We examine spectral operator-theoretic properties of linear and nonlinear dynamical systems with globally stable attractors. Using the Kato decomposition, we develop a spectral expansion for general linear autonomous dynamical systems with analytic observables and define the notion of generalized eigenfunctions of the associated Koopman operator. We interpret stable, unstable and center subspaces in terms of zero-level sets of generalized eigenfunctions. We then utilize conjugacy properties of Koopman eigenfunctions and the new notion of open eigenfunctions—defined on subsets of state space—to extend these results to nonlinear dynamical systems with an equilibrium. We provide a characterization of (global) center manifolds, center-stable, and center-unstable manifolds in terms of joint zero-level sets of families of Koopman operator eigenfunctions associated with the nonlinear system. After defining a new class of Hilbert spaces, that capture the on- and off-attractor properties of dissipative dynamics, and introducing the concept of modulated Fock spaces, we develop spectral expansions for a class of dynamical systems possessing globally stable limit cycles and limit tori, with observables that are square-integrable in on-attractor variables and analytic in off-attractor variables. We discuss definitions of stable, unstable, and global center manifolds in such nonlinear systems with (quasi)-periodic attractors in terms of zero-level sets of Koopman operator eigenfunctions. We define the notion of isostables for a general class of nonlinear systems. In contrast with the systems that have discrete Koopman operator spectrum, we provide a simple example of a measure-preserving system that is not chaotic but has continuous spectrum, and discuss experimental observations of spectrum on such systems. We also provide a brief characterization of the data types corresponding to the obtained theoretical results and define the coherent principal dimension for a class of datasets based on the lattice-type principal spectrum of the associated Koopman operator.

我们研究具有全局稳定吸引子的线性和非线性动力学系统的谱算子理论性质。使用Kato分解，我们为带有解析可观测量的一般线性自治动力系统开发了一个谱展开，并定义了相关Koopman算子的广义本征函数的概念。我们根据零阶广义本征函数解释稳定，不稳定和中心子空间。然后，我们利用Koopman特征函数的共轭特性和在状态空间子集上定义的开放特征函数的新概念，将这些结果扩展到具有平衡的非线性动力学系统。我们提供（整体）中心歧管，中心稳定，非线性系统相关联的Koopman算子本征函数族的零级联合集和中心不稳定流形。在定义了一类新的希尔伯特空间后，它捕获了耗散动力学的吸引子性质和吸引者性质，并引入了调制的福克空间的概念，我们为一类具有全局稳定极限环和极限环的动力学系统开发了谱展开式，可观察变量在吸引子变量上可平方积分，而在吸引子变量中可分析。我们讨论了具有（准）周期吸引子的此类非线性系统中的稳定，不稳定和全局中心流形的定义，涉及的是零级Koopman算子本征函数。我们为一类通用的非线性系统定义了等耗量的概念。与具有离散Koopman运算符频谱的系统相比，我们提供了一个简单的示例，该系统不是混沌的而是具有连续频谱，并且讨论了在此类系统上频谱的实验观察结果。我们还简要描述了与获得的理论结果相对应的数据类型，并根据相关的库普曼算子的格型主谱为一类数据集定义了一致的主维。