Type: Article
Publication Date: 2016-02-20
Citations: 37
DOI: https://doi.org/10.1515/crelle-2015-0097
Abstract By the von Neumann inequality for homogeneous polynomials there exists a positive constant <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>C</m:mi> <m:mrow> <m:mi>k</m:mi> <m:mo>,</m:mo> <m:mi>q</m:mi> </m:mrow> </m:msub> </m:math> C_{k,q} ( n ) such that for every k -homogeneous polynomial p in n variables and every n -tuple of commuting operators ( <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>T</m:mi> <m:mn>1</m:mn> </m:msub> </m:math> T_{1} ,…, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>T</m:mi> <m:mi>n</m:mi> </m:msub> </m:math> T_{n} ) with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:msubsup> <m:mo>∑</m:mo> <m:mrow> <m:mi>i</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mi>n</m:mi> </m:msubsup> <m:msup> <m:mrow> <m:mo>∥</m:mo> <m:msub> <m:mi>T</m:mi> <m:mi>i</m:mi> </m:msub> <m:mo>∥</m:mo> </m:mrow> <m:mi>q</m:mi> </m:msup> </m:mrow> <m:mo>≤</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> {\sum_{i=1}^{n}\|T_{i}\|^{q}\leq 1} we have <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mo>∥</m:mo> <m:mrow> <m:mi>p</m:mi> <m:mo></m:mo> <m:mrow> <m:mo>(</m:mo> <m:msub> <m:mi>T</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo>,</m:mo> <m:mi>…</m:mi> <m:mo>,</m:mo> <m:msub> <m:mi>T</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mo>∥</m:mo> </m:mrow> <m:mrow> <m:mi>ℒ</m:mi> <m:mo></m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>ℋ</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:msub> <m:mo>≤</m:mo> <m:mrow> <m:msub> <m:mi>C</m:mi> <m:mrow> <m:mi>k</m:mi> <m:mo>,</m:mo> <m:mi>q</m:mi> </m:mrow> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>n</m:mi> <m:mo>)</m:mo> </m:mrow> <m:mo></m:mo> <m:mrow> <m:mo>sup</m:mo> <m:mo></m:mo> <m:mrow> <m:mo>{</m:mo> <m:mrow> <m:mo>|</m:mo> <m:mrow> <m:mi>p</m:mi> <m:mo></m:mo> <m:mrow> <m:mo>(</m:mo> <m:msub> <m:mi>z</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo>,</m:mo> <m:mi>…</m:mi> <m:mo>,</m:mo> <m:msub> <m:mi>z</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mo>|</m:mo> </m:mrow> <m:mo>:</m:mo> <m:mrow> <m:mrow> <m:munderover> <m:mo>∑</m:mo> <m:mrow> <m:mi>i</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mi>n</m:mi> </m:munderover> <m:msup> <m:mrow> <m:mo>|</m:mo> <m:msub> <m:mi>z</m:mi> <m:mi>i</m:mi> </m:msub> <m:mo>|</m:mo> </m:mrow> <m:mi>q</m:mi> </m:msup> </m:mrow> <m:mo>≤</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo>}</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:mrow> <m:mo>.</m:mo> </m:mrow> </m:math> \|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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>C</m:mi> <m:mrow> <m:mi>k</m:mi> <m:mo>,</m:mo> <m:mi>q</m:mi> </m:mrow> </m:msub> </m:math> C_{k,q} ( n ) as n (the number of variables/operators) tends to infinity. For q = <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>∞</m:mi> </m:math> \infty , we obtain the correct asymptotic behavior of this constant (answering a question posed by Dixon in the 1970s). For 2 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mo>≤</m:mo> </m:math> \leq q <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mo><</m:mo> </m:math> < <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>∞</m:mi> </m:math> \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.