On a Question of N. Th. Varopoulos and the constant $C_2\left(n\right)$

Gupta, Rajeev; Ray, Samya K.

Annales de l'Institut Fourier, Tome 68 (2018) no. 6, pp. 2613-2634.

Soit ${ℂ}_k[Z_1,...,Z_n]$ l’ensemble de tous les polynômes de degré au plus $k$ dans $n$ variables complexes et ${𝒞}_n$ l’ensemble de tous les $n$-tuples $T=(T_1,...,T_n)$ de contractions qui commutent sur un espace de Hilbert $ℍ.$ L’inégalité intéressante

$$K_G^{ℂ}\le \underset{n\to \infty }{lim}{C}_2\left(n\right)\le 2K_G^{ℂ},$$

où

$$C_k\left(n\right)=sup\left\{{\parallel p\left(T\right)\parallel :\parallel p\parallel }_{{𝔻}^n,\infty }\le 1,p\in {ℂ}_k[Z_1,...,Z_n],T\in {𝒞}_n\right\}$$

et $K_G^{ℂ}$ est la constante complexe de Grothendieck, est due à Varopoulos. Nous répondons à une question de longue date en montrant que ${lim}_{n\to \infty }\frac{C_2\left(n\right)}{K_G^{ℂ}}$ est strictement plus grand que $1$. Soit ${ℂ}_2^s[Z_1,...,Z_n]$ l’ensemble des polynômes homogènes complexes $p(z_1,...,z_n)={\sum }_{j,k=1}^n{a}_{jk}{z}_j{z}_k$ de degré deux dans $n$-variables, oú $\left(\left(a_{jk}\right)\right)$ est une matrice symétrique complexe $n\times n$. Pour chaque $n\in \mathbb{N},$ on définit la carte linéaire ${𝒜}_n:({ℂ}_2^s[Z_1,...,Z_n]{,\parallel ·\parallel }_{{𝔻}^n,\infty })\to (M_n{,\parallel ·\parallel }_{\infty \to 1})$ par ${𝒜}_n\left(p\right)=\left(\left(a_{jk}\right)\right).$ Nous montrons que le supremum (sur $n$) de la norme des opérateurs ${𝒜}_n;n\in \mathbb{N},$ est borné inférieurement par la constante ${\pi }^2/8.$ 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 extrémal parmi une grande classe convenablement choisie de polynômes homogènes de degré deux. Nous étudions également le comportement de la constante $C_k\left(n\right)$ lorsque $n\to \infty .$

Let ${ℂ}_k[Z_1,...,Z_n]$ denote the set of all polynomials of degree at most $k$ in $n$ complex variables and ${𝒞}_n$ denote the set of all $n$-tuple $T=(T_1,...,T_n)$ of commuting contractions on some Hilbert space $ℍ.$ The interesting inequality

$$K_G^{ℂ}\le \underset{n\to \infty }{lim}{C}_2\left(n\right)\le 2K_G^{ℂ},$$

where

$$C_k\left(n\right)=sup\left\{{\parallel p\left(T\right)\parallel :\parallel p\parallel }_{{𝔻}^n,\infty }\le 1,p\in {ℂ}_k[Z_1,...,Z_n],T\in {𝒞}_n\right\}$$

and $K_G^{ℂ}$ is the complex Grothendieck constant, is due to Varopoulos. We answer a long–standing question by showing that the limit ${lim}_{n\to \infty }\frac{C_2\left(n\right)}{K_G^{ℂ}}$ is strictly bigger than $1.$ Let ${ℂ}_2^s[Z_1,...,Z_n]$ denote the set of all complex valued homogeneous polynomials $p(z_1,...,z_n)={\sum }_{j,k=1}^n{a}_{jk}{z}_j{z}_k$ of degree two in $n$-variables, where $\left(\left(a_{jk}\right)\right)$ is a $n\times n$ complex symmetric matrix. For each $n\in \mathbb{N},$ define the linear map ${𝒜}_n:({ℂ}_2^s[Z_1,...,Z_n]{,\parallel ·\parallel }_{{𝔻}^n,\infty })\to (M_n{,\parallel ·\parallel }_{\infty \to 1})$ to be ${𝒜}_n\left(p\right)=\left(\left(a_{jk}\right)\right).$ We show that the supremum (over $n$) of the norm of the operators ${𝒜}_n;n\in \mathbb{N},$ is bounded below by the constant ${\pi }^2/8.$ 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 polynomial is extremal among a, suitably chosen, large class of homogeneous polynomials of degree two. We also study the behaviour of the constant $C_k\left(n\right)$ as $n\to \infty .$

Classification : 47A13, 47A25, 47A60, 47A63
Keywords: Grothendieck Inequality, von Neumann Inequality, Varopoulos Operator, Grothendieck Constant, Positive Grothendieck Constant
Mot clés : Inégalité de Grothendieck, inégalité de von Neumann, opérateur de Varopoulos, Constante de Grothendieck, Constante de Grothedieck positive

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