Asymptotic estimates on the von Neumann inequality for homogeneous polynomials

Abstract

Abstract By the von Neumann inequality for homogeneous polynomials there exists a positive constant C_{k,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 {\sum_{i=1}^{n}\|T_{i}\|^{q}\leq 1} we have \|p(T_{1},\ldots,T_{n})\|_{\mathcal{L}(\mathcal{H})}\leq C_{k,q}(n)\sup\Biggl{% \{}|p(z_{1},\ldots,z_{n})|:\sum_{i=1}^{n}|z_{i}|^{q}\leq 1\Biggr{\}}. For fixed k and q, we study the asymptotic growth of the smallest constant C_{k,q} (n) as n (the number of variables/operators) tends to infinity. For q = \infty , we obtain the correct asymptotic behavior of this constant (answering a question posed by Dixon in the 1970s). For 2 \leq q < \infty 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.