  Recent years have seen rapid advances in the data-driven analysis of
dynamical systems based on Koopman operator theory and related approaches. On
the other hand, low-rank tensor product approximations -- in particular the
tensor train (TT) format -- have become a valuable tool for the solution of
large-scale problems in a number of fields. In this work, we combine
Koopman-based models and the TT format, enabling their application to
high-dimensional problems in conjunction with a rich set of basis functions or
features. We derive efficient algorithms to obtain a reduced matrix
representation of the system's evolution operator starting from an appropriate
low-rank representation of the data. These algorithms can be applied to both
stationary and non-stationary systems. We establish the infinite-data limit of
these matrix representations, and demonstrate our methods' capabilities using
several benchmark data sets.
