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Crouzeix’s Conjecture and Related Problems

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Abstract

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

$$\begin{aligned} \Vert f(A)\Vert \le 2\,\sup \{|f(z)|:\ z \in W(A)\}, \end{aligned}$$
(1)

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.

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Notes

  1. From now on \(\Vert \cdot \Vert \) will denote the 2-norm for vectors and the corresponding operator norm for matrices: \(\Vert A \Vert = \sup _{\Vert x \Vert = 1} \Vert Ax \Vert \), which is also the largest singular value of A.

  2. The name of this basis is not standard. According to [18] these appeared in Takenaka’s 1925 paper, [37]. The text [30] discusses this basis for the case including infinite Blaschke products and uses the term “Malmquist–Walsh” basis.

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Acknowledgements

The authors of this note are grateful to the American Institute of Mathematics (AIM) and to the organizers of the Workshop on Crouzeix’s conjecture for bringing us together. We are also indebted to Banff International Research Station (BIRS) for providing us with the opportunity to work as a team at BIRS as part of a focused research group. We also thank Łukasz Kosiński for helpful discussions. Lastly, we are grateful to the anonymous referee for useful comments, in particular on Theorems  and 3.8 and their proofs.

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Correspondence to Thomas Ransford.

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Communicated by Vladimir V. Andrievskii.

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To the memory of Stephan Ruscheweyh.

Since August 2018, Pamela Gorkin has been serving as a Program Director in the Division of Mathematical Sciences at the National Science Foundation (NSF), USA, and as a component of this position, she received support from NSF for research, which included work on this paper. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

Bickel’s research is supported in part by the National Science Foundation DMS grant #2000088.

Ransford’s research is supported by grants from NSERC and the Canada Research Chairs program.

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Bickel, K., Gorkin, P., Greenbaum, A. et al. Crouzeix’s Conjecture and Related Problems. Comput. Methods Funct. Theory 20, 701–728 (2020). https://doi.org/10.1007/s40315-020-00350-9

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  • DOI: https://doi.org/10.1007/s40315-020-00350-9

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