5.1 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Crouzeix's conjecture | 1/1 | https://en.wikipedia.org/wiki/Crouzeix's_conjecture | reference | science, encyclopedia | 2026-05-05T11:02:53.424444+00:00 | kb-cron |
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}
. More recently, dimension-dependent improvements have been obtained: Malman, Mashreghi, O'Loughlin and Ransford showed that for each fixed dimension
N
{\displaystyle N}
there exists a constant
C
N
<
1
+
2
{\displaystyle C_{N}<1+{\sqrt {2}}}
such that the inequality holds for all
N
×
N
{\displaystyle N\times N}
matrices. Related work connects the constant in Crouzeix-type inequalities to configuration constants arising from the Neumann–Poincaré operator and yields domain-dependent improvements of the Crouzeix–Palencia bound in certain settings.
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 [math.FA].
== References ==
== See also == Von Neumann's inequality