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on Crouzeix constant ($C_{2}$)

The statement

$C_{2}$ is the Crouzeix constant (sometimes denoted $Q$ ). It is the smallest constant $C$ such that for every $n \ge 1$ , every complex matrix $A \in \mathbb{C}^{n \times n}$ , and every complex polynomial $p$ one has

\|p(A)\| \ \le\ C \ \max_{z \in W(A)} |p(z)|,

where $\|\cdot\|$ is the operator norm induced by the Euclidean norm (i.e. the spectral norm), and

W(A) := \{ v^\ast A v : v \in \mathbb{C}^n,\ \|v\|_2 = 1\}

is the numerical range (field of values) of $A$ .

Equivalently,

C_{2} = \sup_{n \ge 1}\ \sup_{A \in \mathbb{C}^{n\times n}}\ \sup_{p \not\equiv 0} \frac{\|p(A)\|}{\max_{z \in W(A)} |p(z)|}.

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.