  We study a class of dynamical systems modelled as Markov chains that admit an
invariant distribution via the corresponding transfer, or Koopman, operator.
While data-driven algorithms to reconstruct such operators are well known,
their relationship with statistical learning is largely unexplored. We
formalize a framework to learn the Koopman operator from finite data
trajectories of the dynamical system. We consider the restriction of this
operator to a reproducing kernel Hilbert space and introduce a notion of risk,
from which different estimators naturally arise. We link the risk with the
estimation of the spectral decomposition of the Koopman operator. These
observations motivate a reduced-rank operator regression (RRR) estimator. We
derive learning bounds for the proposed estimator, holding both in i.i.d. and
non i.i.d. settings, the latter in terms of mixing coefficients. Our results
suggest RRR might be beneficial over other widely used estimators as confirmed
in numerical experiments both for forecasting and mode decomposition.
