Crouzeix's Conjecture and related problems

Abstract

In this paper, we establish several results related to Crouzeix's conjecture. We show that the conjecture holds for contractions with eigenvalues that are sufficiently well-separated. This separation is measured by the so-called separation constant, which is defined in terms of the pseudohyperbolic metric. Moreover, we study general properties of related extremal functions and associated vectors. Throughout, compressions of the shift serve as illustrating examples which also allow for refined results.

24 pages

References in corpus (2)

• Remarks on the Crouzeix-Palencia proof that the numerical range is a $(1+\sqrt{2})$-spectral set
• Spectral sets and operator radii

Cited by in corpus (5)

• Dilation theory in finite dimensions and matrix convexity
• Schwarz-Pick type inequalities from an operator theoretical point of view
• The algebraic numerical range as a spectral set in Banach algebras
• Black Holes, Complex Curves, and Graph Theory: Revising a Conjecture by Kasner
• Contractive upper triangular matrices with prescribed diagonals and superdiagonals