摘要
We prove that every matrix </span><i><span style="font-family:"">F</span></i><span style="font-size:6.5pt;line-height:102%;font-family:宋体;">∈</span><i><span style="font-family:"">M</span></i><sub><span style="font-family:"">k </span></sub><span style="font-family:"">(P<sub>n</sub>)</span><span style="font-family:""> is associated </span><span style="font-family:"">with</span><span style="font-family:""> </span><span style="font-family:"">the</span><span style="font-family:""> smallest positive integer </span><i><span style="font-family:"">d</span></i><span style="font-family:""> (<i>F</i>)</span><span style="font-size:8.0pt;line-height:102%;font-family:宋体;">≠</span><span style="font-family:"">1</span><span style="font-family:""> such that </span><i><span style="font-family:"">d </span></i><span style="font-family:"">(<i>F</i>)</span><span style="font-family:宋体;">‖</span><i><span style="font-family:"">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:""> </span><span style="font-family:"">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:"">s</span><span style="font-family:""> up to the constant </span><span style="font-family:"">2<sup>n </sup></span><span style="font-family:"">for matrices of the algebra</span><span style="font-family:""> <i>M</i><sub>k </sub>(P<sub>n</sub>).</span><span style="font-family:""></span> </p> <br /> <span style="font-family:;" "=""></span>
We prove that every matrix </span><i><span style="font-family:"">F</span></i><span style="font-size:6.5pt;line-height:102%;font-family:宋体;">∈</span><i><span style="font-family:"">M</span></i><sub><span style="font-family:"">k </span></sub><span style="font-family:"">(P<sub>n</sub>)</span><span style="font-family:""> is associated </span><span style="font-family:"">with</span><span style="font-family:""> </span><span style="font-family:"">the</span><span style="font-family:""> smallest positive integer </span><i><span style="font-family:"">d</span></i><span style="font-family:""> (<i>F</i>)</span><span style="font-size:8.0pt;line-height:102%;font-family:宋体;">≠</span><span style="font-family:"">1</span><span style="font-family:""> such that </span><i><span style="font-family:"">d </span></i><span style="font-family:"">(<i>F</i>)</span><span style="font-family:宋体;">‖</span><i><span style="font-family:"">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:""> </span><span style="font-family:"">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:"">s</span><span style="font-family:""> up to the constant </span><span style="font-family:"">2<sup>n </sup></span><span style="font-family:"">for matrices of the algebra</span><span style="font-family:""> <i>M</i><sub>k </sub>(P<sub>n</sub>).</span><span style="font-family:""></span> </p> <br /> <span style="font-family:;" "=""></span>
作者
Joachim Moussounda Mouanda
Joachim Moussounda Mouanda(Mathematics Department, Blessington Christian University, Nkayi, Republic of Congo)
