LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-12

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.

Compact intertwiners produce finite dimensional invariant subspaces

Statement

Assume AC. A nonzero invariant square-integrable kernel with zero marginals for a measure-preserving transformation of a completed Lebesgue probability space produces a nonzero finite-dimensional $U_{T}$ -invariant subspace $E$ of $H_{0}$ . The restriction $U_{T} ∣_{E}$ is unitary, even when $T$ is not invertible.

Facts & Assumptions

[F1]

The associated $S = K^{*} K$ is nonzero, positive, self-adjoint and compact on $H_{0}$ , and commutes with $U_{T}$ Nonzero compact kernel operators yield nonzero positive compact k star k.

[F2]

Such an $S$ has a nonzero finite-dimensional positive eigenspace The positive norm eigenvalue of a nonzero positive compact self-adjoint operator has a nonzero finite-dimensional eigenspace.

[F3]

Invariant zero-marginal kernels give commuting operators preserving $H_{0}$ Invariant square integrable kernel produces a compact intertwiner.

[F4]

Koopman is an isometry and $H_{0}$ is closed Eigenfunction for a probability system.

[F5]

A finite-dimensional linear map satisfies rank-nullity Rank-nullity: $\dim_{F} V = nullity T + rank T$ .

[F6]

Assume AC The Axiom of Choice.

Proof

Given: The kernel in the statement and AC.

1.1F1F2F3F4F6

F3 supplies the nonzero restriction of $K$ to the closed space $H_{0}$ , so this space is nonzero. Apply F1 there, then F2, to obtain $α > 0$ and the nonzero finite-dimensional subspace $E = \ker (S - α I) \subseteq H_{0}$ . AC supplies all inherited projection and compactness selections.

2.1F1F4F5step 1.1∎

For $f \in E$ , commutation gives $S (U_{T} f) = U_{T} (S f) = α U_{T} f$ , so $U_{T} f \in E$ . The restriction is injective because $U_{T}$ preserves norm. Rank-nullity gives image dimension equal to $\dim E$ , and a subspace of a finite-dimensional space with full dimension equals that space: a basis of a proper subspace could be enlarged by a vector outside it, contradicting the dimension. Thus the restriction is surjective. Being a surjective isometry, it is unitary.

Depends on

Used by

Weak mixing is equivalent to absence of nonconstant eigenfunctions Theorem

Dependency tree · two levels

26 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

Sarig Theorem 3.2 p.91, local compact-intertwiner proof of its difficult implication (standard reference, not scraped)
Axler 10.99 p.326 (local positive version) (standard reference, not scraped)