  This manuscript revisits theoretical assumptions concerning dynamic mode
decomposition (DMD) of Koopman operators, including the existence of lattices
of eigenfunctions, common eigenfunctions between Koopman operators, and
boundedness and compactness of Koopman operators. Counterexamples that
illustrate restrictiveness of the assumptions are provided for each of the
assumptions. In particular, this manuscript proves that the native reproducing
kernel Hilbert space (RKHS) of the Gaussian RBF kernel function only supports
bounded Koopman operators if the dynamics are affine. In addition, a new
framework for DMD, that requires only densely defined Koopman operators over
RKHSs is introduced, and its effectiveness is demonstrated through numerical
examples.
