%0 Journal Article
%T On Von Neumann’s Inequality for Matrices of Complex Polynomials
%A Joachim Moussounda Mouanda
%J American Journal of Computational Mathematics
%P 289-303
%@ 2161-1211
%D 2021
%I Scientific Research Publishing
%R 10.4236/ajcm.2021.114019
%X We prove that every
matrix </span><i><span style=\"font-family:&quot;\">F</span></i><span style=\"font-size:6.5pt;line-height:102%;font-family:宋体;\">∈</span><i><span style=\"font-family:&quot;\">M</span></i><sub><span style=\"font-family:&quot;\">k </span></sub><span style=\"font-family:&quot;\">(P<sub>n</sub>)</span><span style=\"font-family:&quot;\"> is associated </span><span style=\"font-family:&quot;\">with</span><span style=\"font-family:&quot;\"> </span><span style=\"font-family:&quot;\">the</span><span style=\"font-family:&quot;\"> smallest positive integer </span><i><span style=\"font-family:&quot;\">d</span></i><span style=\"font-family:&quot;\"> (<i>F</i>)</span><span style=\"font-size:8.0pt;line-height:102%;font-family:宋体;\">≠</span><span style=\"font-family:&quot;\">1</span><span style=\"font-family:&quot;\"> such that </span><i><span style=\"font-family:&quot;\">d </span></i><span style=\"font-family:&quot;\">(<i>F</i>)</span><span style=\"font-family:宋体;\">‖</span><i><span style=\"font-family:&quot;\">F</span></i><span style=\"font-family:宋体;\">‖</span><sub><span style=\"font-size:9px;line-height:102%;font-family:宋体;\">∞</span></sub><span style=\"font-family:&quot;\"> </span><span style=\"font-family:&quot;\">is always bigger than the sum of the operator norms of
the Fourier coefficients of <i>F</i>. We
establish some inequalities for matrices of complex polynomials. In application,
we show that von Neumann’s inequality hold</span><span style=\"font-family:&quot;\">s</span><span style=\"font-family:&quot;\"> up to the constant </span><span style=\"font-family:&quot;\">2<sup>n </sup></span><span style=\"font-family:&quot;\">for matrices of the algebra</span><span style=\"font-family:&quot;\"> <i>M</i><sub>k </sub>(P<sub>n</sub>).</span><span style=\"font-family:&quot;\"></span>
</p>
<br />
<span style=\"font-family:;\" \"=\"\"></span>
%K Fourier Coefficients
%K Operator Theory
%K Polynomials
%U http://www.scirp.org/journal/PaperInformation.aspx?PaperID=113769