发表机构
University of Reading; Université de Lille（雷丁大学; 里尔大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究接近恒等算子的有界算子，通过位移界和外围谱条件刻画其Ritt性质，给出最优预解界、角逃逸不变量及定量估计，并建立算子值函数的范数间隙。
AI 中文摘要
我们研究有界算子，其指定的幂次保持一致接近恒等算子，以及沿幂次序列采样的预解条件。位移界 $\sup_{n\ge1}\\|I-T^n\\|\le q<2$ 迫使有限外围谱由奇数阶单位根组成，并使 $T$ 的一个明确确定的奇数次幂成为Ritt算子。对于 $T$ 本身，尖锐的无条件阈值是 $\sqrt3$。若 $Φ_*(q)$ 表示低于此阈值的最优通用Ritt预解界，则当 $q\uparrow\sqrt3$ 时，$Φ_*(q)=2/(\sqrt3-q)+O(1)$。在每个更高的外围阈值处，最优有限外围预解界具有倒数的增长阶。我们还获得了一般加权Wiener符号的构造性界。对于每个 $1<q<2$，一个角逃逸不变量精确刻画了采样序列，使得位移界 $q$ 连同外围谱条件迫使Ritt性质成立。我们为渐近几何序列计算了此不变量。在端点 $q=1$ 处，适当的相位条件意味着完整的位移界，而无需假设幂有界性；正常压缩算子允许精确的标量判据。最后，我们建立了定量的Hilbert空间和 $L^p$ 预解估计，一致凸空间上具有奇数Ritt幂的算子的刻画，以及算子值圆盘代数函数的严格范数间隙，并给出来自Fejér–Riesz分解的显式矩阵多项式界。
英文摘要
We study bounded operators whose prescribed powers remain uniformly close to the identity, and resolvent conditions sampled along sequences of powers. The displacement bound $\sup_{n\ge1}\|I-T^n\|\le q<2$ forces a finite peripheral spectrum consisting of odd-order roots of unity and makes an explicitly determined odd power of $T$ a Ritt operator. The sharp unconditional threshold for $T$ itself is $\sqrt3$. If $Φ_*(q)$ denotes the optimal universal Ritt resolvent bound below this threshold, then $Φ_*(q)=2/(\sqrt3-q)+O(1)$ as $q\uparrow\sqrt3$. At every higher peripheral threshold the optimal finite-peripheral resolvent bound has reciprocal order of growth. We also obtain constructive bounds for general weighted Wiener symbols. For each $1<q<2$, an angular escape invariant characterises exactly the sampling sequences for which a displacement bound by $q$, together with the peripheral spectral condition, forces the Ritt property. We compute this invariant for asymptotically geometric sequences. At the endpoint $q=1$, suitable phase conditions imply the full displacement bound without assuming power boundedness; normal contractions admit an exact scalar criterion. Finally, we establish quantitative Hilbert-space and $L^p$ resolvent estimates, a characterisation of operators with odd Ritt powers on uniformly convex spaces, and strict norm gaps for operator-valued disc-algebra functions, with explicit matrix-polynomial bounds from Fejér--Riesz factorisation.
Comments45 pages
