  The Koopman operator is a mathematical tool that allows for a linear
description of non-linear systems, but working in infinite dimensional spaces.
Dynamic Mode Decomposition and Extended Dynamic Mode Decomposition are amongst
the most popular finite dimensional approximation. In this paper we capture
their core essence as a dual version of the same framework, incorporating them
into the Kernel framework. To do so, we leverage the RKHS as a suitable space
for learning the Koopman dynamics, thanks to its intrinsic finite-dimensional
nature, shaped by data. We finally establish a strong link between kernel
methods and Koopman operators, leading to the estimation of the latter through
Kernel functions. We provide also simulations for comparison with standard
procedures.
