谱集、极值函数与例外矩阵
Spectral sets, extremal functions and exceptional matrices
摘要：
设$A$是一个方阵，$Ω$是包含$A$的谱的平面上的开集。考虑了从$Ω$到闭单位圆盘的所有全纯函数$f$中算子范数$\|f(A)\|$的最大化问题.如果$f_0$是这个问题的极值，如果$\|f_0(A)\|>1$，那么矩阵$f_0(A)$有特殊的性质，其中之一是它的主左和右奇异向量是相互正交的。研究了这类例外矩阵$f_0(A)$ .特别是，我们感兴趣的程度，他们的特点是由上述正交属性。
英文摘要：
Let $A$ be a square matrix and let $Ω$ be an open set in the plane containing the spectrum of $A$. We consider the problem of maximizing the operator norm $\|f(A)\|$ amongst all holomorphic functions $f$ from $Ω$ into the closed unit disk. If $f_0$ is extremal for this problem and if $\|f_0(A)\|>1$, then it turns out that the matrix $f_0(A)$ has special properties, among them the fact that its principal left and right singular vectors are mutually orthogonal. We study this class of exceptional matrices $f_0(A)$. In particular, we are interested in the extent to which they are characterized by the aforementioned orthogonality property.
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