  In this work, the problem of learning Koopman operator of a discrete-time
autonomous system is considered. The learning problem is formulated as a
constrained regularized empirical loss minimization in the infinite-dimensional
space of linear operators. We show that under certain but general conditions, a
representer theorem holds for the learning problem. This allows reformulating
the problem in a finite-dimensional space without any approximation and loss of
precision. Following this, we consider various cases of regularization and
constraints in the learning problem, including the operator norm, the Frobenius
norm, rank, nuclear norm, and stability. Subsequently, we derive the
corresponding finite-dimensional problem. Furthermore, we discuss the
connection between the proposed formulation and the extended dynamic mode
decomposition. Finally, we provide an illustrative numerical example.
