In this work, we study the convergence of Hermitian Dynamic Mode Decomposition (DMD) to the spectral properties of self-adjoint Koopman operators. Hermitian DMD is a data-driven method for approximating the Koopman operator associated with an unknown nonlinear dynamical system from discrete-time snapshots, while preserving the self-adjointness of the operator on its finite-dimensional approximations. We show that, under suitable conditions, the eigenvalues and eigenfunctions of HDMD converge to the spectral properties of the underlying Koopman operator. Along the way, we establish a general theorem on the convergence of spectral measures, and demonstrate our results numerically on the two-dimensional Schr\"odinger equation.

翻译：本文研究Hermitian动态模态分解（DMD）在自伴Koopman算子谱性质上的收敛性。Hermitian DMD是一种数据驱动方法，用于从离散时间快照中近似未知非线性动力系统对应的Koopman算子，同时在其有限维近似中保持算子的自伴性。我们证明，在适当条件下，HDMD的特征值与特征函数收敛于底层Koopman算子的谱性质。在此过程中，我们建立了关于谱测度收敛性的通用定理，并通过二维薛定谔方程对结果进行了数值验证。