TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedverified 2026-10-15 (gpt-6-luna, gpt-6.1-sol, and delegated Codex review/repair agents; see completed run evidence)

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.

Trace-class Gibbs normalization for a semibounded self-adjoint operator

Statement

Assume the Axiom of Choice. Let $H \neq {0}$ be a complex Hilbert space and let $A$ be self-adjoint with regular Borel projection-valued measure $E$ from Spectral theorem for unbounded self-adjoint operators (PVM form), with $E ((- \infty, a)) = 0$ for finite $a \in R$ . For each $γ > 0$ , define the bounded Gibbs operator $K_{γ} : = Φ_{E} (f_{γ})$ using the bounded Borel PVM integral of Bounded borel pvm integral, where $f_{γ} (λ) = e^{- γ λ}$ for $λ \geq a$ and $f_{γ} (λ) = e^{- γ a}$ for $λ < a$ . Fix $β > 0$ and suppose $K_{β}$ is trace class in the sense of Trace class operator. Then $K_{β}$ is injective and has a finite or countable orthonormal basis $(u_{j})$ of $H$ with $A u_{j} = E_{j} u_{j}$ , $E_{j} \geq a$ , and $K_{β} u_{j} = e^{- β E_{j}} u_{j}$ . Its partition function $Z_{β} : = Tr K_{β} = \sum_{j} e^{- β E_{j}}$ is positive and finite, and $ρ_{β} : = K_{β} / Z_{β}$ is positive with $Tr ρ_{β} = 1$ . For every $B \in B (H)$ , $Tr (ρ_{β} B) = \sum_{j} (e^{- β E_{j}} / Z_{β}) ⟨ B u_{j}, u_{j} ⟩$ absolutely. Further, let $J \subset (0, \infty)$ be open and suppose that for every compact $C \subset J$ there is a summable family $a_{j} \geq 0$ with $(1 + ∣ E_{j} ∣^{2}) e^{- γ E_{j}} \leq a_{j}$ for all $j$ and $γ \in C$ . Then for each $γ \in J$ , $K_{γ}$ is trace class with partition function $Z_{γ} : = Tr K_{γ} = \sum_{j} e^{- γ E_{j}}$ . With $p_{j} (γ) : = e^{- γ E_{j}} / Z_{γ}$ , define ${\overset{ˉ}{E}}_{γ} : = \sum_{j} p_{j} (γ) E_{j}$ , ${Var}_{γ} (E) : = \sum_{j} p_{j} (γ) (E_{j} - {\overset{ˉ}{E}}_{γ})^{2}$ , and $S_{γ} : = - \sum_{j} p_{j} (γ) \log p_{j} (γ)$ . These quantities are finite and satisfy $- \partial_{γ} \log Z_{γ} = {\overset{ˉ}{E}}_{γ}$ , $\partial_{γ}^{2} \log Z_{γ} = {Var}_{γ} (E)$ , and $S_{γ} = \log Z_{γ} + γ {\overset{ˉ}{E}}_{γ}$ as in Summable canonical differentiation. Countable nonnegative sums use the finite-subsum supremum convention of Countable canonical model.

Facts & Assumptions

Given: AC, a nonzero complex Hilbert space, a self-adjoint semibounded operator, the specified Borel functional calculus, trace-class $K_{β}$ , and a bounded operator $B$ .

[A1]

The Hilbert space is nonzero; $A$ is self-adjoint with spectral support in $[a, \infty)$ ; AC is adopted; $β > 0$ ; and $K_{β}$ is trace class. For the interval clause, the same list $(E_{j})$ has the local summable moment envelope stated in the theorem.

[A2]

$B$ is a bounded operator on $H$ .

[F1]

The Axiom of Choice and Spectral theorem for unbounded self-adjoint operators (PVM form): under AC a self-adjoint $A$ has a regular PVM $E$ , domain $D (A) = {x : \int λ^{2} d E_{x} < \infty}$ , and $A x = \int λ d E (λ) x$ .

[F2]

Bounded borel pvm integral: for bounded Borel $f$ , $K = Φ_{E} (f)$ satisfies $∥ K x ∥^{2} = \int ∣ f ∣^{2} d E_{x}$ and nonnegative $f$ gives a positive operator.

[F3]

Spectral theorem for compact self adjoint operators and Orthonormal eigenbasis for a compact self adjoint operator: compact self-adjoint operators have finite-dimensional nonzero eigenspaces whose closed span is $(\ker K)^{⊥}$ ; under AC their union has a complete orthonormal eigenbasis when $\ker K = 0$ .

[F4]

Trace class operator and Trace of a positive operator is the sum of its eigenvalues: positive trace-class operators are compact and their trace is the sum of their nonnegative eigenvalues with multiplicity.

[F5]

Trace is absolutely convergent and basis independent and Trace class is a two sided Banach operator ideal: the trace is basis-independent and linear in trace-class operators, and multiplying a trace-class operator by a bounded operator preserves trace class.

[F6]

Countable canonical model and Summable canonical differentiation: nonnegative countable sums are suprema of finite subsums, and the stated local moment envelope gives a finite partition sum, finite mean and variance, and the canonical derivative and entropy identities.

[F7]

Cyclicity of the trace: for trace-class $S$ and bounded $B$ , $Tr (S B) = Tr (B S)$ .

[F8]

Real and complex inner-product spaces and their induced length: the inner product is linear in the first argument and conjugate-linear in the second.

[F9]

Absolute value and singular values of a compact operator: for compact $T$ , its positive singular values are the positive eigenvalues of the positive operator $∣ T ∣ = (T^{*} T)^{1 / 2}$ counted with multiplicity. In particular, if $T$ is positive then its absolute value equals $T$ .

[F10]

Parseval equivalences for an orthonormal family: a complete orthonormal family has Parseval equality and its finite-coordinate projections converge in norm to each vector.

[F11]

The exponential is a continuous bijection from $R$ onto $(0, \infty)$ and The exponential function is strictly increasing: the real exponential is continuous, positive, injective, and strictly increasing.

[F12]

The natural logarithm as the inverse of the exponential function: for $x > 0$ , $\log x$ is the unique real $y$ satisfying $e^{y} = x$ , so $\exp (\log x) = x$ and $\log (\exp y) = y$ .

[F13]

Integral of a simple function against a pvm, Projection valued measure, and Scalar and complex measures from a pvm: for a simple function $s = \sum_{k} c_{k} 1_{D_{k}}$ in disjoint normal form, $\int s d E = \sum_{k} c_{k} E (D_{k})$ ; the PVM satisfies $E (B \cap C) = E (B) E (C)$ and $E (R) = I$ , each $E (D_{k})$ is self-adjoint with $⟨ E (D_{k}) x, x ⟩ = ∥ E (D_{k}) x ∥^{2} \geq 0$ , and $E_{x} (B) = ⟨ E (B) x, x ⟩$ is a positive countably additive measure with $E_{x} (R) = ∥ x ∥^{2}$ .

[F14]

Cauchy–Schwarz: $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥ ∥ y ∥$ , with equality exactly for dependent pairs: $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥ ∥ y ∥$ , so inner products are continuous under norm convergence.

[F15]

Bounded finite rank operators are compact and Norm limit of compact operators is compact: bounded finite-rank operators are compact, and under AC a norm limit of compact operators into the Hilbert space $H$ is compact.

[F16]

Monotonicity and nonnegative homogeneity of the nonnegative integral, The nonnegative integral agrees with the simple integral on simple functions, and The integral of a nonnegative simple function: the nonnegative integral is monotone and homogeneous for positive scalars, and $\int 1_{C} d μ = μ (C)$ for measurable $C$ .

Proof

Given: AC, a nonzero complex Hilbert space, self-adjoint semibounded $A$ , its spectral PVM, a parameter $β > 0$ , and trace-class $K_{β}$ .

1.1A1F1F2F11F13F14F16

Since $β > 0$ , $- β λ \leq - β a$ for $λ \geq a$ ; strict increase and positivity from [F11] show that $f_{β}$ is bounded and strictly positive. Each branch is continuous and their values agree at $a$ , so $f_{β}$ is Borel. For each $n$ , partition its bounded range into finitely many intervals of length at most $1 / n$ and take their Borel preimages; choosing a nonnegative value in each interval gives a nonnegative real simple $s_{n}$ converging uniformly to $f_{β}$ . By [F13], $T_{n} : = \int s_{n} d E$ is a finite real linear combination of self-adjoint projections, hence self-adjoint and has nonnegative quadratic form. By [F2], $T_{n} \to K_{β}$ in operator norm. Therefore $⟨ K_{β} x, y ⟩ = ⟨ x, K_{β} y ⟩$ and $⟨ K_{β} x, x ⟩ \geq 0$ by [F14] and passage to the limit, so $K_{β}$ is self-adjoint and positive. If $K_{β} x = 0$ , [F2] gives $\int f_{β}^{2} d E_{x} = 0$ . For each $n \geq 1$ , $f_{β} \geq 1 / n$ on $C_{n} : = {λ : f_{β} (λ) \geq 1 / n}$ ; [F16] gives $0 = \int f_{β}^{2} d E_{x} \geq n^{- 2} E_{x} (C_{n})$ , so $E_{x} (C_{n}) = 0$ . The sets $C_{n}$ increase to $R$ ; set $D_{1} = C_{1}$ and $D_{n} = C_{n} ∖ C_{n - 1}$ for $n > 1$ . They are disjoint, cover $R$ , and satisfy $E_{x} (D_{n}) \leq E_{x} (C_{n}) = 0$ . Countable additivity in [F13] gives $E_{x} (R) = \sum_{n} E_{x} (D_{n}) = 0$ , hence $∥ x ∥ = 0$ and $K_{β}$ is injective.

2.1A1F3F4step 1.1

By the trace-class definition [F4], $K_{β}$ is compact; together with step 1.1 it is compact self-adjoint and positive. Apply [F3]: its nonzero eigenspaces have finite multiplicity and span $(\ker K_{β})^{⊥} = H$ , so AC supplies a finite or countable complete orthonormal family $(u_{j})$ of eigenvectors with eigenvalues $κ_{j} > 0$ . By [F4], $Z_{β} = Tr K_{β} = \sum_{j} κ_{j} < \infty$ ; because $H \neq 0$ and $K_{β}$ is injective, this sum is positive.

3.1A1F1F2F3F4F11F12F13F16step 2.1

Fix $j$ . To justify the constant shift in the calculus, choose bounded simple $s_{n} \to f_{β}$ uniformly. By [F13], $\int (s_{n} - κ_{j}) d E = \int s_{n} d E - κ_{j} I$ ; [F2] passes both uniform limits, so $Φ_{E} (f_{β} - κ_{j}) = K_{β} - κ_{j} I$ . Applying the norm identity in [F2] to this difference and the eigenvector gives $\int ∣ f_{β} - κ_{j} ∣^{2} d E_{u_{j}} = ∥ (K_{β} - κ_{j} I) u_{j} ∥^{2} = 0$ . For every $n$ , [F16] shows that the set where $∣ f_{β} - κ_{j} ∣ \geq 1 / n$ has zero $E_{u_{j}}$ -measure; their union is the complement of the level set. Hence $E_{u_{j}}$ is supported on $f_{β}^{- 1} (κ_{j}) \cap [a, \infty) = {E_{j}}$ : injectivity of the exponential gives uniqueness, and [F12] gives $E_{j} = - β^{- 1} \log κ_{j} \geq a$ . Thus $\int λ^{2} d E_{u_{j}} = E_{j}^{2} < \infty$ and [F1] yields $u_{j} \in D (A)$ and $A u_{j} = E_{j} u_{j}$ .

4.1A1F1F2F4F6F8F9F10F13F14F15step 1.1step 3.1

The spectral measure of each $u_{j}$ is the unit mass at $E_{j}$ by step 3.1. Hence $∥ (I - E ({E_{j}})) u_{j} ∥^{2} = E_{u_{j}} (R ∖ {E_{j}}) = 0$ , so $E ({E_{j}}) u_{j} = u_{j}$ ; for Borel $D$ not containing $E_{j}$ , $E (D) u_{j} = E (D) E ({E_{j}}) u_{j} = E (\emptyset) u_{j} = 0$ by [F13]. For every $γ > 0$ , choose nonnegative simple $s_{n} \to f_{γ}$ uniformly as in step 1.1. In each disjoint normal form, [F13] then gives $(\int s_{n} d E) u_{j} = s_{n} (E_{j}) u_{j}$ ; the uniform-limit clause of [F2] yields $K_{γ} u_{j} = f_{γ} (E_{j}) u_{j} = e^{- γ E_{j}} u_{j}$ . Fix $γ \in J$ and enumerate that basis; let $P_{N}$ project onto its first $N$ vectors, taking $P_{N} = I$ once a finite basis is exhausted. Parseval gives $∥ P_{N} x ∥ \leq ∥ x ∥$ , so $K_{γ} P_{N}$ is bounded and finite rank. Fix the summable envelope $a_{j}$ for ${γ} \subset J$ ; every finite subsum of $t_{j} : = e^{- γ E_{j}}$ is at most the corresponding sum of $a_{j}$ , hence $S : = \sum_{j} t_{j} < \infty$ by the finite-subsum definition [F6]. This sum is positive because the basis is nonempty and every term is positive. For $ε > 0$ , choose a finite set $F$ with $S - ε < \sum_{j \in F} t_{j}$ ; once $N$ contains $F$ , every finite tail subsum is at most $S - \sum_{j \in F} t_{j} < ε$ , so $\sum_{j > N} t_{j} \to 0$ . Again by Parseval, $∥ (K_{γ} - K_{γ} P_{N}) x ∥^{2} = \sum_{j > N} e^{- 2 γ E_{j}} ∣ ⟨ x, u_{j} ⟩ ∣^{2} \leq (\sum_{j > N} e^{- γ E_{j}})^{2} ∥ x ∥^{2},$ so $K_{γ} P_{N}$ converges to $K_{γ}$ in operator norm. For $t_{j} = e^{- γ E_{j}}$ its finite diagonal expansion is $K_{γ} P_{N} x = \sum_{j \leq N} t_{j} ⟨ x, u_{j} ⟩ u_{j}$ ; orthonormality and [F8] give $⟨ K_{γ} P_{N} x, y ⟩ = ⟨ x, K_{γ} P_{N} y ⟩$ and $⟨ K_{γ} P_{N} x, x ⟩ = \sum_{j \leq N} t_{j} ∣ ⟨ x, u_{j} ⟩ ∣^{2} \geq 0$ . Thus each truncation is self-adjoint and positive, and [F14] plus operator-norm convergence preserves both properties. It is compact by [F15], since the truncations are bounded finite-rank operators and $H$ is Banach. Thus $K_{γ}$ is compact self-adjoint and positive. Its absolute value is $K_{γ}$ , by the unique positive square-root definition in [F9]. If $K_{γ} x = λ x$ , taking coordinates in the complete basis gives $(t_{j} - λ) ⟨ x, u_{j} ⟩ = 0$ for every $j$ ; thus its eigenvalues and their multiplicities are exactly the diagonal values $t_{j}$ . Hence [F9] identifies its positive singular values, with multiplicity, as the list $(t_{j})$ . Their finite subsums are bounded by $\sum_{j} a_{j}$ , so the trace-class definition [F4] applies and gives $Tr K_{γ} = \sum_{j} t_{j}$ .

5.1A1F5F6step 2.1step 4.1

At the fixed $β$ , $ρ_{β} = K_{β} / Z_{β}$ is positive trace class and [F5] gives $Tr ρ_{β} = Z_{β}^{- 1} Tr K_{β} = 1$ . For each $γ \in J$ , $ρ_{γ} = K_{γ} / Z_{γ}$ likewise has trace one; its eigenvalues $p_{j} = e^{- γ E_{j}} / Z_{γ}$ form the canonical law, so [F6] gives the finite mean, variance, and identity $- \sum_{j} p_{j} \log p_{j} = \log Z_{γ} + γ \sum_{j} p_{j} E_{j}$ .

6.1A1A2F3F5F7F8step 5.1∎

For bounded $B$ , [F5] makes $ρ_{β} B$ and $B ρ_{β}$ trace class and [F7] gives $Tr (ρ_{β} B) = Tr (B ρ_{β})$ . Evaluate the second trace in the complete basis $(u_{j})$ : $Tr (B ρ_{β}) = \sum_{j} ⟨ B ρ_{β} u_{j}, u_{j} ⟩ = \sum_{j} p_{j} ⟨ B u_{j}, u_{j} ⟩$ , where [F8] and real $p_{j}$ justify pulling the scalar out. This series is absolutely convergent since $∣ ⟨ B u_{j}, u_{j} ⟩ ∣ \leq ∥ B ∥$ and $\sum_{j} p_{j} = 1$ .

Depends on

Dependency tree · 2 levels

Sources

Daniel Arovas, Lecture Notes on Thermodynamics and Statistical Mechanics (work in progress, dated 12 October 2026)