Crouzeix’s Conjecture and Related Problems
Crouzeix’s Conjecture and Related Problems
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DOI：
10.1007/s40315-020-00350-9
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发表时间：
2020-06
影响因子：
2.1
通讯作者：
K. Bickel;P. Gorkin;A. Greenbaum;T. Ransford;Felix L. Schwenninger;E. Wegert
中科院分区：
文献类型：
--
作者：
K. Bickel;P. Gorkin;A. Greenbaum;T. Ransford;Felix L. Schwenninger;E. Wegert
Crouzeix 的猜想断言，对于任何多项式 f 和任何方阵 A，算子范数 off(A) 满足估计 $$\begin{aligned} \Vert f(A)\Vert \le 2\,\sup \{|f(z)|:\ z \in W(A)\}， \end{aligned}$$其中表示A的数值范围。这也适用于在 W(A) 邻域中解析的所有函数。我们对与该猜想相关的最新调查进行了调查，并得出了特定类别算子 A 的界限。这使我们能够陈述明确的条件，保证 Crouzeix 的估计 (1) 成立。我们描述相关极值函数（Blaschke 产品）和相关极值向量的属性。更详细地研究了 A 是压缩移位算子的矩阵表示的情况。
Crouzeix’s conjecture asserts that, for any polynomialfand any square matrixA, the operator norm off(A) satisfies the estimate $$\begin{aligned} \Vert f(A)\Vert \le 2\,\sup \{|f(z)|:\ z \in W(A)\}, \end{aligned}$$wheredenotes the numerical range ofA. This would then also hold for all functionsfwhich are analytic in a neighborhood ofW(A). We provide a survey of recent investigations related to this conjecture and derive bounds forfor specific classes of operatorsA. This allows us to state explicit conditions that guarantee that Crouzeix’s estimate (1) holds. We describe properties of related extremal functions (Blaschke products) and associated extremal vectors. The case whereAis a matrix representation of a compressed shift operator is studied in some detail.