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Crouzeix’s Conjecture and Related Problems

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

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

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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  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.


  1. Agler, J., McCarthy, J.E.: Pick interpolation and Hilbert function spaces. Graduate Studies in Mathematics, vol. 44. American Mathematical Society, Providence (2002)

  2. Badea, C., Crouzeix, M., Delyon, B.: Convex domains and K-spectral sets. Math. Z. 252(2), 345–365 (2006)

    Article  MathSciNet  Google Scholar 

  3. Badea, C., Crouzeix, M., Klaja, H.: Spectral sets and operator radii. Bull. Lond. Math. Soc. 50(6), 986–996 (2018)

    Article  MathSciNet  Google Scholar 

  4. Caldwell, T., Greenbaum, A., Li, K.: Some extensions of the Crouzeix–Palencia result. SIAM J. Matrix Anal. Appl. 39, 769–780 (2018)

    Article  MathSciNet  Google Scholar 

  5. Choi, D.: A proof of Crouzeix’s conjecture for a class of matrices. Linear Algebra Appl. 438, 3247–3257 (2013)

    Article  MathSciNet  Google Scholar 

  6. Choi, D., Greenbaum, A.: Roots of matrices in the study of GMRES convergence and Crouzeix’s conjecture. SIAM J. Matrix Anal. Appl. 36(1), 289–301 (2015)

    Article  MathSciNet  Google Scholar 

  7. Crabb, M.: The powers of an operator of numerical radius one. Mich. Math. J. 18, 253–256 (1971)

    Article  MathSciNet  Google Scholar 

  8. Crouzeix, M.: Bounds for analytical functions of matrices. Integr. Equ. Oper. Theory 48, 461–477 (2004)

    Article  MathSciNet  Google Scholar 

  9. Crouzeix, M.: Numerical range and functional calculus in Hilbert space. J. Funct. Anal. 244(2), 668–690 (2007)

    Article  MathSciNet  Google Scholar 

  10. Crouzeix, M., Gilfeather, M., Holbrook, J.: Polynomial bounds for small matrices. Linear Multilinear Algebra 62(5), 614–625 (2014)

    Article  MathSciNet  Google Scholar 

  11. Crouzeix, M., Palencia, C.: The numerical range is a \((1 + \sqrt{2})\) spectral set. SIAM J. Matrix Anal. Appl. 38, 649–655 (2017)

    Article  MathSciNet  Google Scholar 

  12. Daepp, U., Gorkin, P., Shaffer, A., Voss, K.: Finding Ellipses: What Blaschke Products, Poncelet’s Theorem, and the Numerical Range Know about Each Other, The Carus Mathematical Monographs, vol. 34 (2018)

  13. Delyon, B., Delyon, F.: Generalization of von Neumann’s spectral sets and integral representation of operators. Bull. Soc. Math. Fr. 127, 25–41 (1999)

    Article  MathSciNet  Google Scholar 

  14. Drury, S.W.: A counterexample to a conjecture of Matsaev. Linear Algebra Appl. 435(2), 323–329 (2011)

    Article  MathSciNet  Google Scholar 

  15. Earl, J.P.: On the interpolation of bounded sequences by bounded functions. J. Lond. Math. Soc. 2, 544–548 (1970)

    Article  MathSciNet  Google Scholar 

  16. Garcia, S.R., Mashreghi, J., Ross, W.T.: Finite Blaschke Products and their Connections. Springer, Cham (2018)

    Book  Google Scholar 

  17. Garcia, S.R., Ross, W.T.: A non-linear extremal problem on the Hardy space. Comput. Methods Funct. Theory 9(2), 485–524 (2009)

    Article  MathSciNet  Google Scholar 

  18. Garcia, S.R., Ross, W.T.: Model spaces: a survey. Contemp. Math. 638, 197–245 (2015)

    Article  MathSciNet  Google Scholar 

  19. Garnett, J.B.: Bounded Analytic Functions, Graduate Texts in Mathematics, vol. 236. Springer, Berlin (2007)

    Google Scholar 

  20. Gau, H.-L., Wu, P.-Y.: Numerical range of S(\(\phi \)). Linear Multilinear Algebra 45(1), 49–73 (1998)

    Article  MathSciNet  Google Scholar 

  21. Gau, H.-L., Wu, P.-Y.: Numerical range and Poncelet property. Taiwan. J. Math. 7(2), 173–193 (2003)

    Article  MathSciNet  Google Scholar 

  22. Glader, C., Kurula, M., Lindström, M.: Crouzeix’s conjecture holds for tridiagonal \(3 \times 3\) matrices with elliptic numerical range centered at an eigenvalue. SIAM J. Matrix Anal. Appl. 39(1), 346–364 (2018)

    Article  MathSciNet  Google Scholar 

  23. Greenbaum, A., Choi, D.: Crouzeix’s conjecture and perturbed Jordan blocks. Linear Algebra Appl. 436(7), 2342–2352 (2012)

    Article  MathSciNet  Google Scholar 

  24. Greenbaum, A., Overton, M.L.: Numerical investigation of Crouzeix’s conjecture. Linear Algebra Appl. 542, 225–245 (2018)

    Article  MathSciNet  Google Scholar 

  25. Li, K.: On the uniqueness of extremal Blaschke products. arXiv:2002.01027

  26. Mergelyan, S.: On the representation of functions by series of polynomials on closed sets. Transl. Am. Math. Soc. 3, 287–293 (1962)

    Google Scholar 

  27. Mergelyan, S.: Uniform approximations to functions of a complex variable. Transl. Am. Math. Soc. 3, 294–391 (1962)

    Google Scholar 

  28. Mirman, B.: UB-matrices and conditions for Poncelet polygon to be closed. Linear Algebra Appl. 360, 123–150 (2003)

    Article  MathSciNet  Google Scholar 

  29. Nikolski, N.K.: Five problems on invariant subspaces. J. Sov. Math. 2, 441–450 (1974)

    Article  Google Scholar 

  30. Nikolski, N.K.: Treatise on the Shift Operator. Springer, Berlin (1986)

    Book  Google Scholar 

  31. Ransford, T., Schwenninger, F.L.: Remarks on the Crouzeix–Palencia proof that the numerical range is a \((1+\sqrt{2})\)-spectral set. SIAM J. Matrix Anal. Appl. 39(1), 342–345 (2018)

    Article  MathSciNet  Google Scholar 

  32. Sarason, D.: Generalized interpolation in \(H^\infty \). Trans. Am. Math. Soc. 127, 179–203 (1967)

    MathSciNet  MATH  Google Scholar 

  33. Shapiro, H.S., Shields, A.L.: On some interpolation problems for analytic functions. Am. J. Math. 83, 513–532 (1961)

    Article  MathSciNet  Google Scholar 

  34. Smith, R.A.: The condition numbers of the matrix eigenvalue problem. Numer. Math. 10, 232–240 (1967)

    Article  MathSciNet  Google Scholar 

  35. Sz.-Nagy, B., Foias, C., Bercovici, H., Kérchy, L.: Harmonic Analysis of Operators on Hilbert Space, 2nd edn. Universitext, Springer, New York (2010)

  36. Takahashi, S.: Extension of the theorems of Carathéodory–Toeplitz–Schur and Pick. Pac. J. Math. 138, 391–399 (1989)

    Article  Google Scholar 

  37. Takenaka, S.: On the orthonormal functions and a new formula of interpolation. Jpn. J. Math. 2, 129–145 (1925)

    Article  Google Scholar 

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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).

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