论文标题
Crouzeix猜想的抽象方法
An abstract approach to the Crouzeix conjecture
论文作者
论文摘要
令$ a $为统一的代数，$θ：a \ to m_n（\ mathbb {c}）$是连续的同构态，$α： \ quad（a）。 \]我们表明$ \ |θ\ | \ le 1+ \ sqrt {2} $，而$ 1+\ sqrt2 $是尖锐的。我们推测，如果进一步的$α（1）= 1 $，那么我们可以得出结论，$ \ |θ\ | \ le2 $。这将为Crouzeix猜想在数值范围内产生积极的解决方案。为了支持我们的猜想，我们证明这在两种特殊情况下是正确的。我们还讨论了我们的猜想的完全有限的版本，该版本将扩张理论带入戏剧思想。
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.
