem and Dirichlet’s theorem there exists a prime p3 > p2 such that
p3 ≡−1 (mod 2a+2), p3 ≡ 1 (mod m1), p3 ≡−1 (mod p1), p3 ≡ 1 (mod p2).
Jin-Hui Fang and Fang-Gang Xue 277
By the Chinese remainder theorem and Dirichlet’s theorem there exists a prime p4 > p3 such that
p4 ≡−1 (mod 2a+2), p4 ≡−1 (mod m1), p4 ≡ 1 (mod p1), p4 ≡−1 (mod p2), p4 ≡ 1 (mod p3).
For i ⩾ 5, we will choose pi successively according to the parity of i . For j ⩾ 3, we firstly choose
p2 j−1, where 3 ⩽ j ⩽ 2k. By the Chinese remainder theorem and Dirichlet’s theorem there exists
a prime p2 j−1 > p2 j−2 such that
p2 j−1 ≡−1 (mod 2a+2), p2 j−1 ≡−1 (mod m1),
p2 j−1 ≡−1 (mod p1), p2 j−1 ≡ 1 (mod p2), p2 j−1 ≡ 1 (mod p3), p2 j−1 ≡ 1 (mod p4),
p2 j−1 ≡ 1 (mod p2s−1), p2 j−1 ≡−1 (mod p2s ) for s = 3,4, . . . , j −1.
(If j = 3, then we omit the last procedure).
Now we choose p2 j . For 3 ⩽ j ⩽ 2k, by the Chinese remainder theorem and Dirichlet’s
theorem there exists a prime p2 j > p2 j−1 such that
p2 j ≡−1 (mod 2a+2), p2 j ≡ 1 (mod m1),
p2 j ≡ 1 (mod p1), p2 j ≡−1 (mod p2), p2 j ≡−1 (mod p3), p2 j ≡−1 (mod p4),
p2 j ≡−1 (mod p2s−1), p2 j ≡ 1 (mod p2s ) for s = 3,4, . . . , j −1, p2 j ≡−1 (mod p2 j−1).
Furthermore, we add the restriction
p2 ≡ p3 ≡ 1 (mod 3), p4 ≡ 2 (mod 3), p2 j ≡ 1 (mod 5) for j = 3,4, . . . ,2k. (3)
(If 3 | m1, then the congruences p2 ≡ p3 ≡ 1 (mod 3) and p4 ≡ 2 (mod 3) follow from the condition
p2 ≡ p3 ≡ 1 (mod m1) and p4 ≡ −1 (mod m1); if 5 | m1, then the congruences p2 j ≡ 1 (mod 5)
follow from the condition p2 j ≡ 1 (mod m1) for j = 3,4, . . . ,2k; if (3,m1) = 1 or (5,m1) = 1, then (3)
follows from Dirichlet’s theorem.)
By the Chinese remainder theorem and Dirichlet’s theorem there exists a prime p4k+1 > p4k
such that
p4k+1 ≡−1 (mod 2a+2), p4k+1 ≡ 1 (mod m1),
p4k+1 ≡−1 (mod p1), p4k+1 ≡ 1 (mod p2), p4k+1 ≡ 1 (mod p3), p4k+1 ≡ 1 (mod p4),
p4k+1 ≡ 1 (mod p2s−1), p4k+1 ≡−1 (mod p2s ) for s = 3,4, . . . ,2k.
Finally, we will choose p4k+2. By the Chinese remainder theorem and Dirichlet’s theorem there
exists a prime p4k+2 > p4k+1 such that
p4k+2 ≡ 4k −1 (mod 2a+2), p4k+2 ≡−2 (mod m1),
p4k+2 ≡−1 (mod p1), p4k+2 ≡−3 (mod p2), p4k+2 ≡−3 (mod p3),
p4k+2 ≡−3 (mod p4), p4k+2 ≡ 1 (mod p5),
p4k+2 ≡−5 (mod p2s ), p4k+2 ≡ 1 (mod p2s+1) for s = 3,4, . . . ,2k.
(If k = 1, then we delete the middle stage and choose p5 and p6 by the last process).
Obviously,
p4k+2 > p4k+1 > ·· · > p2 > p1 > max{2a+2,m1,5}.
Noting that p1 ≡ 1 (mod 4), p4k+2 ≡ 4k −1 ≡ 3 (mod 4) and pi ≡ 3 (mod 4) (i = 2, . . . ,4k +1), we
could deduce from the law of quadratic reciprocity that(
p1
p j
)
=
(
p j
p1
)
for j = 2, . . . ,4k +2,
(
pi
p j
)
=−
(
p j
pi
)
for 2⩽ i < j ⩽ 4k +2. (4)
It follows from the definition of αi and
p
1
2 (p j −1)
i ≡
(
pi
p j
)
(mod p j ), 1⩽ i < j ⩽ 4k +2
278 Jin-Hui Fang and Fang-Gang Xue
that
pαii ≡
(
pi
p j
)
(mod p j ), 1⩽ i < j ⩽ 4k +2.
Thus, for each positive integer n with the form (2), we could deduce from (4) and the fact αi is
odd that
n ≡ 0+1+ (−1)+·· ·+ (−1)︸ ︷︷ ︸
4k times
+(4k −1) ≡ 0 (mod 2a+2),
n ≡ 1+ (−1)+1+1+ (−1)+ (−1)+1+·· ·+ (−1)+1+1+ (−2) ≡ 0 (mod m1),
n ≡ 1+0+1+ (−1)+1+ (−1)+·· ·+1+ (−1)+ (−1) ≡ 0 (mod p1),
n ≡ 1+
(
p1
p2
)
+0+1+ (−1)+·· ·+1+ (−1)+1+ (−3) ≡ 0 (mod p2),
n ≡ 1+
(
p1
p3
)
+
(
p2
p3
)
+0+1+1+ (−1)+·· ·+1+ (−1)+1+ (−3) ≡ 0 (mod p3),
n ≡ 1+
(
p1
p4
)
+
(
p2
p4
)
+
(
p3
p4
)
+0+1+ (−1)+·· ·+1+ (−1)+1+ (−3) ≡ 0 (mod p4).
For j ⩾ 3, we firstly consider p2 j−1, where 3⩽ j ⩽ 2k +1, we have
n ≡ 1+
(
p1
p2 j−1
)
+·· ·+
(
p2 j−2
p2 j−1
)
+0+ (−1)+1+·· ·+ (−1)+1+1 ≡ 0 (mod p2 j−1).
Now we consider p2 j , where 3⩽ j ⩽ 2k, we have
n ≡ 1+
(
p1
p2 j
)
+·· ·+
(
p2 j−1
p2 j
)
+0+ (−1)+1+·· ·+1+ (−1)+ (−5) ≡ 0 (mod p2 j ).
Finally, we consider p4k+2, we could deduce from (3) and (4) that
n ≡ 1+
(
p1
p4k+2
)
+
(
p2
p4k+2
)
+·· ·+
(
p4k
p4k+2
)
+
(
p4k+1
p4k+2
)
≡ 1+1−
( p2
3
)
−
( p3
3
)
−
( p4
3
)
+ (−1)+
( p6
5
)
+·· ·+ (−1)+
( p4k
5
)
+ (−1)
≡ 0 (mod p4k+2).
To sum up,
n = 2u +pα11 +pα22 +·· ·+p
α4k+1
4k+1 +p4k+2
and 4mp1p2 . . . p4k+1p4k+2 = 2a+2m1p1p2 . . . p4k+1p4k+2 | n.
This completes the proof of Theorem 3. □
Acknowledgments
The authors sincerely thank the referees for their invaluable suggestions which improve the
quality of this manuscript to a large extent.
References
[1] J.-M. De Koninck, F. Luca, “Integers representable as the sum of powers of their prime factors”, Funct. Approximatio,
Comment. Math. 33 (2005), p. 57-72.
[2] P. Erdős, N. Hegyvári, “On prime-additive numbers”, Stud. Sci. Math. Hung. 27 (1992), no. 1-2, p. 207-212.
[3] J.-H. Fang, “Note on the weakly prime-additive numbers”, J. Nanjing Norm. Univ., Nat. Sci. Ed. 41 (2018), no. 4, p. 26-
28.
[4] ——— , “A note on weakly prime-additive numbers”, Int. J. Number Theory 18 (2022), no. 1, p. 175-178.
[5] J.-H. Fang, Y.-G. Chen, “On the shortest weakly prime-additive numbers”, J. Number Theory 182 (2018), p. 258-270.


On weakly prime-additive numbers with length $4k+3$

Mao Hoshino

Kan Kitamura

Comptes Rendus
Mathématique
Mao Hoshino and Kan Kitamura
A note on quantum subgroups of free quantum groups
Volume 362 (2024), p. 1327-1330
Online since: 14 November 2024
https://doi.org/10.5802/crmath.641
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
C EN T R E
MER S ENN E
The Comptes Rendus. Mathématique are a member of the
Mersenne Center for open scientific publishing
www.centre-mersenne.org — e-ISSN : 1778-3569
Comptes Rendus. Mathématique
2024, Vol. 362, p. 1327-1330
https://doi.org/10.5802/crmath.641
Research article / Article de recherche
Functional analysis / Analyse fonctionnelle
A note on quantum subgroups of free
quantum groups
Une note sur les sous-groupes quantiques des groupes
quantiques libres
MaoHoshino a and Kan Kitamura a
a Graduate School of Mathematical Sciences, the University of Tokyo, 3-8-1 Komaba
Meguro-ku Tokyo 153-8914, Japan
E-mails: mhoshino@ms.u-tokyo.ac.jp, kankita@ms.u-tokyo.ac.jp
Abstract. In this short note, quantum subgroups in finite free products of the Pontryagin duals of free unitary
quantum groups are classified. They correspond to pairs of a subgroup Γ and a subset S of the free group Fn
such that S is Γ-invariant, containing Γ, and connected in the Cayley graph of Fn .
Résumé. Dans cette courte note, les sous-groupes quantiques dans les produits libres finis des duaux de
Pontryagin des groupes quantiques unitaires libres sont classifiés. Ils correspondent à des paires formées
d’un sous-groupe Γ et d’un sous-ensemble S du groupe libre Fn tel que S est Γ-invariant, contenant Γ, et
connexe dans le graphe de Cayley de Fn .
Keywords. free unitary quantum group, quantum subgroup, Cayley graph.
Mots-clés. groupe quantique unitaire libéré, sous-groupe quantique, graphe de Cayley.
2020 Mathematics Subject Classification. 46L67, 05C25.
Funding. MH was supported by Forefront Physics and Mathematics Program and JSPS KAKENHI Grant
Number JP23KJ0695. KK was supported by JSPS KAKENHI Grant Number JP22KJ0618.
Manuscript received 25 March 2024, revised 15 April 2024, accepted 17 April 2024.
1. Introduction
In this note, we classify quantum subgroups in the Pontryagin duals of free unitary quantum
groups or, more generally, in their finite free products. Associated with Q ∈ GLN (C) for N ≥ 2,
van Daele–Wang [4] defined a compact quantum group called a free unitary quantum group,
which shall be denoted by U+Q . It is constructed by forgetting the commutativity relations for the
coefficients of a unitary matrix. Variations of this construction have provided abundant examples
of compact quantum groups.
For general conventions and fundamental facts on compact quantum groups and tensor
categories, we refer to [3]. For a compact quantum group G , its finite dimensional unitary
representations form a rigid semisimple C*-tensor category with a simple unit object, denoted
by RepG . We write IrrG for the set of all isomorphism classes of simple objects in RepG . We write
Ĝ for the Pontryagin dual of G , which is a discrete quantum group.
ISSN (electronic): 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
1328 Mao Hoshino and Kan Kitamura
The notion of free product for (the C*-algebras of) discrete quantum groups was studied
by Wang [5]. The main object of this note is the free product ∗nj=1Û+Q j for n ∈ Z≥1, Q j ∈
GL(N j ,C), N j ≥ 2, defined by the Pontryagin dual F̂ of the compact quantum group F given
by Cu(F ) = ∗nj=1Cu(U+Q j ) with a canonical comultiplication. This quantum group F̂ has the
following remarkable property analogous to the free group Fn with n generators: for any compact
quantum group G , any n-tuple of unitary representations (u j )nj=1 of G with dimCu j = N j
and Q∗−1j u
⊤
j Q
∗
j being a unitary uniquely induces a unital ∗-homomorphism Cu(F ) → Cu(G)
preserving comultiplications. Quantum subgroups of F̂ has been recently classified by Freslon–
Weber [2] when F̂ = Û+Q , together with rich structural results for such quantum subgroups.
We classify quantum subgroups of F̂ in general, based on a simple observation relating IrrF
to the path space of the Cayley graph of Fn . Our classification exhibits to what extent more
quantum subgroups exist than mere subgroups of free groups and thus could be regarded as a
complementary result to [2] towards the quantum analogue of Kurosh’s theorem.
2. Classification
Note that the classification of quantum subgroups in F̂ is equivalent to the classification of idem-
potent complete full rigid C*-tensor subcategories in RepF , as a consequence of Woronowicz’s
Tannaka–Krein duality. Since the latter is a purely categorical problem and in fact relies only on
the fusion rule of RepF , we work with RepF rather than F̂ and restrict to a fixed skeleton of RepF
for simplicity.
The theory of unitary representations of U+Q was investigated by Banica [1]. We recall its fusion
rule. Consider the free product of two copies of monoids M := {αr |r ∈ Z≥0}∗ {βr |r ∈ Z≥0} ∼=
Z≥0∗Z≤0. By abusing notation, we write [ar ] :=αr and [a−r ] :=βr in M for r ∈Z≥0. Then, we can
identify IrrU+Q with M so that the unit object is [a
0], [ar ] = [a−r ] for r ∈Z, (x y) = y x for x, y ∈ M ,
and the tensor product is determined recursively by
(x[ar ])⊗ ([as ]y) =
{
x[ar ][as ]y = x[ar+s ]y if r s = 1,
x[ar ][as ]y ⊕ (x ⊗ y) if r s =−1 (x, y ∈ M ,r, s ∈ {±1}),
with the aid of the fact that any element in M is expressed as a word of [a1] and [a−1].
Combined with [5], we can canonically identify IrrF with the monoid M∗n as a set. For
1 ≤ j ≤ n, we put α j , β j for the generators of the j th component M = {αrj |r ∈Z≥0}∗ {βrj |r ∈Z≥0}
in M∗n , and a j for the generator of the j th component Z ∼= {arj |r ∈ Z} in Fn . We shall write
[arj ] :=αrj and [a−rj ] :=βrj in M∗n for r ∈Z≥0.
Consider the Cayley graph G of Fn , where the set of its oriented edges is {(g , g a
±1
j ) | g ∈ Fn ,1 ≤
j ≤ n}. Let Path(G ) be the set of all finite paths in G starting from 1Fn , possibly with turning-
backs. We can identify Path(G ) with M∗n by assigning each path of the form (g0, g1, . . . , gk ) ∈ Fk+1n
with k ≥ 0, g0 = 1Fn , gl = gl−1arljl , 1 ≤ jl ≤ n, rl = ±1 for 1 ≤ l ≤ k to [a
r1
j1
][ar2j2 ] · · · [a
rk
jk
] ∈ M∗n .
Then, extending our convention of [arj ] for r ∈ Z, we define [g ] ∈ M∗n for g ∈ Fn by the shortest
path from 1Fn to g via the identification M
∗n = Path(G ). From now, the symbol k, j and r ,
possibly accompanied with indices, stand for elements in Z≥0, Z∩ [1,n], and {±1} respectively,
unless clarified otherwise. Note that any element in IrrF = M∗n can be expressed in the form
[ar1j1 ][a
r2
j2
] · · · [arkjk ] without redundancy.
Remark. We reinterpret the fusion rule of RepF in terms of G . Take any e,e ′ ∈ Path(G ), whose
endpoints are denoted by g , g ′ ∈ Fn , respectively. We write g e ′ for the path from g to g g ′ given
by the left translation of Fn on G and ee ′ ∈ Path(G ) for the concatenation of e and g e ′. Note
that ee ′ coincides with the product of the monoid M∗n . We set e0 := e, e ′0 := e ′ and define
el+1,e ′l+1 ∈ Path(G ) = M∗n recursively on l ∈Z≥0 so that el+1[arj ] = el and [a−rj ]e ′l+1 = e ′l as long as
Mao Hoshino and Kan Kitamura 1329
such j and r exist. If we regard e,e ′ ∈ IrrF , the tensor product e ⊗ e ′ is the direct sum of all paths
of the form el e
′
l ∈ Path(G ) = IrrF . Also, when we write e ∈ Path(G ) for the path from 1Fn to g−1
given by reversing e, the involution e 7→ e corresponds to the conjugate operation on IrrF . If we
put V (e) ⊂ Fn for the set of all vertices appearing in e, we have V (el e ′l ) ⊂ V (ee ′) = V (e)∪ gV (e ′)
for el and e
′
l above, and V (e) = g−1V (e).
Our main result is as follows.
Main Theorem. For any pair (Γ,S) of a subgroup Γ ⊂ Fn and a left Γ-invariant subset S ⊂ Fn
containing Γ and connected in the Cayley graph G , there is an idempotent complete full rigid C*-
tensor subcategory of RepF such that the set of its irreducible objects equals{
[ar1j1 ][a
r2
j2
] · · · [arkjk ] ∈ IrrF
∣∣ar1j1 ar2j2 · · ·arkjk ∈ Γ, ar1j1 ar2j2 · · ·arljl ∈ S for all 0 ≤ l ≤ k}. (I)
Conversely, any idempotent complete full rigid C*-tensor subcategory of RepF is of this form for
a pair (Γ,S) with the conditions above, which must be necessarily unique.
Proof. It is not hard to see the first half of the statement with the aid of equations on V (e) in the
remark above. We show the latter half. Note that (Γ,S) can be reconstructed from (I) since S is
connected and contains the unit element. This shows the uniqueness. Let C be an idempotent
complete full rigid C*-tensor subcategory of RepF and I be the set of irreducible objects in C .
For any [ar1j1 ] · · · [a
rk
jk
] ∈ I \ {[1Fn ]}, inductively on l = 1, . . . ,k, we see
[ar1j1 · · ·a
rl
jl
][arl+1jl+1 ] · · · [a
rk
jk
] ≤ ([ar1j1 · · ·arl−1jl−1 ][a−rl−1jl−1 · · ·a−r1j1 ])⊗ ([ar1j1 · · ·arl−1jl−1 ][arljl ] · · · [arkjk ]) ∈C , (1)
and
[ar1j1 · · ·a
rl
jl
][a−rljl · · ·a
−r1
j1
] ≤ ([ar1j1 · · ·arljl ][arl+1jl+1 ] · · · [arkjk ])⊗ ([a−rkjk ] · · · [a−rl+1jl+1 ][a−rljl · · ·a−r1j1 ]) ∈C . (2)
Indeed, (1) for l = 1 is trivial, and (2) for l can be obtained by tensoring (1) for l with its conjugate,
while (1) for l +1 can be seen by tensoring (2) for l and (1) for l .
Clearly, S := {ar1j1 · · ·a
rl
jl
| [ar1j1 ] · · · [a
rk
jk
] ∈ I ,0 ≤ l ≤ k} ⊂ Fn is connected in G as every element in S
can be connected to 1Fn by a path within S, and Γ := {ar1j1 · · ·a
rk
jk
| [ar1j1 ] · · · [a
rk
jk
] ∈ I } is a subgroup of
Fn contained in S. Then, (1) and (2) for l = k imply [g ] ∈ I for all g ∈ Γ and [g ][g−1] ∈ I for all g ∈ S,
respectively. For all g ∈ Γ and h ∈ S, it holds [g ][h][h−1] ∈ I as a direct summand of [g ]⊗([h][h−1])
and thus g h ∈ S. By construction, the set defined by (I) contains I . They coincide because for any
[ar1j1 ] · · · [a
rk
jk
] ∈ IrrF satisfying the conditions in (I),
[ar1j1 ] · · · [a
rk
jk
] ≤ ([ar1j1 ] · · · [arkjk ][a−rkjk · · ·a−r1j1 ])⊗ [ar1j1 · · ·arkjk ]
≤ ([ar1j1 ][a−r1j1 ])⊗ ([ar1j1 ar2j2 ][a−r2j2 a−r1j1 ])⊗·· ·⊗ ([ar1j1 · · ·arkjk ][a−rkjk · · ·a−r1j1 ])⊗ [ar1j1 · · ·arkjk ] ∈C . □
Acknowledgments
The authors would like to thank Amaury Freslon and Moritz Weber for their comments on the
draft and their encouragement to release the content. They would like to thank their advisor
Yasuyuki Kawahigashi for his invaluable support.
Declaration of interests
The authors do not work for, advise, own shares in, or receive funds from any organization
that could benefit from this article, and have declared no affiliations other than their research
organizations.
1330 Mao Hoshino and Kan Kitamura
References
[1] T. Banica, “Le groupe quantique compact libre U (n)”, Commun. Math. Phys. 190 (1997),
no. 1, pp. 143–172.
[2] A. Freslon and M. Weber, “Discrete quantum subgroups of free unitary quantum groups”,
2024. https://arxiv.org/abs/2403.01526.
[3] S. Neshveyev and L. Tuset, Compact quantum groups and their representation categories,
Société Mathématique de France, 2013, pp. vi+169.
[4] A. Van Daele and S. Wang, “Universal quantum groups”, Int. J. Math. 7 (1996), no. 2, pp. 255–
263.
[5] S. Wang, “Free products of compact quantum groups”, Commun. Math. Phys. 167 (1995),
no. 3, pp. 671–692.


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