ABSTRACT

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 $\parallel f\left(A\right)\parallel$ amongst all holomorphic functions f from Ω into the closed unit disk. If $f_0$ is extremal for this problem and if $\parallel f_0\left(A\right)\parallel >1$, then it turns out that the matrix $f_0\left(A\right)$ 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 $f_0\left(A\right)$. In particular, we are interested in the extent to which they are characterized by the aforementioned orthogonality property.

中文翻译：

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谱集、极值函数和异常矩阵

摘要

令A为方阵，令 Ω 为包含A谱的平面中的开集。我们考虑最大化算子范数的问题$\parallel F\left(\mathrm{一个}\right)\parallel$在从 Ω 到封闭单位圆盘的所有全纯函数f中。如果$F_0$对于这个问题是极值的，如果$\parallel F_0\left(\mathrm{一个}\right)\parallel >1$, 那么矩阵$F_0\left(\mathrm{一个}\right)$具有特殊性质，其中主要的左右奇异向量是相互正交的。我们研究这类异常矩阵$F_0\left(\mathrm{一个}\right)$. 特别是，我们对它们在多大程度上具有上述正交性特性感兴趣。