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Asymptotic estimates on the von Neumann inequality for homogeneous polynomials

  • Daniel Galicer EMAIL logo , Santiago Muro und Pablo Sevilla-Peris
Veröffentlicht/Copyright: 20. Februar 2016

Abstract

By the von Neumann inequality for homogeneous polynomials there exists a positive constant Ck,q(n) such that for every k-homogeneous polynomial p in n variables and every n-tuple of commuting operators (T1,…,Tn) with i=1nTiq1 we have

p(T1,,Tn)()Ck,q(n)sup{|p(z1,,zn)|:i=1n|zi|q1}.

For fixed k and q, we study the asymptotic growth of the smallest constant Ck,q(n) as n (the number of variables/operators) tends to infinity. For q = , we obtain the correct asymptotic behavior of this constant (answering a question posed by Dixon in the 1970s). For 2 q< 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.

Funding statement: The first two named authors were supported by CONICET projects PIP 0624 and PICT 2011-1456, and by UBACyT projects 20020130300057BA and 20020130300052BA. The third named author was supported by MICINN project MTM2014-57838-C2-2-P.

Acknowledgements

We wish to thank the referee for her/his comments. Also, we would like to warmly thank our friend Michael Mackey for his careful reading and suggestions that improved considerably the final presentation of the paper.

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Received: 2014-10-06
Revised: 2015-06-24
Published Online: 2016-02-20
Published in Print: 2018-10-01

© 2018 Walter de Gruyter GmbH, Berlin/Boston

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