Английская Википедия:Crouzeix's conjecture

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Шаблон:Short description 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:

<math>\|f(A)\| \le 2 \sup_{z\in W(A)} |f(z)|,</math>

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

<math>\|p(A)\| \le 2 \sup_{z\in W(A)} |p(z)|</math>

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

History

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

<math>\|f(A)\| \le 11.08 \sup_{z\in W(A)} |f(z)|</math>

(the constant <math>11.08</math> 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 <math>1+\sqrt{2}</math>, improving the original constant of <math>11.08</math>. The not yet proved conjecture states that the constant can be refined to <math>2</math>.

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

• Шаблон:Cite journal
• Шаблон:Cite arXiv

See also

• Von Neumann's inequality