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 that approximates the Koopman operator associated with an unknown nonlinear dynamical system, using discrete-time snapshots. This approach preserves the self-adjointness of the operator in its finite-dimensional approximations. \rev{We prove that, under suitably broad conditions, the spectral measures corresponding to the eigenvalues and eigenfunctions computed by Hermitian DMD converge to those of the underlying Koopman operator}. This result also applies to skew-Hermitian systems (after multiplication by $i$), applicable to generators of continuous-time measure-preserving systems. Along the way, we establish a general theorem on the convergence of spectral measures for finite sections of self-adjoint operators, including those that are unbounded, which is of independent interest to the wider spectral community. We numerically demonstrate our results by applying them to two-dimensional Schr\"odinger equations.

翻译：本文研究了Hermitian动态模态分解（DMD）向自伴Koopman算子谱性质的收敛性。Hermitian DMD是一种数据驱动方法，它利用离散时间快照来逼近与未知非线性动力系统相关联的Koopman算子。该方法在其有限维近似中保持了算子的自伴性。我们证明，在适当广泛的条件下，由Hermitian DMD计算得到的特征值和特征函数所对应的谱测度会收敛于底层Koopman算子的谱测度。该结果同样适用于斜Hermitian系统（乘以$i$后），可应用于连续时间保测系统的生成元。在此过程中，我们建立了一个关于自伴算子有限截断谱测度收敛性的通用定理，该定理涵盖无界算子情形，对更广泛的谱学界具有独立意义。我们通过将方法应用于二维薛定谔方程，对理论结果进行了数值验证。