  This paper demonstrates the benefits of imposing stability on data-driven
Koopman operators. The data-driven identification of stable Koopman operators
(DISKO) is implemented using an algorithm \cite{mamakoukas_stableLDS2020} that
computes the nearest \textit{stable} matrix solution to a least-squares
reconstruction error. As a first result, we derive a formula that describes the
prediction error of Koopman representations for an arbitrary number of time
steps, and which shows that stability constraints can improve the predictive
accuracy over long horizons. As a second result, we determine formal conditions
on basis functions of Koopman operators needed to satisfy the stability
properties of an underlying nonlinear system. As a third result, we derive
formal conditions for constructing Lyapunov functions for nonlinear systems out
of stable data-driven Koopman operators, which we use to verify stabilizing
control from data. Lastly, we demonstrate the benefits of DISKO in prediction
and control with simulations using a pendulum and a quadrotor and experiments
with a pusher-slider system. The paper is complemented with a video:
\url{https://sites.google.com/view/learning-stable-koopman}.
