Crouzeix's conjecture

Crouzeix's conjecture is an unsolved (as of 2018) problem in matrix theory proposed by Michel Crouzeix, which refines his following earlier proven Crouzeix's theorem:

$|f(A)|\leq 11.08\sup _{z\in W(A)}|f(z)|$

Here, $|f|$ denotes any function analytic on W(A), the field of values of a n×n (i.e. square) matrix A. Notoriously, the constant 11.08 is independent of the matrix dimension, thus transferable to infinite-dimensional settings. The not yet proved conjecture states that the constant is sharpable to 2:

$|f(A)|\leq 2\sup _{z\in W(A)}|f(z)|$

Slightly reformulated, the conjecture can be stated as, that for all square matrices A and all polynomials p:

$|p(A)|\leq 2\max\{p(z):z\in W(A)\}$

Here, $|p(A)|_{W(A)}$ denotes the maximum of $|p(z)|$, where z ranges over the numerical range of A. W(A) is defined as the set of Rayleigh quotients associated with A:

Or it can be reformulated as:

$|p(A)|\leq 2|p|_{W(A)}$

Here, the left part of the inequality denotes the spectral operator 2-norm and the right part denotes ∞-norm on the field of values:

$W(A)=\{q^{*}Aq:q\in \mathbb {C} ^{n},q^{*}q=1\}$

While the general case is unknown, it's known that the conjecture holds for tridiagonal 3×3 matrices with elliptic field of values centered at an eigenvalue.

Further reading

• M. Crouzeix, "Bounds for analytical functions of matrices", Integral Equations and Operator Theory, issue 48 (2004), pp. 461–477