Crouzeix's Conjecture and Related Problems

Publication Date

Fall 10-17-2020

Description

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

||f(A)|| \le 2 sup{|f(z)|: z in W(A)}

where W(A) = {: ||x|| = 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 ‖f(A)‖ 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.

Journal

Computational Methods and Function Theory

Volume

20

First Page

701

Last Page

728

Department

Mathematics

Second Department

Mathematics

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