  Koopman operators are infinite-dimensional operators that linearize nonlinear
dynamical systems, facilitating the study of their spectral properties and
enabling the prediction of the time evolution of observable quantities. Recent
methods have aimed to approximate Koopman operators while preserving key
structures. However, approximating Koopman operators typically requires a
dictionary of observables to capture the system's behavior in a
finite-dimensional subspace. The selection of these functions is often
heuristic, may result in the loss of spectral information, and can severely
complicate structure preservation. This paper introduces Multiplicative Dynamic
Mode Decomposition (MultDMD), which enforces the multiplicative structure
inherent in the Koopman operator within its finite-dimensional approximation.
Leveraging this multiplicative property, we guide the selection of observables
and define a constrained optimization problem for the matrix approximation,
which can be efficiently solved. MultDMD presents a structured approach to
finite-dimensional approximations and can more accurately reflect the spectral
properties of the Koopman operator. We elaborate on the theoretical framework
of MultDMD, detailing its formulation, optimization strategy, and convergence
properties. The efficacy of MultDMD is demonstrated through several examples,
including the nonlinear pendulum, the Lorenz system, and fluid dynamics data,
where we demonstrate its remarkable robustness to noise.
