发表机构
University of Ottawa; University of Wyoming; Central South University（渥太华大学; 怀俄明大学; 中南大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文在弱Lorentz理想上建立Pietsch对应的谱形式，引入超可测性并证明其与谱可测性不可比较，应用于非经典Weyl定律相关例子。
AI 中文摘要
这是关于弱Lorentz理想上Connes积分的一系列论文中的第二篇。基于Dixmier迹理论、Birman--Solomyak扰动理论以及第一部分中发展的强可测性，我们在这些理想上建立了Pietsch对应的谱形式，将Semenov--Sukochev--Usachev--Zanin关于弱迹类的结果推广到这一设定。该对应关系通过Banach极限描述了正规范化迹，并给出了强可测性的完整谱刻画。我们还引入了超可测性（相对于每个规范化迹的可测性）。与环境理想不同，超可测性依赖于所选的特定正则变化函数，我们通过特征值和对它进行了谱刻画。我们进一步证明了超可测性与谱可测性是不可比较的。最后，我们将这些结果应用于由Simon意义上的非经典Weyl定律产生的例子，包括闭流形上Laplacian的对数、双Laplacian、Laplacian的多重张量积，以及由Connes关于黎曼猜想的方法产生的一个例子。
英文摘要
This is the second in a series of papers on Connes' integration in weak Lorentz ideals. Building on the Dixmier trace theory, Birman--Solomyak perturbation theory, and strong measurability developed in Part 1, we establish a spectral form of Pietsch's correspondence for traces on these ideals, extending to this setting results of Semenov--Sukochev--Usachev--Zanin for the weak trace-class. The correspondence describes the positive normalized traces in terms of Banach limits and yields a complete spectral characterization of strong measurability. We also introduce hypermeasurability (measurability with respect to every normalized trace). Unlike the ambient ideal, it depends on the specific regularly varying function chosen, and we characterize it spectrally by means of eigenvalue sums. We further show that hypermeasurability and spectral measurability are incomparable. Finally, we apply these results to examples arising from nonclassical Weyl laws in the sense of Simon, including the logarithm of the Laplacian on a closed manifold, the double Laplacian, multi-tensor products of Laplacians, and an example arising from Connes' approach to the Riemann Hypothesis.
Comments40 pages
