2016/11/21 by Gupta, Rajeev, Ray, Samya Kumar
#FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.1611.06726
Let \mathbb Ck[Z1,…, Zn] denote the set of all polynomials of degree at most k in n complex variables and \mathscrCn denote the set of all n - tuple \boldsymbol T=(T1,…,Tn) of commuting contractions on some Hilbert space ℍ. The interesting inequality KG\mathbb C≤ limn→ ∞C2(n)≤ 2 K^\mathbb CG, where Ck(n)=sup\‖p(\boldsymbol T)‖:‖p‖\mathbb Dn,∞≤ 1, p∈ \mathbb Ck[Z1,…,Zn],\boldsymbol T∈\mathscrCn \ and KG\mathbb 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^\mathbb CG) is strictly bigger than 1. Let \mathbb C2s[Z1,… , Zn] denote the set of all complex valued homogeneous polynomials p(z1,…,zn) =∑j,k=1najkzjzk of degree two in n - variables, where ( (ajk) ) is a n× n complex symmetric matrix. For each n∈ℕ, define the linear map \mathscrAn: (\mathbb C2s[Z1,… , Zn],‖⋅‖\mathbb Dn, ∞ ) → (Mn, ‖⋅ ‖∞ → 1 ) to be \mathscrAn (p) = ( (ajk) ). We show that the supremum (over n) of the norm of the operators \mathscrAn; n∈ℕ, is bounded below by the constant π2/8. 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 Ck(n) as n → ∞.