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Atoms of the spectral measure and eigenvalues of a self-adjoint operator

Statement

Assume the Axiom of Choice and the Axiom of Countable Choice (The Axiom of Choice, The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $H \neq {0}$ be a complex Hilbert space (Hilbert space), let $A$ be a self-adjoint operator on $H$ with spectral projection-valued measure $E_{A}$ on the Borel $σ$ -algebra of $R$ (Spectral theorem for unbounded self-adjoint operators (PVM form), Projection valued measure), and let $E_{A, x} (B) : = ⟨ E_{A} (B) x, x ⟩$ be its scalar measures (Scalar and complex measures from a pvm). For $λ \in R$ say that $λ$ is an eigenvalue of $A$ when $A x = λ x$ for some nonzero $x \in D (A)$ , the domain-restricted reading of Eigenvalues, eigenvectors, eigenspaces $E_{λ} (T) = \ker (T - λ I)$ , and the spectrum $σ_{F} (T)$ of an endomorphism. Then:

(a) The orthogonal projection $E_{A} ({λ})$ is the projection of $H$ onto the eigenspace $\ker (A - λ) = {x \in D (A) : A x = λ x}$ ; in particular $E_{A} ({λ}) \neq 0$ if and only if $λ$ is an eigenvalue of $A$ .

(b) For every density operator $ρ$ on $H$ , that is every self-adjoint positive trace-class $ρ$ with $tr ρ = 1$ (Density operators and scalar spectral probability clause (a)), the number $μ_{ρ}^{A} ({λ}) = tr (ρ E_{A} ({λ}))$ is the weight that the scalar law $μ_{ρ}^{A}$ assigns to the singleton ${λ}$ : it vanishes for every $ρ$ whenever $λ$ is not an eigenvalue of $A$ , and it equals $1$ for the pure state $T_{u, u}$ of every unit eigenvector $u \in \ker (A - λ)$ .

(c) If $A$ has no eigenvalues at all, then $μ_{ρ}^{A} ({λ}) = 0$ for every density operator $ρ$ and every $λ \in R$ : the scalar law of every state carries no point mass, while its values on general Borel sets remain $tr (ρ E_{A} (B))$ .

Clauses (a)-(c) are mathematical statements about $A$ , $E_{A}$ and the density operators; no physical interpretation of $μ_{ρ}^{A}$ is asserted here.

Facts & Assumptions

Given: The Axiom of Choice and the Axiom of Countable Choice (The Axiom of Choice, The Axiom of Countable Choice ( ${A C}_{ω}$ )); a nonzero complex Hilbert space $H$ ; a self-adjoint operator $A$ on $H$ with spectral projection-valued measure $E_{A}$ and scalar measures $E_{A, x} (B) = ⟨ E_{A} (B) x, x ⟩$ ; and a real number $λ$ .

[F1]

Spectral representation. $E_{A}$ is the unique regular projection-valued measure on the Borel $σ$ -algebra of $R$ with $D (A) = {x \in H : \int_{R} λ^{2} d E_{A, x} (λ) < \infty}$ and $A x = \int_{R} λ d E_{A} (λ) x$ for every $x \in D (A)$ (Spectral theorem for unbounded self-adjoint operators (PVM form),).

[F2]

Functional calculus. For every Borel function $f : R \to C$ the operator $f (A) = f (E_{A})$ has domain $D (f (A)) = {x \in H : \int_{R} ∣ f ∣^{2} d E_{A, x} < \infty}$ , is closed and densely defined, satisfies $∥ f (A) x ∥^{2} = \int_{R} ∣ f ∣^{2} d E_{A, x}$ and $f (A)^{*} = \overline{f} (A)$ , and is the norm limit of its bounded truncations $f 1_{{∣ f ∣ \leq n}} (A)$ on that domain; on $D (A)$ the functions $f (λ) = λ$ and $f (λ) = λ - μ$ give $A$ and $A - μ I$ (Unbounded Borel functional calculus: domains, products, spectral mapping,).

[F3]

Projection-valued measure. $E_{A}$ takes values in the orthogonal projections, with $E_{A} (\emptyset) = 0$ , $E_{A} (R) = I$ and $E_{A} (B \cap C) = E_{A} (B) E_{A} (C)$ for Borel $B, C$ ; for pairwise disjoint Borel sets $B_{n}$ with union $B$ one has $\sum_{n} E_{A} (B_{n}) x = E_{A} (B) x$ in norm for every $x \in H$ ; and $B \subseteq C$ implies $E_{A} (B) = E_{A} (B) E_{A} (C) = E_{A} (C) E_{A} (B)$ and $E_{A} (C) - E_{A} (B) = E_{A} (C ∖ B)$ (Projection valued measure, Scalar and complex measures from a pvm).

[F4]

Density operators and their scalar law. A density operator $ρ$ admits an expansion $ρ = \sum_{n} w_{n} T_{u_{n}, u_{n}}$ with $w_{n} \geq 0$ , $\sum_{n} w_{n} = 1$ and $(u_{n})$ a finite or countable orthonormal family; for every PVM $E$ on a measurable space the assignment $B \mapsto tr (ρ E (B))$ is a probability measure with $tr (ρ E (B)) = \sum_{n} w_{n} ∥ E (B) u_{n} ∥^{2}$ , and for a unit vector $u$ one has $μ_{T_{u, u}} (B) = ∥ E (B) u ∥^{2}$ (Density operators and scalar spectral probability clauses (a) and (b)).

[F5]

Orthogonal projections. An orthogonal projection $P$ satisfies $P^{2} = P = P^{*}$ and $∥ P x ∥^{2} = ⟨ P x, x ⟩ \leq ∥ x ∥^{2}$ for every $x \in H$ , and has zero range exactly when $P = 0$ ; moreover $∥ x - P x ∥^{2} = ∥ x ∥^{2} - ∥ P x ∥^{2}$ (The Hilbert orthogonal projection onto a closed subspace, Hilbert projections are linear, self-adjoint and contractive).

Proof

1.1givenF1F2F3algebra

[ $E_{A} ({λ}) H$ lies in the eigenspace] Let $x \in H$ satisfy $E_{A} ({λ}) x = x$ . For every Borel set $B$ the projection identity gives $E_{A, x} (B) = ⟨ E_{A} (B) E_{A} ({λ}) x, x ⟩ = ⟨ E_{A} (B \cap {λ}) x, x ⟩$ , so the finite measure $E_{A, x}$ is concentrated on ${λ}$ with total mass $∥ x ∥^{2}$ ; hence $\int λ^{2} d E_{A, x} = λ^{2} ∥ x ∥^{2} < \infty$ and $x \in D (A)$ by [F1]. By [F2] applied to $f (μ) = μ - λ$ one has $∥ (A - λ) x ∥^{2} = \int (μ - λ)^{2} d E_{A, x} (μ)$ , and the integrand vanishes $E_{A, x}$ -almost everywhere because $E_{A, x}$ is concentrated on ${λ}$ ; hence $A x = λ x$ .

1.2givenF2F3F5algebra

[Every eigenvector is fixed by $E_{A} ({λ})$ ] Conversely let $x \in D (A)$ satisfy $A x = λ x$ . By [F2] with $f (μ) = μ - λ$ , $0 = ∥ (A - λ) x ∥^{2} = \int (μ - λ)^{2} d E_{A, x} (μ)$ . For $n \geq 1$ put $C_{n} : = {μ \in R : ∣ μ - λ ∣ \geq 1 / n}$ ; on $C_{n}$ the integrand is at least $1 / n^{2}$ , so $E_{A, x} (C_{n}) \leq n^{2} \int (μ - λ)^{2} d E_{A, x} = 0$ . Set $D_{1} : = C_{1}$ and $D_{n} : = C_{n} ∖ C_{n - 1}$ for $n \geq 2$ . The sets $D_{n}$ are pairwise disjoint, their union is $R ∖ {λ}$ , and $E_{A, x} (D_{n}) = 0$ because $D_{n} \subseteq C_{n}$ ; hence countable additivity [F3] gives $E_{A, x} (R ∖ {λ}) = 0$ . Therefore $E_{A, x} ({λ}) = E_{A, x} (R) = ∥ x ∥^{2}$ , and since $(I - E_{A} ({λ}))$ is an orthogonal projection, [F5] gives $∥ x - E_{A} ({λ}) x ∥^{2} = ∥ x ∥^{2} - ∥ E_{A} ({λ}) x ∥^{2} = ∥ x ∥^{2} - E_{A, x} ({λ}) = 0$ , so $x = E_{A} ({λ}) x$ .

2.1step 1.1step 1.2F5

[Clause (a)] Steps 1.1 and 1.2 show that the range of the orthogonal projection $E_{A} ({λ})$ is exactly $\ker (A - λ)$ . A nonzero eigenspace contains a nonzero vector, and a projection with nonzero range is nonzero by [F5]; conversely $E_{A} ({λ}) \neq 0$ supplies a nonzero element of its range, which is an eigenvector by step 1.1. Hence $E_{A} ({λ}) \neq 0$ exactly when $λ$ is an eigenvalue of $A$ .

3.1givenstep 1.2step 2.1F4algebra

[Clause (b)] Let $ρ = \sum_{n} w_{n} T_{u_{n}, u_{n}}$ be the expansion of [F4]. Then $μ_{ρ}^{A} ({λ}) = tr (ρ E_{A} ({λ})) = \sum_{n} w_{n} ∥ E_{A} ({λ}) u_{n} ∥^{2} \geq 0$ by [F4], and the sum is zero if and only if $∥ E_{A} ({λ}) u_{n} ∥ = 0$ for every $n$ of positive weight. If $λ$ is not an eigenvalue then $E_{A} ({λ}) = 0$ by step 2.1, so every term vanishes and $μ_{ρ}^{A} ({λ}) = 0$ . If $u$ is a unit eigenvector at $λ$ , then $u = E_{A} ({λ}) u$ by step 1.2 and [F4] gives $μ_{T_{u, u}}^{A} ({λ}) = ∥ E_{A} ({λ}) u ∥^{2} = ∥ u ∥^{2} = 1$ .

4.1step 2.1F4algebra∎

[Clause (c)] If $A$ has no eigenvalues then $E_{A} ({λ}) = 0$ for every $λ$ by step 2.1, so $μ_{ρ}^{A} ({λ}) = tr (ρ 0) = 0$ for every density operator and every $λ$ : the scalar law assigns no mass to any singleton. The values of $μ_{ρ}^{A}$ on general Borel sets are unchanged and remain those of [F4].

Atoms of the spectral measure and eigenvalues of a self-adjoint operator

Statement

Facts & Assumptions

Proof

Depends on

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