L’INSTITUTFOURIER DE ANNALES

A L

E S

E D L ’IN IT ST T U

F O U R

ANNALES

DE

L’INSTITUT FOURIER

Rajeev GUPTA & Samya K. RAY

On a Question of N. Th. Varopoulos and the constantC₂(n) Tome 68, n^o6 (2018), p. 2613-2634.

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ON A QUESTION OF N. TH. VAROPOULOS AND THE CONSTANT C

(n)

by Rajeev GUPTA & Samya K. RAY (*)

Abstract. — LetCk[Z1, . . . , Zn] denote the set of all polynomials of degree at mostkinncomplex variables andCⁿdenote the set of alln-tupleT= (T1, . . . , Tn) of commuting contractions on some Hilbert spaceH.The interesting inequality

K_G^C 6 lim

n→∞C2(n)62K_G^C, where

Ck(n) = sup

kp(T)k:kpk_Dn,∞61, p∈Ck[Z1, . . . , Zn],T∈Cn

and K_G^C is the complex Grothendieck constant, is due to Varopoulos. We answer a long–standing question by showing that the limit limn→∞C₂(n)

K^C G

is strictly big- ger than 1.LetC^s2[Z1, . . . , Zn] denote the set of all complex valued homogeneous polynomials p(z1, . . . , zn) =Pn

j,k=1a_jkzjz_k of degree two inn-variables, where ((ajk)) is an×ncomplex symmetric matrix. For each n∈ N,define the linear map An : (C^s2[Z1, . . . , Zn],k · k_Dn,∞) → (Mn,k · k∞→1) to be An(p) = ((ajk)).

We show that the supremum (overn) of the norm of the operatorsAn;n∈N,is bounded below by the constantπ²/8.Using a class of operators, first introduced by Varopoulos, we also construct a large class of explicit polynomials for which the von Neumann inequality fails. We prove that the original Varopoulos–Kaijser polyno- mial is extremal among a, suitably chosen, large class of homogeneous polynomials of degree two. We also study the behaviour of the constantCk(n) asn→ ∞.

Résumé. — Soit Ck[Z1, . . . , Zn] l’ensemble de tous les polynômes de degré au plus kdansnvariables complexes etCn l’ensemble de tous lesn-tuplesT = (T1, . . . , Tn) de contractions qui commutent sur un espace de HilbertH.L’inégalité intéressante

K_G^C 6 lim

n→∞C2(n)62K_G^C, où

Ck(n) = sup

kp(T)k:kpk_Dn,∞61, p∈Ck[Z1, . . . , Zn],T∈Cn

Keywords: Grothendieck Inequality, von Neumann Inequality, Varopoulos Operator, Grothendieck Constant, Positive Grothendieck Constant.

2010Mathematics Subject Classification:47A13, 47A25, 47A60, 47A63.

(*) The first named author is supported by National Postdoctoral Fellowship (Ref. No.

PDF/2016/000094), SERB, DST, Government of India. The second named author is supported by Council for Scientific and Industrial Research, MHRD, Government of India.

et K^C_G est la constante complexe de Grothendieck, est due à Varopoulos. Nous répondons à une question de longue date en montrant que limn→∞C₂(n)

K^C G

est stric- tement plus grand que 1. Soit C^s2[Z1, . . . , Zn] l’ensemble des polynômes homo- gènes complexes p(z1, . . . , zn) =Pn

j,k=1ajkzjzkde degré deux dansn-variables, oú ((ajk)) est une matrice symétrique complexe n×n. Pour chaquen ∈ N,on définit la carte linéaire An : (C^s2[Z1, . . . , Zn],k · k_Dn,∞) → (Mn,k · k∞→1) par An(p) = ((a_jk)).Nous montrons que le supremum (sur n) de la norme des opé- rateursAⁿ;n ∈ N,est borné inférieurement par la constanteπ²/8.En utilisant une classe d’opérateurs, introduite par Varopoulos, nous construisons aussi une grande classe de polynômes explicites pour laquelle l’inégalité de von Neumann n’est pas satisfaite. Nous prouvons que le polynôme de Varopoulos–Kaijser est ex- trémal parmi une grande classe convenablement choisie de polynômes homogènes de degré deux. Nous étudions également le comportement de la constanteCk(n) lorsquen→ ∞.

1. Introduction

Let C[Z1, . . . , Zn] denote the set of all complex valued polynomials in n-variables. For any continuous functionf :X →C,on a compact set X, we letkfkX,∞denote its supremum norm, namely,

kfkX,∞= sup{|f(x)|:x∈X}.

LetDⁿ denote the unit polydisc inCⁿ. The von Neumann inequality [15]

states thatkp(T)k6kpk_D,∞ for allp∈C[Z] and for any contractionT on a complex Hilbert space. For any pair of commuting contractions T1, T2, a generalization of the von Neumann inequality: kp(T₁, T₂)k 6 kpk_D2,∞, p∈C[Z1, Z2] follows from a deep theorem of Ando [2] on unitary dilation of a pair of commuting contractions. Forn∈N,letCndenote the set of alln- tupleT = (T₁, . . . , T_n) of commuting contractions on some Hilbert spaceH. In the paper [22], Varopoulos showed that the von Neumann inequality fails forT inCn, for somen >2.He along with Kaijser [22] and simultaneously Crabb and Davie [6] produced an explicit example of three commuting con- tractionsT₁, T₂, T₃and a polynomialpfor whichkp(T₁, T₂, T₃)k>kpk_D3,∞. For a fixedk∈N,define (see [23] and [19, p. 24]):

Ck(n) = sup

kp(T)k:kpk_Dⁿ,∞61, p∈Ck[Z1, . . . , Zn],T ∈Cn

and letC(n) denote lim_k→∞Ck(n).

Since the counterexample to the von Neumann inequality in three vari- ables, due to Varopoulos and Kaijser [22], involves a (explicit) homoge- neous polynomial of degree two therefore C₂(3) > 1. From the von Neu- mann inequality and its generalization to two variables, it follows that

C(1) =C(2) = 1.In the paper [23], Varopoulos shows that

(1.1) K_G^C6 lim

n→∞C2(n)62K_G^C,

whereK_G^Cis the complex Grothendieck constant defined below.

Definition 1.1(Grothendieck Constant). — For a complex (real) array A:= a_jk

n×n,define the following norm (1.2) kAk∞→1:= sup

|hAv, wi|:kvk`<sup>∞</sup>(n)61,kwk`^∞(n)61 , wherev andware vectors inCⁿ (resp.Rⁿ). There exists a finite constant K >0such that for any choice of unit vectors(x_j)ⁿ₁ and(y_k)ⁿ₁ in a complex (resp. real) Hilbert spaceH, we have

(1.3)

n

X

j,k=1

ajkhxj, yki

6KkAk∞→1

for all n ∈ N and A = a_jk

. The least such constant is denoted by KG and is known as the Grothendieck constant. Note that the definition of KG depends on the underlying field. When it is the field of complex numbers C (resp. R), this constant is known as the complex (resp. real) Grothendieck constant and is denoted byK_G^C(resp.(K_G^R)). It is known that 1.338< K_G^C61.4049,and 1.666K_G^R6 _2log(1+^{π √}₂₎, see[20, §4]. Recently, it has been proved in[5]that this upper bound ofK_G^Ris strict which settles a long–standing conjecture of Krivine [12, 13]. We refer the reader to [4]

for some explicit computations of this constant for small values ofn. For more on Grothedieck constant, we refer the reader to[20].

Since it is known that K_G^C >1, the inequality (1.1) is yet another way to see that the von Neumann inequality fails eventually. We refer to the inequality (1.1) asthe Varopoulos inequality. In the paper [23], Varopoulos had implicitly asked if lim_n→∞C₂(n) =K_G^C? Recently, first named author of this paper, has improved the Varopoulos inequality (1.1):

(1.4) K_G^C6 lim

n→∞C₂(n)6 3√ 3 4 K_G^C.

This inequality is proved by first obtaining a bound for the second derivative of any holomorphic map f : Dⁿ → D, namely, kD²f(0)k∞→1 6 ³

√ 3 2 . In this paper, we answer the question of Varopoulos in the negative by improving the lower bound in the inequality (1.1). Indeed, we prove that lim_n→∞C2(n)>1.118K_G^C.

In what follows, for each p∈ [1,∞] and n∈ N, we denote the normed linear space (Cⁿ,k · kp) by `^p(n) and when the space is (Rⁿ,k · kp) then

we choose to denote it by `^p

R(n). Let C^s2[Z₁, . . . , Z_n] denote the set of all homogeneous polynomials of degree two in n-variables. A homogeneous polynomial of degree two inn-variables is of the form

p(z1, . . . , zn) =

n

X

j,k=1

ajkzjzk,

whereAp:= ajk

is a symmetric matrix associated to p. Define the map An : C^s2[Z1, . . . , Zn] → M_n^s by the rule An(p) = Ap, where M_n^s is the set of all symmetric matrices of order n. Equip C^s2[Z1, . . . , Zn] with the supremum normk · k_Dⁿ,∞andM_n^swith the normk · k∞→1.For eachn∈N, kAn⁻¹k61,thereforekAnk>1.

In [9], it is shown that lim_n→∞kAnk63√

3/4.In this paper, we prove limn→∞kAnk > π²/8, improving the bound limn→∞kAnk >1.2323, ob- tained earlier in an unpublished article by Holbrook and Schoch (see their related work [11]).

In Section 3, we investigate, in some detail, the constant C2(n). We exhibit a large class of examples of Varopoulos–Kaijser type and show that the original Varopoulos–Kaijser example is extremal, in an appropriate sense, in this large class of polynomials.

Let M_n⁺(C) (resp. M_n⁺(R)) denote the set of all n×n complex (real) non-negative definite matrices.

Definition 1.2 (Positive Grothendieck Constant). — Suppose A :=

ajk

n×n is a complex (real) non-negative definite array then there exists K > 0, independent of n and A, such that (1.3) holds. The least such constant is denoted by K_G⁺(C) (resp. K_G⁺(R)) and called complex (resp.

real) Positive Grothendieck constant. The values of K_G⁺(C) and K_G⁺(R) are exactly4/πandπ/2 respectively, see[20, p. 259–260]and[18, Remark following Theorem 5.4]. To the reader, we also refer[17, Theorem 1.3] for these constants.

The non-negative definite Grothendieck constant plays an important role in operator theory. We refer [3, Theorem 1.9] for some important connec- tions.

For anyn×ncomplex matrix A:= ajk

,we associate a homogeneous polynomial of degree two denoted byp_A and defined by p_A(z1, . . . , zn) = Pn

j,k=1ajkzjzk. Suppose p is the following polynomial of degree at most two inn-variables and of the form

p(z1, . . . , zn) =a0+

n

X

j=1

ajzj+

n

X

j,k=1

ajkzjzk

witha_jk =a_kj (can be assumed without loss of generality) for allj, k = 1, . . . , n.Corresponding top,one can define the following (n+ 1)×(n+ 1) symmetric complex matrix

(1.5) A(p) =













 a0 1

2a1 1

2a2 · · · ¹₂an 1

2a₁ a₁₁ a₁₂ · · · a1n 1

2a2 a12 a22 · · · a2n

... ... ... ...

1

2a_n a_1n a_2n · · · a_nn















It can be seen thatkpk_Dn,∞=kp_A(p)k_Dn+1,∞.We define the following quan- tity:

C₂⁺(n) = sup

kp_A(T)k kp_Ak_Dⁿ,∞

: 06=A∈M_n⁺(C),T ∈ Cn

and let C₂⁺ denote the quantity lim_n→∞C₂⁺(n). It is clear from the def- initions that C2(n) > C₂⁺(n) for each n. In this paper, we prove that lim_n→∞C₂⁺(n) = π/2. Since K_G^C 6 1.4049 therefore lim_n→∞C2(n) >

π/2>1.118K_G^C.

Acknowledgement. We are very grateful to Prof. Gadadhar Misra and Prof. Gilles Pisier for several fruitful discussions and suggestions. The au- thors also express their sincere gratitude to Prof. Sameer Chavan and Prof.

Parasar Mohanty for their constant support. We also thank the referee and the editor for several constructive suggestions which significantly improved the presentation of the paper.

2. Improvement of the Lower Bound in the Varopoulos inequality

In this section we improve the lower bound of the Varopoulos inequality and as a consequence, we answer negatively a question of Varopoulos posed in the paper [23]. The following theorem (see [9]) concerns an improvement in the upper bound of the Varopoulos inequality.

Theorem 2.1. — Suppose p is a polynomial of degree at most 2 in n-variables andT ∈Cn.Thenkp(T)k6 ³

√3

4 K_G^Ckpk_Dn,∞.

Throughout this paper, the vectors are assumed to be row vectors unless specified otherwise. The following series of lemmas are the key ingredients of this paper.

Lemma 2.2. — For any symmetric n×n matrix A = a_jk

, we have the equality:

sup

kxjk_`261

n

X

j,k=1

ajkhxj, xki

= sup

kRjk_`2 R

61

n

X

j,k=1

ajkhRj, Rki .

Proof. — Fix xj ∈ C^m with kxjk_`2(m) 6 1 for j = 1, . . . , n. Define Rj= ^xj+x₂ ^j, i^xj−x₂ ^j

forj= 1, . . . , n.We see that hRj, Rki=

m

X

p=1

x^(p)_j +x^(p)_j 2

! x^(p)_k +x^(p)_k 2

!

+ i²

m

X

p=1

x^(p)_j −x^(p)_j 2

! x^(p)_k −x^(p)_k 2

!

=

m

X

p=1

<x^(p)_j <x^(p)_k +=x^(p)_j =x^(p)_k

=<

m

X

p=1

x^(p)_j x^(p)_j

!

= hxj, xki+hxk, xji

2 ,

where<z and=zdenote the real and imaginary part of zrespectively. In particular, we get thatkRjk_`2

R(2m)=kxjk_`2(m)for eachj= 1, . . . , n.Since Ais a symmetric matrix therefore

n

X

j,k=1

ajkhxj, xki=

n

X

j,k=1

ajkhxk, xji

=

n

X

j,k=1

a_jkhxj, xki+hxk, xji 2

=

n

X

j,k=1

ajkhRj, Rki.

This shows that for eachm ∈ N and symmetric matrix A, one gets the following identity

sup

kxjk_{`2 (m)}61

Xajkhxj, xki

= sup

kRjk_`2 R(2m)61

XajkhRj, Rki .

This completes the proof.

Remark 2.3. — For any matrixA,one can associate a symmetric matrix S(A) = (A+A^t)/2,which has the propertykp_Ak_Dⁿ_,∞=kp_S(A)k_Dⁿ_,∞.More- over, ifAis a non-negative definite matrix thenS(A) is a real non-negative definite matrix.

Lemma 2.4. — Suppose A = ajk

is a non-negative definite n×n matrix. Then

sup

z_j∈T

n

X

j,k=1

ajkzjzk

= sup

s_j=±1 n

X

j,k=1

ajksjsk.

Proof. — Suppose A is a n×n complex non-negative definite matrix.

From Remark 2.3, we know thatS(A) is a real non-negative definite matrix withkp_Ak_Dⁿ_,∞=kp_S(A)k_Dⁿ_,∞.Thus, to prove this lemma, it suffices to work with real non-negative definite matrices only.

Let (z₁⁰, . . . , z⁰_n) be a maximizing vector for kp_Ak_Dⁿ,∞ i.e. (z₁⁰, . . . , z⁰_n) satisfies

sup

z_j∈T

n

X

j,k=1

a_jkz_jz_k

=

n

X

j,k=1

a_jkz_j⁰z⁰_k .

Define ˜zj = e^−iφ/2z⁰_j for j = 1, . . . , n, where φ = arg Pn

j,k=1ajkz_j⁰z_k⁰ . Then, we have the following

n

X

j,k=1

a_jkz˜_j˜z_k =

n

X

j,k=1

a_jkz⁰_jz_k⁰ .

Therefore

n

X

j,k=1

ajkz˜jz˜k+

n

X

j,k=1

ajkz˜j˜zk = 2

n

X

j,k=1

ajkz_j⁰z_k⁰

and hence one concludes the following identity

sup

zj∈T

n

X

j,k=1

ajkzjzk+

n

X

j,k=1

ajkzjzk

= 2 sup

zj∈T

n

X

j,k=1

ajkzjzk

.

SinceA is real non-negative definite matrix therefore we observe the fol- lowing

1 2 sup

z_j∈T

n

X

j,k=1

a_jkz_jz_k+

n

X

j,k=1

a_jkz_jz_k

= sup

θ_j∈R

n

X

j,k=1

a_jkcos(θ_j+θ_k)

= sup

θj∈R

n

X

j,k=1

ajk(cosθjcosθk−sinθjsinθk)

=

n

X

j,k=1

ajkcosδjcosδk−

n

X

j,k=1

ajksinδjsinδk

6

n

X

j,k=1

ajkcosδjcosδk

6 sup

sj=±1 n

X

j,k=1

ajksjsk,

whereδ_j, j = 1, . . . , n,are chosen such thatPn

j,k=1a_jkcos(θ_j+θ_k) is pos- itive and attains maximum in modulus at θj = δj for j = 1, . . . , n. The last inequality in above computation can be deduced from the fact that

“For any convex subset Ω ofRⁿ and for any non-negative definite matrix A, the functionf : Ω→Rdefined by f(x) =hAx, xi is convex” (see [16, Corollary 3.9.5]). Thus we get the identity

sup

z_j∈T

n

X

j,k=1

ajkzjzk

= sup

sj=±1 n

X

j,k=1

ajksjsk.

This proves the claim.

We now prove the main theorem as a corollary of Lemma 2.2 and Lem- ma 2.4. For the proof, it would be convenient to introduce, what we call, Varopoulos operators.

LetHbe a separable Hilbert space and{ej}j∈Nbe an orthonormal basis forH. For anyx∈H, definex^]:H→Cby x^](y) =P

jxjyj, where x= P

jxjej andy=P

jyjej. Forx, y∈H, we set x^], y

=x^](y).ThenH^]:=

x^]:x∈H is a Hilbert space when equipped with the operator norm.

Since the mapφ:H→H^] defined byφ(x) =x^] is a linear onto isometry, thereforeH^] is linearly (as opposed to the usual anti-linear identification) isometrically isomorphic toH. The following definition is taken from the Ph.D. thesis of the first named author of this paper [8] submitted to the Indian Institute of Science in 2015.

Definition 2.5(Varopoulos Operator). — LetHbe a separable Hilbert space. Forx, y ∈ H, define the Varopoulos operator Tx,y : C⊕H⊕C → C⊕H⊕C, corresponding to the pair(x, y), to be the linear transformation with the matrix representation:

Tx,y =





0 x^] 0

0 0 y

0 0 0



.

Notice that for any pairs (x₁, y₁) and (x₂, y₂) inH⊕H,the corresponding Varopoulos operators Tx₁,y₁ and Tx₂,y₂ commute if and only if [x^]₁, y2] = [x^]₂, y₁]. If x = y, then we set T_x := T_x,x. Since for any x₁, x₂ ∈ H, we have [x^]₁, x₂] = [x^]₂, x₁], the corresponding Varopoulos operators T_x1 and Tx₂ commute.

Theorem 2.6. — C₂⁺=π/2.

Proof. — SupposeAis n×n non-negative definite matrix and suppose that (T1, . . . , Tn) is a tuple inCn,then,

p_A(T₁, . . . , T_n) =p_S(A)(T₁, . . . , T_n).

Ifb_jk denotes the (j, k) entry of the matrixS(A) then for everyx, y∈H, we have the following

sup

kxk61,kyk61

|hp_S(A)(T1, . . . , Tn)x, yi|= sup

kxk61,kyk61

n

X

j,k=1

bjkhTjx, T_k^∗yi

6 sup

kxjk61,kykk61

n

X

j,k=1

bjkhxj, yki

6 sup

kxjk61

n

X

j,k=1

bjkhxj, xki .

The last inequality is explained by the following computation, which can essentially be found in [1]. Since ((b_jk)) is a non-negative definite matrix therefore there exist V1, . . . , Vn ∈ Cⁿ such that bjk = hVj, Vki for each

j, k= 1, . . . , n.

sup

kx_jk61 kykk61

n

X

j,k=1

bjkhxj, yki

= sup

kx_jk61 kykk61

m

X

p=1

* _n X

j=1

xjpVj,

n

X

k=1

ykpVk

+

6 sup

kxjk61 ky_kk61

m

X

p=1

* _n X

j=1

xjpVj,

n

X

k=1

ykpVk

+

6 sup

kxjk61





m

X

p=1

n

X

j=1

xjpVj

2



1/2

sup

kykk61





m

X

p=1

n

X

k=1

y_kpV_k

2



1/2

= sup

kxjk61 m

X

p=1

n

X

j=1

xjpVj

2

= sup

kxjk61 m

X

p=1

* _n X

j=1

xjpVj,

n

X

k=1

xkpVk

+

= sup

kxjk61

n

X

j,k=1

b_jkhxj, x_ki .

Therefore, we get the following inequality sup

kxk61,kyk61

|hp_S(A)(T1, . . . , Tn)x, yi|6 sup

kxjk61

n

X

j,k=1

bjkhxj, xki .

Also note that k((bjk))k_`^∞

R (n)→`¹

R(n)= sup

s_j,t_k∈[−1,1]

n

X

j,k=1

bjksjtk

= sup

s_j,t_k∈[−1,1]

* _n X

j=1

sjVj,

n

X

k=1

tkVk

+

6 sup

sj,tk∈[−1,1]





n

X

j=1

s_jV_j

2



1/2



n

X

k=1

t_kV_k

2



1/2

= sup

s_j∈[−1,1]

n

X

j=1

sjVj

2

= sup

sj∈[−1,1]

n

X

j,k=1

b_jks_js_k . Thus we get the identity k((bjk))k_`^∞

R (n)→`¹

R(n) = kp_S(A)k_[−1,1]ⁿ_,∞. By this identity and Lemma 2.4 we get the following

(2.1) sup

A∈M_n⁺(C)\{0}

sup_T∈Cnkp_A(T1, . . . , Tn)k kp_Ak_Dⁿ_,∞

6 sup

B∈M_n⁺(R)\{0}

sup_kxjk

`261

Pn

j,k=1bjkhxj, xki kBk_`^∞

R (n)→`¹

R(n)

. Using the definition ofC₂⁺,Lemma 2.2 and the inequality (2.1), we get the following

C₂⁺6 sup

B∈M_n⁺(R)\{0}

sup_kRjk

`2 R

61

Pn

j,k=1bjkhRj, Rki kBk_`^∞

R (n)→`¹

R(n)

=K_G⁺(R) =π/2.

Fix ann×nnon-negative definite matrixAandx_j ∈C^m,j= 1, . . . , n,for somem∈N.Define the Varopoulos operatorTj(=TR_j) corresponding to the vectorR_j∈R^2m(⊂C^2m),whereR_j= ^xj+x₂ ^j, i^xj−x₂ ^j

forj= 1, . . . , n.

Then, taking the form ofp_A(T1, . . . , Tn) into account, we get kp_A(T1, . . . , Tn)k=

n

X

j,k=1

ajkhRj, Rki=

n

X

j,k=1

ajk+akj

2

hRj, Rki.

We use Lemma 2.4 to subsequently obtain sup

A∈M_n⁺(C)\{0}

sup_T∈Cnkp_A(T1, . . . , Tn)k kp_Ak_Dⁿ_,∞

> sup

B∈M_n⁺(R)\{0}

sup_kRjk

`2 R

61

Pn

j,k=1bjkhRj, Rki kBk_`∞

R (n)→`¹

R(n)

.

This proves the theorem.

Theorem 2.6 shows that if we restrict ourselves to the set of homogeneous polynomials of degree two coming from the real non-negative definite ma- trices, then we obtain an analogous inequality to that of Varopoulos, where the factor 2 on the right disappears and the constantK_G^Cgets replaced by K_G⁺(R),that is,

sup

n,p_A

kp_A(T₁, . . . , T_n)k=K_G⁺(R),

where supremum is taken over all homogeneous polynomialsp_A of supre- mum norm at most 1 with A being a non-negative definite matrix, the tuple (T₁, . . . , T_n) being arbitrary in Cn and n ∈ N. The next corollary shows that limn→∞C2(n)/K_G^C>1.This, in turn, answers a long–standing question of Varopoulos, raised in [23], in negative.

Corollary 2.7. — For some >0,we have the inequality

n→∞lim C2(n)>(1 +)K_G^C.

Proof. — We know thatC₂(n)>C₂⁺(n) for eachn∈NandK_G^C61.4049 therefore we have the following

n→∞lim C₂(n)> lim

n→∞C₂⁺(n) =π/2> K_G^C.

To complete the proof one can take= 0.118.

A variant of Corollary 2.7 onL^p-spaces can be found in [21].

In the next theorem, we compute a lower bound for the norm ofAn asn tends to infinity. This improves a bound obtained earlier in an unpublished paper of Holbrook and Schoch where they had proved that limn→∞kAnk>

1.2323.

Theorem 2.8. — lim_n→∞kAnk> ^π₈².

Proof. — Fix a natural number l. Since limn→∞C₂⁺(n) = π/2, there existn ∈Nand a real non-negative definite matrix Al = ((a^(l)_jk)) of sizen such that

sup_T∈C

n

Pa^(l)_jkT_jT_k sup_zj∈T

Pa^(l)_jkzjzk

>π/2−1/l.

By Lemma 2.2 and the fact thatK_G⁺(C) = 4/π,we get the following for the matrixA_l

4/π>sup_kxjk

`261

Pn

j,k=1a^(l)_jkhxj, xki kA_lk<sub>`</sub>∞(n)→`¹(n)

=

sup_kRjk

`2 R

61Pn

j,k=1a^(l)_jkhRj, Rki kAlk<sub>`</sub>∞(n)→`¹(n)

=sup_T∈C

n

Pa^(l)_jkT_jT_k sup_zj∈T

Pa^(l)_jkzjzk

·sup_z

j∈T

Pa^(l)_jkz_jz_k kAlk_{`</sub><sup>∞</sup><sub>(n)→`}1(n)

>(π/2−1/l) kp_Alk_Dⁿ_,∞

kA_lk<sub>`</sub>∞(n)→`¹(n)

.

Rewriting the inequality appearing above, we see that, for eachl∈N,there exists a symmetric matrixAl such that for the corresponding polynomial p_Al,we have the following estimate

kA_lk<sub>`</sub>∞(n)→`¹(n)

kp_Alk_Dⁿ,∞ >(π/2−1/l)

4/π .

Taking supremum over all the natural numberslon both the sides, we get the following inequality

sup

l∈N

kAlk_{`</sub><sup>∞</sup><sub>(n)→`}1(n)

kp_Alk_Dn,∞ > π²

8 ≈1.2337.

The result follows immediately from here.

3. Varopoulos–Kaijser type examples and the constant C₂(n)

In this section, we focus on the constantC2(n) in more detail. We discuss the asymptotic behaviour ofC₂(n) and exhibit an explicit class of exam- ples of Varopoulos–Kaijser type. For this, we mainly rely on a construction which appeared in [7]. In this case, our main tool is a very general max- imizing lemma which can also be of independent interest. For n = 3, we successfully construct a very wide class of examples like Varopoulos–Kaijser for which the von Neumann inequality fails and show that the Varopoulos–

Kaijser example is extremal on the class of certain 3×3 symmetric matrices including symmetric sign matrices.

In [7], the authors produced an explicit set of real non-negative definite matrices for which the positive Grothendieck constant goes up to 1.5. We briefly describe their matrices as follows:

Forl=k(k−1), defineFk ={v1, . . . , v_k(k−1)}, the set of allk-dimensional vectors with two non-zero components, either 1 and 1 or 1 and −1, ap- pearing in that order. Define a real l ×l non-negative definite matrix Ak= ((aij))₁₆i,j6las aij =hvi, vji. In [7], the authors showed that

sup_kXik2=1Pl

i,j=1aijhXi, Xji sup_{ωi∈{1,−1}}Pl

i,j=1aijωiωj

=3k−3 2k−1.

Thus, in view of Lemma 2.4, we get a large class of Varopoulos–Kaijser like examples for which the von Neumann inequality fails andC2(k(k−1))>

3k−3

2k−1. Notice that ^3k−3_2k−1 is an increasing function in k and increases to ³₂ ask tends to infinity. Hence for this explicit class of matrices, we get the

lower bound of limn→∞C₂(n) to be equal to ³₂.Though we already got the lower bound of lim_n→∞C2(n) to be equal to ^π₂,it will be interesting to get an estimate ofC₂(n) as a function ofn.

Motivated by the example of Varopoulos and Kaijser, provided in [22], we develop the following maximizing lemma which enables us to compute the supremum norm of polynomials.

Lemma 3.1 (Maximizing Lemma). — Let Ω be a bounded domain in Rⁿ. Suppose a functionF = (f₁, . . . , f_m) : Ω⊆Rⁿ→C^mis a continuously differentiable and bounded function withfjnon-vanishing forj= 1, . . . , m.

Then we have the following

x∈Ω :kF(x)k_`∞(m)=kFk_Ω,∞ ⊆

m

[

j=1

(

x∈Ω : dim_R

n

_

k=1

∂fj

∂x_k(x)61 )

,

whereW{v₁, . . . , v_n}denotes the subspace spanned by the vectorsv₁, . . . , v_n anddim_Rdenotes the dimension of the space over the field of real numbers R.

Proof. — We notice the following identity {x∈Ω :kF(x)k_`^∞_(m)=kFk_Ω,∞} ⊆

m

[

j=1

x∈Ω :|fj(x)|=kfjk_Ω,∞ . Forf_j: Ω⊆Rⁿ→C,we have

(3.1) |fj|²= (<fj)²+ (=fj)².

Differentiating (3.1) with respect toxk on both the sides, we get the fol- lowing

(3.2) ∂

∂xk

|f_j|²= 2<f_j<∂f_j

∂xk

+ 2=f_j=∂f_j

∂xk

.

By (3.2) and the fact that, at the point of maximum of|fj|,all the partial derivatives of|f_j|² are zero, we obtain

(3.3)

(<fj,=fj),

<∂fj

∂x_k,=∂fj

∂x_k

= 0, ∀16k6n.

From (3.3), we observe that, at the point of maximum of |f_j|², the two dimensional vectors <^∂f_∂x^j

k,=_∂x^∂fj

k

, 16k6n,lie on a line passing through the origin inR².Therefore,

(3.4)

x∈Ω :|fj(x)|=kfjk_Ω,∞ ⊆ (

x∈Ω : dim_R

n

_

k=1

∂fj

∂x_k(x)61 )

.

This completes the proof.