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Strong spectral commutation, joint projection-valued measures and bounded Borel calculus

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 $H$ be a nonzero complex Hilbert space and let $A, B$ be self-adjoint operators on $H$ with spectral projection-valued measures $E_{A}, E_{B}$ (Spectral theorem for unbounded self-adjoint operators (PVM form), Projection valued measure). Then the following are equivalent:

(i) $A$ and $B$ strongly commute in the sense of Strong spectral commutation of self-adjoint operators clause (b), that is $E_{A} (Λ) E_{B} (Σ) = E_{B} (Σ) E_{A} (Λ)$ for all Borel $Λ, Σ \subseteq R$ ;

(ii) there is a regular projection-valued measure $G$ on the Borel sigma-algebra of $R^{2}$ whose values on measurable rectangles (Measurable rectangles in a product of measurable spaces) are $G (Λ \times Σ) = E_{A} (Λ) E_{B} (Σ)$ , and whose marginals are $G (Λ \times R) = E_{A} (Λ)$ and $G (R \times Σ) = E_{B} (Σ)$ ;

(iii) $f (A) g (B) = g (B) f (A)$ for all bounded Borel functions $f, g : R \to C$ , where the operators are those of the bounded Borel functional calculus (Bounded borel pvm integral).

If these conditions hold, the projection-valued measure $G$ of (ii) is unique, and for all bounded Borel $f, g$ one has the representation $f (A) g (B) = g (B) f (A) = \int_{R^{2}} f (a) g (b) d G (a, b) .$

Facts & Assumptions

Given: The Axiom of Choice and the Axiom of Countable Choice; a nonzero complex Hilbert space $H$ ; self-adjoint operators $A, B$ on $H$ with spectral projection-valued measures $E_{A}, E_{B}$ ; Borel sets $Λ, Σ \subseteq R$ ; bounded Borel functions $f, g$ on $R$ .

[F1]

Strong commutation. $A$ and $B$ strongly commute exactly when $E_{A} (Λ) E_{B} (Σ) = E_{B} (Σ) E_{A} (Λ)$ for all Borel $Λ, Σ$ , and this is a symmetric condition in $A$ and $B$ (Strong spectral commutation of self-adjoint operators clause (b)).

[F2]

Bounded calculus of strongly commuting operators. If $A, B$ strongly commute, then $f (A) g (B) = g (B) f (A)$ for all bounded Borel $f, g$ (Bounded Borel calculus of strongly commuting self-adjoint operators clause (a)).

[F3]

Joint projection-valued measure construction. Let $E, F$ be regular projection-valued measures on the Borel sigma-algebra of $R$ with $E (B) F (C) = F (C) E (B)$ for all Borel $B, C \subseteq R$ . Then there is a unique regular projection-valued measure $G$ on the Borel sigma-algebra of $R^{2}$ with $G (B \times C) = E (B) F (C)$ for every measurable rectangle $B \times C$ , and its marginals are $G (B \times R) = E (B)$ and $G (R \times C) = F (C)$ (Joint PVM construction for strongly commuting regular projection-valued measures).

[F4]

Regularity of spectral measures. For every $x \in H$ the set function $E_{A, x} (Λ) = ⟨ E_{A} (Λ) x, x ⟩$ is a finite Borel measure on $R$ with $E_{A, x} (R) = ∥ x ∥^{2}$ (Scalar and complex measures from a pvm). Every Borel measure on $R$ that is finite on compact sets is regular, because $R$ is locally compact Hausdorff and each of its open subsets is sigma-compact (Sigma-compact open sets make locally finite Borel measures regular); hence $E_{A}$ and $E_{B}$ are regular projection-valued measures.

[F5]

Multiplicativity of projection-valued measures. A projection-valued measure $G$ satisfies $G (S \cap T) = G (S) G (T)$ for all measurable $S, T$ , and every value is an orthogonal projection (Projection valued measure).

[F6]

Bounded integral and star homomorphism. For a projection-valued measure $E$ on a measurable space and bounded measurable $h$ , the operator $Φ_{E} (h) = \int h d E$ satisfies $∥ Φ_{E} (h) ∥ \leq ∥ h ∥_{\infty}$ and is the norm limit of the integrals of uniformly convergent simple approximants; the map $h \mapsto Φ_{E} (h)$ is linear, unital, multiplicative and adjoint preserving (Bounded borel pvm integral clauses 1 and 2, Pvm integral is a star homomorphism clauses 1-3).

Proof

technique · prove the equivalence in the order (i) implies (ii) implies (i), (i) implies (iii), (iii) implies (i); obtain the representation of the last display from the unique joint projection-valued measure by simple approximation

Given: The data of the statement and [F1]-[F6].

1.1givenF1F3F4

[(i) implies (ii)] Assume (i). By [F4] the projection-valued measures $E_{A}$ and $E_{B}$ are regular, and by [F1] they commute on all Borel pairs; so [F3] supplies a unique regular projection-valued measure $G$ on the Borel sets of $R^{2}$ with $G (Λ \times Σ) = E_{A} (Λ) E_{B} (Σ)$ for all measurable rectangles and with the stated marginals. This is (ii), including uniqueness.

1.2givenF5

[(ii) implies (i)] Assume (ii). For Borel $Λ, Σ$ , the multiplicativity [F5] and the marginals give $E_{A} (Λ) E_{B} (Σ) = G (Λ \times R) G (R \times Σ) = G (Λ \times Σ) = G (R \times Σ) G (Λ \times R) = E_{B} (Σ) E_{A} (Λ),$ using $(Λ \times R) \cap (R \times Σ) = Λ \times Σ$ . This is (i).

1.3givenF2

[(i) implies (iii)] This is [F2] verbatim.

1.4givenF6

[(iii) implies (i)] Assume (iii). Taking $f = 1_{Λ}$ and $g = 1_{Σ}$ , which are bounded Borel, gives $E_{A} (Λ) E_{B} (Σ) = E_{B} (Σ) E_{A} (Λ)$ . This is (i).

2.1givenstep 1.1F2F5F6constructalgebra∎

[The joint representation] Assume (i)-(iii) and let $G$ be the unique joint projection-valued measure of (ii), so that $G (Λ \times Σ) = E_{A} (Λ) E_{B} (Σ)$ for rectangles. Let $W (h) : = \int h d G$ denote the bounded integral against $G$ for bounded Borel $h : R^{2} \to C$ . For simple $s = \sum_{i} c_{i} 1_{Λ_{i}}$ , $t = \sum_{j} d_{j} 1_{Σ_{j}}$ over disjoint covers of $R$ , the disjointness of the rectangles $Λ_{i} \times Σ_{j}$ and the defining property of the simple integral against a projection-valued measure give $s (A) t (B) = \sum_{i, j} c_{i} d_{j} E_{A} (Λ_{i}) E_{B} (Σ_{j}) = \sum_{i, j} c_{i} d_{j} G (Λ_{i} \times Σ_{j}) = W (s \otimes t) .$ For general bounded Borel $f, g$ choose simple $s_{n}, t_{n}$ with $∥ f - s_{n} ∥_{\infty} \to 0$ and $∥ g - t_{n} ∥_{\infty} \to 0$ (uniform approximation by preimages of finite Borel partitions of bounded squares in $C$ , as in [F6]); then $∥ s_{n} \otimes t_{n} - f \otimes g ∥_{\infty} \to 0$ , so by the norm bound of [F6] both $s_{n} (A) t_{n} (B) \to f (A) g (B)$ and $W (s_{n} \otimes t_{n}) \to W (f \otimes g)$ in operator norm, whence $f (A) g (B) = W (f \otimes g) = \int f (a) g (b) d G (a, b)$ ; combining with [F2] gives the displayed chain.

Strong spectral commutation, joint projection-valued measures and bounded Borel calculus

Statement

Facts & Assumptions

Proof

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