Crouzeix's conjecture explained

Crouzeix's conjecture is an unsolved problem in matrix analysis. It was proposed by Michel Crouzeix in 2004,^[1] and it can be stated as follows:

\|f(A)\| \le 2 \sup _{z\in} |f(z)|,

where the set

W(A) is the field of values of a n×n (i.e. square) complex matrix A and f is a complex function that is analytic in the interior of W(A) and continuous up to the boundary of W(A) . Slightly reformulated, the conjecture can also be stated as follows: for all square complex matrices A and all complex polynomials p :\|p(A)\| \le 2 \sup _{z\in} |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:^[2]

\|f(A)\| \le 11 . 08 \sup _{z\in} |f(z)|

(the constant

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+ \sqrt{ 2 } ,^[3] improving the original constant of 11 . 08 . The not yet proved conjecture states that the constant can be refined to 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^[4] and for general n×n matrices that are nearly Jordan blocks.^[5] Furthermore, Anne Greenbaum and Michael L. Overton provided numerical support for Crouzeix's conjecture.^[6]

Further reading

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

See also

• Von Neumann's inequality

Notes and References

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