Asymptotic estimates on the von Neumann inequality for homogeneous polynomials

Abstract

[EN] By the von Neumann inequality for homogeneous polynomials there exists a positive constant C-k,C-q(n) such that for every k-homogeneous polynomial p in n variables and every n-tuple of commuting operators (T-1, ..., T-n with Sigma(n)(i=1) parallel to T-i parallel to(q) <= 1 we have

parallel to P (T-1, ..., T-n)parallel to L(H) <= C-k,C-q(n) sup {vertical bar p(z(1), ..., z(n))vertical bar: Sigma(n)(i=1) vertical bar(q) <= 1}.

For fixed k and q, we study the asymptotic growth of the smallest constant C-k,C-q(n) as n (the number of variables/operators) tends to infinity. For q = infinity, we obtain the correct asymptotic behavior of this constant (answering a question posed by Dixon in the 1970s). For 2 <= q < infinity we improve some lower bounds given by Mantero and Tonge, and prove the asymptotic behavior up to a logarithmic factor. To achieve this we provide estimates of the norm of homogeneous unimodular Steiner polynomials, i.e. polynomials such that the multi-indices corresponding to the nonzero coefficients form partial Steiner systems.