TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-22

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.

Toeplitz hausdorff

Statement

Assume Countable Choice. The numerical range $W (T)$ of every bounded operator $T$ on a nonzero complex Hilbert space is convex.

Facts & Assumptions

[A1]

$W (T) = {⟨ T x, x ⟩ : ∥ x ∥ = 1}$ (Numerical range and numerical radius).

[A2]

For a complex subspace $V \subseteq H$ of dimension at most two the numerical range of the compression $P_{V} T ∣_{V}$ is convex (Two dimensional numerical range is convex).

[A3]

For the closed subspace $V$ the orthogonal decomposition $H = V \oplus V^{⊥}$ and the projection $P_{V}$ with $P_{V} v = v$ for $v \in V$ are available; a finite-dimensional subspace is closed (Orthogonal decomposition by a closed subspace, The Hilbert orthogonal projection onto a closed subspace).

[A4]

Countable Choice is the hypothesis of the Hilbert-space and compression suppliers (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Proof

technique · direct

Given: A nonzero complex Hilbert space $H$ , a bounded operator $T \in B (H)$ and two points $a = ⟨ T x, x ⟩$ , $b = ⟨ T y, y ⟩$ of $W (T)$ with $∥ x ∥ = ∥ y ∥ = 1$ .

1.1

For a closed subspace $V$ and a unit vector $v \in V$ one has $⟨ P_{V} T ∣_{V} v, v ⟩ = ⟨ T v, v ⟩$ , because $P_{V} v = v$ ; hence the numerical range of the compression is contained in $W (T)$ .

A1A3

2.1

The complex span $V : = C x + C y$ is a closed subspace of dimension at most two containing $x$ and $y$ , so both $a$ and $b$ lie in the numerical range of the compression $P_{V} T ∣_{V}$ .

step 1.1A3

3.1

Since the numerical range of that compression is convex, it contains the whole segment joining $a$ and $b$ ; by step 1.1 that segment lies in $W (T)$ .

step 1.1step 2.1A2

4.1

Every pair of points of $W (T)$ is joined by a segment inside $W (T)$ , so $W (T)$ is convex.

step 3.1A4∎

Sharpness remark

Convexity does not force closedness, and the witness is worth recording even though no item of this pair proves it in detail: for the multiplication operator $M_{t}$ , $(M_{t} f) (t) = t f (t)$ , on the complex Hilbert space $L^{2} (0, 1)$ one has $W (M_{t}) = (0, 1)$ . The two inclusions are the pointwise bounds $0 < \int_{0}^{1} t ∣ f (t) ∣^{2} d t < \int_{0}^{1} ∣ f (t) ∣^{2} d t = 1$ for a unit vector $f$ , together with the explicit unit vectors proportional to the indicators of intervals $[c - ε, c + ε] \subseteq (0, 1)$ , for which the value tends to $c$ ; the exact value $c$ is attained by a continuous tent function concentrated at $c$ . In particular the numerical range of a bounded operator need not be closed, so the convexity conclusion above is not a closedness statement.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

25 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

Joel H. Shapiro, Notes on the Numerical Range, Theorem 6.1, PDF pp.15–17 (standard reference, not scraped)
Theo Bühler and Dietmar Salamon, Functional Analysis, §5.3, printed pp.235–245 (standard reference, not scraped)