Asymptotic estimates on the von Neumann inequality for homogeneous polynomials

GALICER, DANIEL; MURO, SANTIAGO; SEVILLA PERIS, PABLO

JOURNAL FUR DIE REINE UND ANGEWANDTE MATHEMATIK

WALTER DE GRUYTER & CO

Lugar: Berlin; Año: 2016

0075-4102

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 $(T_1, dots, T_n)$ with $sum_{i=1}^{n} Vert T_{i} Vert^{q} leq 1$ we have [ |p(T_1, dots, T_n)|_{mathcal L(mathcal H)} leq C_{k,q}(n) ; sup{ |p(z_1, dots, z_n)| : extstyle sum_{i=1}^{n} ert z_{i} ert^{q} leq 1 },. ] 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 seventies). 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.