Observable Spaces Capturing Koopman Dynamics

Definitions and mathematical statement

For notation, a discrete-time dynamical system consists of a set $X$ and a map $T:X\to X$. An observable is a scalar-valued function on $X$; the source does not specify whether the scalars are real or complex. On a vector space $F$ of observables satisfying $f\circ T\in F$ for every $f\in F$, the Koopman operator is \[ U_T:F\to F,\qquad U_Tf=f\circ T. \] For a continuous-time system $(T^t)$, the corresponding operators have the form $U^tf=f\circ T^t$. These formulas explain the terminology; the supplied record does not select a time model or a class of systems.

Problem. Determine how to choose a space of observables $F$, together with an appropriate topology and operator domain where needed, so that the spectral properties of the associated Koopman operator or operator family meaningfully encode the dynamics.

The phrase "meaningfully encode the dynamics" is not assigned a mathematical criterion in the supplied record. In particular, no specified dynamical information to be recovered, spectral invariant, equivalence criterion, or quantifier over systems is provided. Thus the record states a research question rather than a fully specified conjecture.

The supplied dated assessment reports information loss for $L^2$, an excessively large spectrum for $C(K)$, and construction of a natural observable space only in a hyperbolic-attractor setting. These are source-specific qualifications, not universal assertions established here. The measure underlying $L^2$, the space $K$, the relevant operators, and the hypotheses of the hyperbolic-attractor construction are not supplied.

Short English statement

Choose a space of observables on which Koopman spectral information captures specified features of the underlying dynamics. The supplied question leaves both the class of systems and the meaning of successful recovery unspecified. Its dated assessment identifies limitations of $L^2$ and $C(K)$ and mentions a natural-space construction restricted to a hyperbolic-attractor setting.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003053$, title: Observable Spaces Capturing Koopman Dynamics; source problem code: catalogue entry; collection: Oberwolfach Reports Open Problems. No individual proposer is identified; the supplied proposed year is 2016.
• Mini-Workshop: Applied Koopmanism (2016), Oberwolfach Reports, pp. 297--340, DOI: 10.4171/owr/2016/7. Supplied source URL and literature reference labelled "original OWR report": https://doi.org/10.4171/owr/2016/7. The record attributes the question to this workshop-report provenance but supplies no problem-specific page or named author.
• Literature assessment dated 2026-08-21: the supplied curation labels the problem open and reports the $L^2$, $C(K)$, and hyperbolic-attractor qualifications stated above. This is supplied dated curation, not an independently verified literature assessment. The record explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports a manual semantic, typographic, and LaTeX review. This is supplied registry metadata, not independent mathematical verification of this draft.