  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.
