Crouzeix’s Conjecture and Related Problems

Kelly Bickel, Pamela Gorkin, Anne Greenbaum, Thomas Ransford*, Felix L. Schwenninger, Elias Wegert

*Corresponding author for this work

Research output: Contribution to journalArticleAcademicpeer-review

2 Citations (Scopus)
11 Downloads (Pure)


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)‖≤2sup{|f(z)|:z∈W(A)},where W(A) : = { ⟨ Ax, x⟩ : ‖ 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.

Original languageEnglish
Pages (from-to)701-728
Number of pages28
JournalComputational Methods and Function Theory
Issue number3-4
Publication statusPublished - Nov 2020


  • Blaschke products
  • Crouzeix’s conjecture
  • Nevanlinna–Pick interpolation
  • Numerical radius
  • Numerical range
  • Shift operator
  • 22/2 OA procedure


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