  The estimation for the region of attraction (ROA) of an asymptotically stable
equilibrium point is crucial in the analysis of nonlinear systems. There has
been a recent surge of interest in estimating the solution to Zubov's equation,
whose non-trivial sub-level sets form the exact ROA. In this paper, we propose
a lifting approach to map observable data into an infinite-dimensional function
space, which generates a flow governed by the proposed `Zubov-Koopman'
operators. By learning a Zubov-Koopman operator over a fixed time interval, we
can indirectly approximate the solution to Zubov's equation through iterative
application of the learned operator on certain functions. We also demonstrate
that a transformation of such an approximator can be readily utilized as a
near-maximal Lyapunov function. We approach our goal through a comprehensive
investigation of the regularities of Zubov-Koopman operators and their
associated quantities. Based on these findings, we present an algorithm for
learning Zubov-Koopman operators that exhibit strong convergence to the true
operator. We show that this approach reduces the amount of required data and
can yield desirable estimation results, as demonstrated through numerical
examples.
