Crouzeix’s Conjecture and Related Problems

Crouzeix’s conjecture asserts that, for any polynomial f and any square matrix A, the operator norm of f(A) satisfies the estimate 1 $$\begin{aligned} \Vert f(A)\Vert \le 2\,\sup \{|f(z)|:\ z \in W(A)\}, \end{aligned}$$ 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.