论文标题
关于最大化Crouzeix比率的功能的独特性
On the Uniqueness of Functions that Maximize the Crouzeix Ratio
论文作者
论文摘要
令$ a $为$ n $ by $ n $矩阵，带有数值范围$ w（a）：= \ {q^{*} aq：q \ in \ mathbb {c}^n，〜\ | Q \ | _2 = 1 \} $。我们对功能感兴趣$ \ hat {f} $最大化$ \ | f（a）\ | _2 $（由向量2-norm诱导的矩阵标准）上的所有功能$ f $在$ w（a）$的内部分析并在边界上连续分析并满足$ \ max_ {z \ in W（a）} | f（z）| \ leq 1 $。众所周知，有一些功能$ \ hat {f} $可以达到此最大的功能，并且此类功能是$ b \ circcc $的形式，其中$ ϕ $是从$ w（a）$内部到单位磁盘$ \ mathbb {d} $的任何形式的映射，该函数是$ w（a）$ w（a）$ n是$ n $ n as a a $ n是$ n as a $ n是$ n as a $ bblas and a $ bblas and a $ b。尚不清楚达到此最大值的函数$ \ hat {f} $是唯一的，直到乘以模量的标量。我们表明，当$ a $是$ 2 \ times 2 $非正常矩阵或乔丹块时，我们提供了约3美元的示例，其中有两个不同的功能$ \ hat {f} $，涉及相同的副本映射，但具有相同的MAXIMIMAL IDE MAXIMIM ALCEMIMAL IDE MAXIMIM ALLIPTAL INTER {F HAT $ \ hat $ \ hat abtrical范围$ 3 \ times 3 $矩阵。 $ || f（a）|| _2 $。
Let $A$ be an $n$ by $n$ matrix with numerical range $W(A) := \{ q^{*}Aq : q \in \mathbb{C}^n , ~\| q \|_2 = 1 \}$. We are interested in functions $\hat{f}$ that maximize $\| f(A) \|_2$ (the matrix norm induced by the vector 2-norm) over all functions $f$ that are analytic in the interior of $W(A)$ and continuous on the boundary and satisfy $\max_{z \in W(A)} | f(z) | \leq 1$. It is known that there are functions $\hat{f}$ that achieve this maximum and that such functions are of the form $B\circϕ$, where $ϕ$ is any conformal mapping from the interior of $W(A)$ to the unit disk $\mathbb{D}$, extended to be continuous on the boundary of $W(A)$, and $B$ is a Blaschke product of degree at most $n-1$. It is not known if a function $\hat{f}$ that achieves this maximum is unique, up to multiplication by a scalar of modulus one. We show that this is the case when $A$ is a $2\times 2$ nonnormal matrix or a Jordan block, but we give examples of some $3\times 3$ matrices with elliptic numerical range for which two different functions $\hat{f}$, involving the same conformal mapping but Blaschke products of different degrees, achieve the same maximal value of $||f(A)||_2$.
