DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-10

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The Koopman operator

Definition

For a system in Measure-preserving transformations and systems and $1 \leq p \leq \infty$ , the Koopman operator is $U_{T} : L^{p} (μ) \to L^{p} (μ)$ , $U_{T} [f] = [f \circ T]$ . Scalars can be real or complex, using The space $L^{p} (μ)$ as the quotient by null functions and Complex Lp classes and Euclidean test-function conventions. Membership and representative independence, and hence well-definedness, are proved in Koopman operators are linear isometries ↗.

Depends on

Used by

Eigenfunction for a probability system Definition
The Koopman matrix for a two-point swap Example
Koopman operators are linear isometries Theorem

Dependency tree · two levels

24 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

E–W §2.4 pp.28–29; Sarig Proposition 1.3 (standard reference, not scraped)