Crouzeix猜想的一种抽象方法
An abstract approach to the Crouzeix conjecture
摘要：
设$A$是一致代数，$θ:A\to M_n(\mathbb{C})$是连续同态，$α:A\to A$是反线性压缩，使得\[ \|θ(f)+θ(α(f))^*\|\le 2\|f\| \quad(f\in A). \]我们证明了$\|θ\|\le 1+\sqrt{2}$和$1+\sqrt2$是尖锐的.我们推测，如果进一步$α(1)=1$，那么我们可以得出结论，$\|θ\|\le2$。这将在数值域上产生Crouzeix猜想的正解。为了支持我们的猜想，我们证明了在两种特殊情况下它是真的。我们还讨论了一个完全有界的版本，我们的猜想，发挥膨胀理论的想法。
英文摘要：
Let $A$ be a uniform algebra, $θ:A\to M_n(\mathbb{C})$ be a continuous homomorphism and $α:A\to A$ be an antilinear contraction such that \[ \|θ(f)+θ(α(f))^*\|\le 2\|f\| \quad(f\in A). \] We show that $\|θ\|\le 1+\sqrt{2}$, and that $1+\sqrt2$ is sharp. We conjecture that, if further $α(1)=1$, then we may conclude that $\|θ\|\le2$. This would yield a positive solution to the Crouzeix conjecture on numerical ranges. In support of our conjecture, we prove that it is true in two special cases. We also discuss a completely bounded version of our conjecture that brings into play ideas from dilation theory.
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