Spectra of Koopman Operators for Linear Maps

Definitions and mathematical statement

Let $k\geq 1$ be an integer and let $A\in M_k({\mathbb{C}})$. The source considers the linear map $x\mapsto Ax$ on a unit ball $U\subseteq{\mathbb{C}}^k$ and its Koopman operator on complex-valued continuous functions, given by \[ (T_Af)(x)=f(Ax). \] For this formula to define an operator on $C(U)$, one needs $A(U)\subseteq U$. This necessary interpretation of "acting on the unit ball" is not explicitly stated in the supplied record.

The source does not specify the norm defining the ball, whether the ball is open or closed, or the topology on $C(U)$. If $U$ is a closed unit ball, then it is compact, and the standard Banach-space interpretation is \[ C(U)=\{f:U\to{\mathbb{C}}: f\text{ is continuous}\},\qquad \|f\|_\infty=\sup_{x\in U}|f(x)|. \] Under the invariance condition, $T_A$ is then bounded. If $U$ is open, continuous functions need not be bounded, so this Banach-space interpretation does not automatically apply. The closed-ball interpretation is therefore conditional, not a resolved reading of the source.

For a specified complex Banach space $X=C(U)$ on which $T_A$ is bounded, define \[ \sigma(T_A)=\{\lambda\in{\mathbb{C}}:T_A-\lambda I_X\text{ has no bounded two-sided inverse}\}, \] \[ \sigma_{\mathrm p}(T_A)=\{\lambda\in{\mathbb{C}}:\exists f\in X\setminus\{0\},\ f(Ax)=\lambda f(x)\text{ for every }x\in U\}. \] The matrix spectrum is \[ \sigma(A)=\{\mu\in{\mathbb{C}}:\det(A-\mu I_k)=0\}. \] Problem, subject to the unresolved functional setting. For matrices $A$ acting on the intended unit ball, describe the relation between $\sigma(A)$ and $\sigma(T_A)$, with particular emphasis on $\sigma_{\mathrm p}(T_A)$. Determine in what sense the latter spectra can be determined from the matrix spectrum. The source does not specify whether the matrix spectral data include algebraic multiplicities, or whether additional information about $A$ is permitted. It supplies a question, not an assertion that the set $\sigma(A)$ alone always determines the answer.

Short English statement

The problem asks how the eigenvalues of a linear map on a complex unit ball relate to the spectrum, especially the eigenvalues, of composition by that map on continuous functions. A precise formulation requires specifying the ball, its invariance under the map, and the function-space topology. The supplied record labels the problem partially solved; this is a supplied assessment requiring independent review.

Sources

• Local registry: data/unsolvedmath/problems.json, supplied source record with permanent UnsolvedMath ID $30003052$, title "Spectra of Koopman Operators for Linear Maps," and exact source problem code catalogue entry. The collection is "Oberwolfach Reports Open Problems."
• Supplied report citation: Mini-Workshop: Applied Koopmanism (2016), Oberwolfach Reports, pp. 297--340, DOI 10.4171/owr/2016/7. Source URL and literature-source link labeled "original OWR report": https://doi.org/10.4171/owr/2016/7. The record gives 2016 as the proposed year and supplies no individual proposer.
• Supplied literature assessment, checked 2026-08-21: status "partially solved." The supplied assessment says that the report proves point-spectrum formulas for Koopman operators in several linear systems on a unit ball, but establishes no general relation between the Koopman spectrum and the dynamics spectrum. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment; its status and cited claims require independent verification.
• The registry labels the statement corrected_verified and reports a manual semantic, typographic, and LaTeX review asserting self-containment and preservation of the target. These are supplied editorial claims, not mathematical verification of this draft or resolution of the qualifications identified here.