Crouzeix's conjecture

Unsolved problem in matrix analysis From Wikipedia, the free encyclopedia

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:

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

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]

(the constant 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 ,[3] improving the original constant of . The not yet proved conjecture states that the constant can be refined to .

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,[4] for tridiagonal 3×3 matrices with elliptic field of values centered at an eigenvalue[5] and for general n×n matrices that are nearly Jordan blocks.[4] Furthermore, Anne Greenbaum and Michael L. Overton provided numerical support for Crouzeix's conjecture.[6]

Further reading

  • Ransford, Thomas; Schwenninger, Felix L. (2018-03-01). "Remarks on the Crouzeix–Palencia Proof that the Numerical Range is a -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 [math.FA].

References

See also

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