Crouzeix’s conjecture asserts that, for any polynomial f and any square matrix A, the operator norm of f(A) satisfies the estimate

where \(W(A):=\{\langle Ax,x\rangle : \Vert x\Vert =1\}\) denotes the numerical range of A. This would then also hold for all functions f which are analytic in a neighborhood of W(A). We provide a survey of recent investigations related to this conjecture and derive bounds for \(\Vert f(A)\Vert \) for specific classes of operators A. 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 where A is a matrix representation of a compressed shift operator is studied in some detail.

Crouzeix的猜想断言，对于任何多项式f和任何方阵A，f（A）的算子范数都满足估计

其中\（W（A）：= \ {\ langle Ax，x \ rangle：\ Vert x \ Vert = 1 \} \）表示A的数值范围。对于所有在W（A）附近进行分析的函数f，这也将成立。我们提供了与该猜想相关的最新研究的调查，并为特定类别的算子A得出\（\ Vert f（A）\ Vert \）的边界。这使我们能够陈述明确的条件，以保证Crouzeix的估计（1）成立。我们描述了相关的极值函数（Blaschke乘积）和相关极值向量的属性。详细地研究了A是压缩移位算子的矩阵表示的情况。