AI 中文总结
该研究通过构造反例证明，关于跟踪性的广义双曲性猜想在一般巴拿赫空间上不成立，但在可分希尔伯特空间上成立，明确了两类空间的差异源于满射与右可逆的间隙。
AI 中文摘要
已知在巴拿赫空间上的可逆有界线性算子中，广义双曲性蕴含跟踪性。其逆是否成立是线性动力学中的核心开放问题，且被猜想为成立。我们通过构造反例证明该猜想在一般巴拿赫空间上不成立，但在可分希尔伯特空间上成立。二者的差异源于巴拿赫空间上满射与右可逆之间可能存在的间隙，而该间隙在希尔伯特空间上消失。证明的主要要素是近期基于满射谱的跟踪性谱刻画，以及基于单位圆附近右预解函数的广义双曲性新刻画。
英文摘要
It is known that generalized hyperbolicity implies the shadowing property for invertible bounded linear operators on a Banach space. Whether the converse holds has been a central open problem in linear dynamics and has been conjectured to have a positive answer. We show that this conjecture fails on general Banach spaces by constructing a counterexample, whereas it holds on separable Hilbert spaces. The distinction is explained by the gap that may occur on Banach spaces between surjectivity and right invertibility, a gap that disappears on Hilbert spaces. The main ingredients of the proofs are a recent spectral characterization of shadowing in terms of the surjective spectrum and a new characterization of generalized hyperbolicity in terms of right resolvent functions near the unit circle.
Comments16 pages
