一般Banach空间上在$\ heta$-从属扰动下半群生成元的拟紧性与一致稳定性
Quasi-compactness and uniform stabilization on general Banach spaces under $θ$-subordinate perturbations of semigroup generators
- University of Waterloo（滑铁卢大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文证明在一般Banach空间上，解析半群在沿轨道紧的$\ heta$-从属扰动下保持拟紧性和一致指数稳定性，并推广至Crandall-Pazy半群及立即可微半群，最后通过反例说明$A$-有界扰动下稳定性可能缺失。
AI中文摘要：
我们证明，在一般Banach空间上，若解析半群的生成元受到沿轨道紧的$\ heta$-从属扰动，则其拟紧性保持不变。由此，在类似扰动下，若扰动后的生成元生成强稳定半群，则可证明解析半群的一致指数稳定性具有持久性。随后，我们展示如何将解析性假设放宽到Crandall-Pazy半群类（在更小的$\ heta$-从属扰动类下）以及立即可微半群类（在有界扰动下）。作为稳定性结果的一个应用，我们考虑非自反Banach空间上的一维Neumann拉普拉斯算子。最后，我们给出反例，说明在更大的$A$-有界扰动类下，一致稳定性可能缺失。
英文摘要：
We prove that the quasi-compactness of an analytic semigroup is preserved under $θ$-subordinate perturbations of its generator on general Banach spaces, provided the perturbations are compact along trajectories. This allows us to prove the permanence of uniform exponential stability for an analytic semigroup under similar perturbations of its generator, provided the perturbed generator generates a strongly stable semigroup. We then show how the analyticity assumption can be relaxed to the class of Crandall-Pazy semigroups under a smaller class of $θ$-subordinate perturbations, and to immediately differentiable semigroups under bounded perturbations. As an application of the stabilization result, we consider the one-dimensional Neumann Laplacian on a non-reflexive Banach space. Finally, we present counterexamples that demonstrate the lack of uniform stabilization under the larger class of $A$-bounded perturbations.
