Koopman operators are infinite-dimensional operators that globally linearize nonlinear dynamical systems, making their spectral information useful for understanding dynamics. However, Koopman operators can have continuous spectra and infinite-dimensional invariant subspaces, making computing their spectral information a considerable challenge. This paper describes data-driven algorithms with rigorous convergence guarantees for computing spectral information of Koopman operators from trajectory data. We introduce residual dynamic mode decomposition (ResDMD), which provides the first scheme for computing the spectra and pseudospectra of general Koopman operators from snapshot data without spectral pollution. Using the resolvent operator and ResDMD, we also compute smoothed approximations of spectral measures associated with measure-preserving dynamical systems. We prove explicit convergence theorems for our algorithms, which can achieve high-order convergence even for chaotic systems, when computing the density of the continuous spectrum and the discrete spectrum. We demonstrate our algorithms on the tent map, Gauss iterated map, nonlinear pendulum, double pendulum, Lorenz system, and an $11$-dimensional extended Lorenz system. Finally, we provide kernelized variants of our algorithms for dynamical systems with a high-dimensional state-space. This allows us to compute the spectral measure associated with the dynamics of a protein molecule that has a 20,046-dimensional state-space, and compute nonlinear Koopman modes with error bounds for turbulent flow past aerofoils with Reynolds number $>10^5$ that has a 295,122-dimensional state-space.

翻译：Koopman 操作员是无限的维度操作员, 在全球范围将非线性动态系统线性化, 使其光谱信息对理解动态有用。 然而, Koopman 操作员可以拥有连续的光谱和无限的多维子子空间, 使得计算其光谱信息是一项相当大的挑战。 本文描述数据驱动算法, 并严格保证从轨迹数据计算 Koopman 操作员的光谱信息。 我们引入残余的动态模式分解( ResDMD), 这为普通 Koopman 操作员的光谱和假光谱数据提供了第一个计算光谱和假光谱的图, 没有污染。 然而, Koopman 操作员的光谱操作员和光谱化的光谱性子信息, 我们也可以用光谱化的光谱系统平滑度近近光度的光谱。 最后, 我们的光谱- 直流系统可以实现高顺序趋同, 直线性平面的直径, 直径直径直径直径直径直径直径直径直径直径直径直径直径直径直径直径直径, 和直径直径直径直径直径直径直径直径直径直径直径直的系统,,,, 和直径直径直径直, 向系统, 向系统, 和直径直径直到我们方的系统, 。