Crouzeix's conjecture

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Crouzeix's conjecture is a problem in matrix analysis 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.