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Session 4. Banach Spaces and Operator Theory with Applications
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Asymptotic estimates on an inequality of von Neumann for homogeneous polynomials |
Pablo Sevilla, Universitat Politécnica de Valéncia, Spain
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The talk is based on the joint work with D. Galicer and S. Muro
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We consider the following \(m\)-homogeneous version of an inequality
of von Neumann: 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
\[
\|p(T_1, \dots, T_n)\|_{\mathcal L(\mathcal H)} \leq C_{k,q}(n) \;
\sup\{ |p(z_1, \dots, z_n)| : \textstyle \sum_{i=1}^{n} \vert z_{i}
\vert^{q} \leq 1 \} \, .
\]
A long standing problem is, for fixed \(k\) and \(q\), to study the asymptotic growth of the smallest constant \(C_{k,p}(n)\) as \(n\) (the number of variables/operators) tends to infinity.
Dixon for \(q = \infty\) \cite{Dixon} and Mantero and Tongue for \(1
\leq q < \infty\) \cite{Mantero Tonge1} gave upper and lower bounds
for \(C_{k,p}(n)\). We go on with this study, showing that the upper
bound given by Dixon is optimal and improving the lower bound given
by Mantero and Tonge for \(2 \leq\)<\(\infty\).
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References- P. G. Dixon, The von Neumann inequality for
polynomials of degree greater than two , London Math. Soc, 14,
1976, 369-375.
- A. M. Mantero, and A. Tonge, Banach
algebras and von Neumann's inequality , Proceedings of the
London Mathematical Society, 3(2), 1979, 309-334.
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