Data-driven spectral analysis of Koopman operators is a powerful tool for understanding numerous real-world dynamical systems, from neuronal activity to variations in sea surface temperature. The Koopman operator acts on a function space and is most commonly studied on the space of square-integrable functions. However, defining it on a suitable reproducing kernel Hilbert space (RKHS) offers numerous practical advantages, including pointwise predictions with error bounds, improved spectral properties that facilitate computations, and more efficient algorithms, particularly in high dimensions. We introduce the first general, provably convergent, data-driven algorithms for computing spectral properties of Koopman and Perron--Frobenius operators on RKHSs. These methods efficiently compute spectra and pseudospectra with error control and spectral measures while exploiting the RKHS structure to avoid the large-data limits required in the $L^2$ settings. The function space is determined by a user-specified kernel, eliminating the need for quadrature-based sampling as in $L^2$ and enabling greater flexibility with finite, externally provided datasets. Using the Solvability Complexity Index hierarchy, we construct adversarial dynamical systems for these problems to show that no algorithm can succeed in fewer limits, thereby proving the optimality of our algorithms. Notably, this impossibility extends to randomized algorithms and datasets. We demonstrate the effectiveness of our algorithms on challenging, high-dimensional datasets arising from real-world measurements and high-fidelity numerical simulations, including turbulent channel flow, molecular dynamics of a binding protein, Antarctic sea ice concentration, and Northern Hemisphere sea surface height. The algorithms are publicly available in the software package $\texttt{SpecRKHS}$.

翻译：Koopman算子的数据驱动谱分析是理解众多现实世界动力系统的强大工具，涵盖从神经元活动到海表温度变化等广泛领域。Koopman算子作用于函数空间，最常在平方可积函数空间中被研究。然而，在合适的再生核希尔伯特空间（RKHS）上定义该算子具有诸多实际优势，包括带误差界的逐点预测、改善谱性质以简化计算，以及更高效的算法（尤其在高维情形）。我们首次提出了在RKHS上计算Koopman算子和Perron--Frobenius算子谱性质的通用、可证明收敛的数据驱动算法。这些方法能高效计算带误差控制的谱与伪谱以及谱测度，同时利用RKHS结构规避$L^2$场景所需的大数据极限。函数空间由用户指定的核函数确定，无需像$L^2$空间那样依赖基于积分的采样，从而在有限的外部提供数据集上实现更大灵活性。利用可解复杂性指数层级，我们针对这些问题构造了对抗性动力系统，证明任何算法都无法在更少的极限步骤内成功，由此验证了所提算法的最优性。值得注意的是，这种不可能性同样适用于随机化算法与数据集。我们在源于实际测量和高保真数值模拟的挑战性高维数据集上验证了算法的有效性，包括湍流槽道流、结合蛋白的分子动力学、南极海冰浓度以及北半球海面高度。相关算法已在软件包$\texttt{SpecRKHS}$中公开发布。