Let denote the set of all polynomials of degree at most in complex variables and denote the set of all -tuple of commuting contractions on some Hilbert space The interesting inequality
where
and is the complex Grothendieck constant, is due to Varopoulos. We answer a long–standing question by showing that the limit is strictly bigger than Let denote the set of all complex valued homogeneous polynomials of degree two in -variables, where is a complex symmetric matrix. For each define the linear map to be We show that the supremum (over ) of the norm of the operators is bounded below by the constant 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 as
Soit l’ensemble de tous les polynômes de degré au plus dans variables complexes et l’ensemble de tous les -tuples de contractions qui commutent sur un espace de Hilbert L’inégalité intéressante
où
et est la constante complexe de Grothendieck, est due à Varopoulos. Nous répondons à une question de longue date en montrant que est strictement plus grand que . Soit l’ensemble des polynômes homogènes complexes de degré deux dans -variables, oú est une matrice symétrique complexe . Pour chaque on définit la carte linéaire par Nous montrons que le supremum (sur ) de la norme des opérateurs est borné inférieurement par la constante 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 lorsque
Revised:
Accepted:
Published online:
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
Gupta, Rajeev 1; Ray, Samya K. 1
@article{AIF_2018__68_6_2613_0, author = {Gupta, Rajeev and Ray, Samya K.}, title = {On a {Question} of {N.} {Th.} {Varopoulos} and the constant $C_2(n)$}, journal = {Annales de l'Institut Fourier}, pages = {2613--2634}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {68}, number = {6}, year = {2018}, doi = {10.5802/aif.3218}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3218/} }
TY - JOUR AU - Gupta, Rajeev AU - Ray, Samya K. TI - On a Question of N. Th. Varopoulos and the constant $C_2(n)$ JO - Annales de l'Institut Fourier PY - 2018 SP - 2613 EP - 2634 VL - 68 IS - 6 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3218/ DO - 10.5802/aif.3218 LA - en ID - AIF_2018__68_6_2613_0 ER -
%0 Journal Article %A Gupta, Rajeev %A Ray, Samya K. %T On a Question of N. Th. Varopoulos and the constant $C_2(n)$ %J Annales de l'Institut Fourier %D 2018 %P 2613-2634 %V 68 %N 6 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3218/ %R 10.5802/aif.3218 %G en %F AIF_2018__68_6_2613_0
Gupta, Rajeev; Ray, Samya K. On a Question of N. Th. Varopoulos and the constant $C_2(n)$. Annales de l'Institut Fourier, Volume 68 (2018) no. 6, pp. 2613-2634. doi : 10.5802/aif.3218. https://aif.centre-mersenne.org/articles/10.5802/aif.3218/
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