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Crouzeix's conjecture

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Unsolved problem in matrix analysis

Crouzeix's conjecture is an unsolved problem in matrix analysis. It was proposed by Michel Crouzeix in 2004, and it can be stated as follows:

‖ f ( A ) ‖ ≤ 2 sup z ∈ W ( A ) | f ( z ) | , {\displaystyle \|f(A)\|\leq 2\sup _{z\in W(A)}|f(z)|,}

where the set W ( A ) {\displaystyle W(A)} is the field of values of a n×n (i.e. square) complex matrix A {\displaystyle A} and f {\displaystyle f} is a complex function that is analytic in the interior of W ( A ) {\displaystyle W(A)} and continuous up to the boundary of W ( A ) {\displaystyle W(A)} . Slightly reformulated, the conjecture can also be stated as follows: for all square complex matrices A {\displaystyle A} and all complex polynomials p {\displaystyle p} :

‖ p ( A ) ‖ ≤ 2 sup z ∈ W ( A ) | p ( z ) | {\displaystyle \|p(A)\|\leq 2\sup _{z\in W(A)}|p(z)|}

holds, where the norm on the left-hand side is the spectral operator 2-norm.

History

Crouzeix's theorem, proved in 2007, states that:

‖ f ( A ) ‖ ≤ 11.08 sup z ∈ W ( A ) | f ( z ) | {\displaystyle \|f(A)\|\leq 11.08\sup _{z\in W(A)}|f(z)|}

(the constant 11.08 {\displaystyle 11.08} is independent of the matrix dimension, thus transferable to infinite-dimensional settings).

Michel Crouzeix and Cesar Palencia proved in 2017 that the result holds for 1 + 2 {\displaystyle 1+{\sqrt {2}}} , improving the original constant of 11.08 {\displaystyle 11.08} . The not yet proved conjecture states that the constant can be refined to 2 {\displaystyle 2} .

Special cases

While the general case is unknown, it is known that the conjecture holds for some special cases. For instance, it holds for all normal matrices, for tridiagonal 3×3 matrices with elliptic field of values centered at an eigenvalue and for general n×n matrices that are nearly Jordan blocks. Furthermore, Anne Greenbaum and Michael L. Overton provided numerical support for Crouzeix's conjecture.

Further reading

  • Ransford, Thomas; Schwenninger, Felix L. (2018-03-01). "Remarks on the Crouzeix–Palencia Proof that the Numerical Range is a ( 1 + 2 ) {\displaystyle (1+{\sqrt {2}})} -Spectral Set". SIAM Journal on Matrix Analysis and Applications. 39 (1): 342–345. arXiv:1708.08633. doi:10.1137/17M1143757. S2CID 43945191.
  • Gorkin, Pamela; Bickel, Kelly (2018-10-27). "Numerical Range and Compressions of the Shift". arXiv:1810.11680 .

References

  1. Crouzeix, Michel (2004-04-01). "Bounds for Analytical Functions of Matrices". Integral Equations and Operator Theory. 48 (4): 461–477. doi:10.1007/s00020-002-1188-6. ISSN 0378-620X. S2CID 121371601.
  2. Crouzeix, Michel (2007-03-15). "Numerical range and functional calculus in Hilbert space". Journal of Functional Analysis. 244 (2): 668–690. doi:10.1016/j.jfa.2006.10.013.
  3. Crouzeix, Michel; Palencia, Cesar (2017-06-07). "The Numerical Range is a ( 1 + 2 ) {\displaystyle (1+{\sqrt {2}})} -Spectral Set". SIAM Journal on Matrix Analysis and Applications. 38 (2): 649–655. doi:10.1137/17M1116672.
  4. ^ Choi, Daeshik (2013-04-15). "A proof of Crouzeix's conjecture for a class of matrices". Linear Algebra and Its Applications. 438 (8): 3247–3257. doi:10.1016/j.laa.2012.12.045.
  5. Glader, Christer; Kurula, Mikael; Lindström, Mikael (2018-03-01). "Crouzeix's Conjecture Holds for Tridiagonal 3 x 3 Matrices with Elliptic Numerical Range Centered at an Eigenvalue". SIAM Journal on Matrix Analysis and Applications. 39 (1): 346–364. arXiv:1701.01365. doi:10.1137/17M1110663. S2CID 43922128.
  6. Greenbaum, Anne; Overton, Michael L. (2017-05-04). "Numerical investigation of Crouzeix's conjecture" (PDF). Linear Algebra and Its Applications. 542: 225–245. doi:10.1016/j.laa.2017.04.035.

See also

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