False statementConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-29

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FALSE: every complex-linear operator descends to every chosen real form

Statement

Every complex-linear operator on a complex vector space descends to every chosen real form.

Facts & Assumptions

Given: The complex vector space $C^{2}$ , the coordinatewise conjugation $σ_{0}$ , the real form $R^{2}$ , and the operator $T (z_{1}, z_{2}) = (i z_{1}, z_{2})$ .

[L1]

The complex-linear operator $T (z_{1}, z_{2}) = (i z_{1}, z_{2})$ fails to commute with $σ_{0}$ and does not carry $R^{2}$ into itself (A complex-linear map need not preserve a chosen real form).

[L2]

A complex-linear operator comes from a real operator on the fixed real form exactly when it commutes with the chosen conjugation (A complex-linear operator comes from a real operator exactly when it commutes with the chosen conjugation).

Refutation

technique · direct

1.1L1

By [L1], the operator $T$ is complex-linear but satisfies $T σ_{0} \neq σ_{0} T$ , with the concrete witness $T σ_{0} (1, 0) = (i, 0) \neq (- i, 0) = σ_{0} T (1, 0)$ .

2.1L2step 1.1

By [L2], the failure of commutation means $T$ is not of the form $θ \circ S_{C} \circ θ^{- 1}$ for any real operator $S$ on $R^{2}$ ; hence $T$ does not descend to this chosen real form.

3.1step 1.1step 2.1∎

Steps 1.1 and 2.1 exhibit one complex-linear operator and one chosen real form for which descent fails, refuting the claimed universal statement.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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

Mikhail Troshkin, Real-complex linear algebra and abelian varieties (standard reference, not scraped)