Asymptotic estimates on the von Neumann inequality for homogeneous polynomials

GALICER DANIEL

Valencia

Workshop; Workshop on Functional Analysis Valencia 2015 on the occasion of the 60th birthday of Jos\'e Bonet, Valencia; 2015

By the von Neumann inequality for homogeneous polynomials there exists a constant$C_{k,q}(n) >0$ such that for every $k$-homogeneous polynomial $p$ in $n$ variables and every$n$-tuple of commuting operators $(T_1, \dots, T_n)$ verifying $\sum_{i=1}^{n} \Vert T_{i} \Vert_{B(\mathcal H)}^{q}\leq 1$ we have\[ \|p(T_1, \dots, T_n)\|_{B(\mathcal H)} \leq C_{k,q}(n) \; \sup\{ |p(z_1, \dots, z_n)| : \textstyle \sum_{i=1}^{n} \vert z_{i} \vert^{q} \leq 1 \}\,.\]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 byDixon in the seventies. For $q \in [2, \infty)$ we improve some lower bounds given by Mantero and Tonge, andexhibit the asymptotic growth up to a logarithmic factor.To achieve all this we provide estimates of the norm ofhomogeneous unimodular polynomials with some special combinatorial configuration: Steiner polynomials (i.e., polynomials such that the multi-indices corresponding to the nonzero coefficients form partial Steiner systems).\bigskipJoint work with Santiago Muro (Universidad de Buenos Aires) and Pablo Sevilla-Peris (Universitat Polit\`ecnica de Val\`encia).