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12. Joint numerical ranges

By viewing a matrix $A$ according to the decomposition $A=A_{1}+iA_{2}$ with $A_{1}=\frac{A+A^{*}}{2}$ and $A_{2}=\frac{A-A^{*}}{2i}$, one is lead to the problem for the joint numerical range of two matrices $A_{1}$ and $A_{2}$ and polynomials in two variables (non-commuting).
    1. Problem 12.1.

      For what types of polynomials $p(x,y)$ does it hold that $$\|p(A_{1},A_{2})\|\leq C \sup_{(a_{1},a_{2})\in W(A_{1},A_{2})}|p(a_{1},a_{2})|?$$ A next step is then to consider the question for three matrices $A_{1},A_{2},A_{3}$.

          Cite this as: AimPL: Crouzeix's conjecture, available at http://aimpl.org/crouzeix.