论文标题
c* - 代数smale平均价值猜想和杜宾蛋白 - 瓜川双重价值猜想
C*-algebraic Smale Mean Value Conjecture and Dubinin-Sugawa Dual Mean Value Conjecture
论文作者
论文摘要
基于Smale平均值猜想\ TextIt {[Bull。阿米尔。数学。 Soc。，1981]}和Dubinin-Sugawa双重均值构想\ TextIt {[Proc。日本学院。 ser。数学。 Sci。，2009]}我们制定了以下猜想。 \ textbf {c*-algebraic smale平均值猜想：令$ \ mathcal {a} $为通勤的C*-Algebra。令$ p（z）=（z-a_1）\ cdots（z-a_n）$为$ n \ geq 2 $ over $ \ mathcal {a} $，$ a_1，\ dots，a_n \ in \ mathcal in \ mathcal {a a} $。如果$ z \ in \ Mathcal {a} $不是$ p $的关键点，则存在一个关键点$ w \ in \ nathcal {a} $ $ p $的$ w \ \ frac {\ | p（z）-p（w）\ |} {\ | z-w \ |} \ leq 1 \ | p'（z）\ | \ end {align*}或\ begin {align*} \ frac {\ | p（z）-p（w）\ |} {\ | z-w \ |} \ leq \ frac {n-1} {n} {n} {n} \ | p'（z）\ | = \ | = \ frac {\ frac {\ frac { \ | p'（z）\ |。 \ end {align*}} \ textbf {c*-algebraic dubinin-sugawa双重均值猜想：让$ \ Mathcal {a} $为交换性c*-algebra。令$ p（z）=（z-a_1）\ cdots（z-a_n）$为$ n \ geq 2 $ over $ \ mathcal {a} $，$ a_1，\ dots，a_n \ in \ mathcal in \ mathcal {a a} $。如果$ z \ in \ Mathcal {a} $不是$ p $的关键点，则存在一个关键点$ w \ in \ nathcal {a} $ of $ p $，这样 \ begin {align*} \ frac {\ | p'（z）\ |} {\ operatorName {deg}（p）} = \ frac {\ | p'（z）\ |} {n} \ leq \ leq \ frac {\ frac {\ | p（z）-p（z）-p（z）-p（w）-p（w）\ |} {\ | | w \ | - | w \ | |} \ end {align*}} 我们表明，（甚至是强烈的）C* - 代数smale平均值构想和C* - 代数dubinin-sugawa双重平均值构想的二重性值构想，用于2 C* - 代数多项式在交换c*-ergebras上。
Based on Smale mean value conjecture \textit{[Bull. Amer. Math. Soc., 1981]} and Dubinin-Sugawa dual mean value conjecture \textit{[Proc. Japan Acad. Ser. A Math. Sci., 2009]} we formulate the following conjectures. \textbf{C*-algebraic Smale Mean Value Conjecture : Let $\mathcal{A}$ be a commutative C*-algebra. Let $P(z)= (z-a_1)\cdots (z-a_n)$ be a polynomial of degree $n\geq 2$ over $\mathcal{A}$, $a_1, \dots, a_n \in \mathcal{A}$. If $z\in\mathcal{A}$ is not a critical point of $P$, then there exists a critical point $w\in \mathcal{A}$ of $P$ such that \begin{align*} \frac{\|P(z)-P(w)\|}{\|z-w\|}\leq 1 \|P'(z)\| \end{align*} or \begin{align*} \frac{\|P(z)-P(w)\|}{\|z-w\|}\leq \frac{n-1}{n} \|P'(z)\|=\frac{\operatorname{deg} (P)-1}{\operatorname{deg} (P)} \|P'(z)\|. \end{align*}} \textbf{C*-algebraic Dubinin-Sugawa Dual Mean Value Conjecture : Let $\mathcal{A}$ be a commutative C*-algebra. Let $P(z)= (z-a_1)\cdots (z-a_n)$ be a polynomial of degree $n\geq 2$ over $\mathcal{A}$, $a_1, \dots, a_n \in \mathcal{A}$. If $z\in \mathcal{A}$ is not a critical point of $P$, then there exists a critical point $w\in \mathcal{A}$ of $P$ such that \begin{align*} \frac{\|P'(z)\|}{\operatorname{deg} (P)} =\frac{\|P'(z)\|}{n} \leq \frac{\|P(z)-P(w)\|}{\|z-w\|}. \end{align*}} We show that (even a strong form of) C*-algebraic Smale mean value conjecture and C*-algebraic Dubinin-Sugawa dual mean value conjecture hold for degree 2 C*-algebraic polynomials over commutative C*-algebras.
