发表机构
Czech Academy of Sciences; Institute of Mathematics and Computer Science Jagiellonian University（捷克科学院; 雅盖隆大学数学与计算机科学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究单位Banach代数中代数数值域的谱集常数，给出次数依赖的上下界，证明C*-代数中的间隙定理，并构造无穷常数例子，解决三个公开问题。
AI 中文摘要
我们研究了单位Banach代数中与代数数值域相关的谱集常数。若$\Gamma_n$表示次数至多为$n$的代数元素的普适常数，则\\[ \Gamma_1=1, \qquad 2n-1\leqslant\Gamma_n<\infty\qquad(n\geqslant2). \\] 因此，代数数值域是一个谱集，其常数仅依赖于代数次数。在次数为二的情形，我们得到\\[ 3\leqslant\Gamma_2 \leqslant\sqrt{1+(2\mathrm e-1)^2}<4.55. \\] 对于单位$C^*$-代数，我们证明了一个间隙定理：在交换情形下，代数级常数恰好为1，而每个非交换代数的常数至少为2。此外，Jiang-Su代数的常数恰好是普适Crouzeix常数。在相反方向上，我们构造了一个范数为一的算子，其在单位圆盘上具有压缩多项式演算，但数值域谱常数为无穷大，并且我们证明了在广泛类中的每个蔓延组合空间上的典型左移位具有无穷常数。这些结果解决了Blazhko、Homza、Schwenninger、de Vries和Wojtylak提出的三个问题。
英文摘要
We study spectral-set constants associated with the algebraic numerical range in unital Banach algebras. If $Γ_n$ denotes the universal constant for algebraic elements of degree at most $n$, then \[ Γ_1=1, \qquad 2n-1\leqslantΓ_n<\infty\qquad(n\geqslant2). \] Thus the algebraic numerical range is a spectral set with a constant depending only on the algebraic degree. In degree two we obtain \[ 3\leqslantΓ_2 \leqslant\sqrt{1+(2\mathrm e-1)^2}<4.55. \] For unital $C^*$-algebras we prove a gap theorem: the algebra-level constant is one precisely in the commutative case, whereas every non-commutative algebra has constant at least two. Moreover, the constant of the Jiang--Su algebra is exactly the universal Crouzeix constant. In the opposite direction, we construct a norm-one operator with a contractive polynomial calculus on the unit disc but infinite numerical-range spectral constant, and we show that the canonical left shift on every spreading combinatorial space in a broad class has infinite constant. These results settle the three questions posed by Blazhko, Homza, Schwenninger, de Vries, and Wojtylak.
Comments29 pp
