Edit: Rephrasing my post to make my statement clearer.
In $1931$, Bernard Koopman known for his work in ergodic theory, proposed a linear operator that describes the evolution of scalar observables (i.e., measurement functions of the states) in an infinite dimensional Hilbert space. In other words, if we have a non-linear system then we can shoot it to an infinite-dimensional function space where the evolution of the original system becomes linear. This has now become known as Koopman Operator Theory (KOT). This is a relatively new (for me) topic
$80$ years later his work resurged and became an important contribution in the study of stability analysis and in providing alternative formalism for study of dynamical systems mainly in non-linear control systems with applications ranging in adaptive control, non-linear control, system identification, deep learning, reinforcement learning which is the source of my motivation (to model non-linear dynamics using advanced and rigorous mathematics).
I myself have no rigorious understanding in much of functional analysis except from an engineering point of view in topics such as signal processing, control theory, but I questioned how can non-linear systems from an engineering perspective be dealt in an "infinite"-dimensional space.
With that being said, I am looking for a formal roadmap that can help me establish a proper and rigorous understanding of Koopman Operator Theory. While being obvious that this topic is just a fraction of what operator theory contains, I wish to know what concepts are needed in measure theory, functional analysis to reach the level of understanding required to deal with KOP from a mathematician perspective.
I am asking this question due to my engineering background with a rich background in applied mathematics but limited knowledge in analysis. Therefore, this question targets individuals who have knowledge in functional analysis, hilbert space,… or individuals who are active in the research community in particular in KOP theory.
Best Answer
Here are a few references wtih different perspectives:
First item is from the Hilbert space/ spectral theory perspective. The second item contains references that often emphasize relations with geometry and measure theory. The remainder are from a more applied perspective (though from what you wrote it seems you are already familiar with the literature from this perspective).
Also related are the discussions at Understanding the Definition of the Koopman Operator and Mathematical framework of the Koopman operator.