Mathematical framework of the Koopman operator

Question

I'm trying to understand the relation between the study of a dynamical system, seen as a pair $(X,\varphi_t)$ where $\varphi_t: X \rightarrow X$ represents a dynamics on the (measurable) space of states $X$ and the 'spectrum' of the associated Koopman operator

$$f \longmapsto f \circ \varphi$$

Basically the Koopman operator allow us to see the states involved in terms of their measurements.

Where $f$ is an observable, namely a linear positive functional $f: X \rightarrow \mathbb{C}$. Physically, I undestand that the dynamics $\varphi_t(x) := (x_t)_{t \in \mathbb{R}}$ describes the transitions of the physical system by time $t$ evolution running on $\mathbb{R}$, for example the position and velocity of gas particles in a three-dimensional space. And the observable $f$ is just a measurement of some quantity, for instance the temperature.

What I'm trying to understand is, what is the main 'mathematical' advantage and relation of studying operator properties of $\{T_{\varphi_t}(f) \, : \, t \in \mathbb{R}\}$ and why are we interested on studying the spectrum of such a thing?

Answer 1

I do ergodic theory, so I will speak from my experience in that area, using the notation of that area. As such, we tend to look at things through the perspective of a measure. So we suppose that $m$ is a natural measure on your space (normally just Lebesgue) which is non-singular with respect to your dynamical system. I will assume for simplicity that your dynamics have discrete indices, that is, we just iterate a single map $T\colon X \to X$. The non-singularity means that $m(T^{-1} A) =0$ if and only if $m(A)=0$. Note that this is much weaker than invariance. We assume this for technical reasons, although it is reasonable to think that for a natural measure, the transformation doesn't turn sets of measure zero into sets of positive measure.

The Koopman operator $U$ is defined then as an operator acting on $L^\infty(\mu) $, that is, the set of essentially bounded functions. It is defined as composition with $T$, that is, $U(f) = f\circ T$. It seems like we have not achieved anything, but some interesting things happen with this operator. Some dynamical properties of $T$ are translated to spectral properties of $U$. For instance, ergodicity means that the only fixed points of $U$ are the constant functions (see Walters book). You can go further and look at the 'dual' operator $P$ of $U$, acting on $L^1(\mu)$ and defined by the duality relationship

$$ \int_X Uf g dm = \int_X f Pg dm $$

for all $f\in L^\infty$ and $g\in L^1$. Here things get really interesting, as more dynamical properties of $T$ are translated into spectral properties of $P$. For instances, you can construct absolutely continuous invariant measures densities by looking at the fixed points of $P$, you can characterize mixing in terms of the spectrum of $P$ and many other things. There is a lot of spectral theory you can use to gain dynamycal insights using these operators.

Answer 2

Perhaps this is too obvious but I believe it is worth stating: The Koopman operator allows one to attach a linear system (albeit acting on a big space) to a possibly nonlinear system in accordance with general categorical principles (pullbacks in this case). This is done to get extra structure to use; in this case e.g. spectral theory.

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This fits into the framework of attaching to an anonymous dynamical system some other not-as-anonymous dynamical system. For $X$ measurable there are essentially three popular such categorical constructions. There is:

• An induced action on the space of probability measures on $X$ by affine automorphisms,
• An induced action on some space of observables on $X$ by (typically bounded) linear operators, given by the Koopman operator,
• An induced action on the Boolean $\sigma$-algebra of $X$ by Boolean $\sigma$-algebra automorphisms.

(Halmos in his Lectures on Ergodic Theory calls the first the geometric approach, the second the analytic approach and the third the algebraic approach (p. 42).)

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Likewise, for instance if $X$ has some topological structure one looks at associated homological/homotopical actions and if $X$ has some differential structure one looks at actions on vector fields or forms. Often there are many structures one considers in tandem as these different structures have implications for each other and the original anonymous dynamics.