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Weak and strong additivity of orthogonal projections

Statement

Assume Countable Choice. Let $(X, Σ)$ be a measurable space, let $H$ be a complex Hilbert space, and let $E : Σ \to B (H)$ take values in orthogonal projections and satisfy $E (\emptyset) = 0$ and $E (B \cap C) = E (B) E (C)$ for all $B, C \in Σ$ . Then the following two properties are equivalent:

(weak countable additivity) for every pairwise disjoint sequence $(B_{n})_{n \in N}$ in $Σ$ with union $B$ and all $x, y \in H$ , $⟨ E (B) x, y ⟩ = \sum_{n = 0}^{\infty} ⟨ E (B_{n}) x, y ⟩;$
(strong countable additivity) for every such sequence and every $x \in H$ , $E (B) x = \sum_{n = 0}^{\infty} E (B_{n}) x$ with the series converging in norm.

Facts & Assumptions

[A1]

An orthogonal projection value $P = E (B)$ satisfies $P^{2} = P = P^{*}$ and $⟨ P x, x ⟩ = ∥ P x ∥^{2} \geq 0$ , and it is contractive, $∥ P u ∥ \leq ∥ u ∥$ for all $u$ (Projection valued measure, Hilbert projections are linear, self-adjoint and contractive).

[A2]

The Hilbert adjoint satisfies $⟨ S u, v ⟩ = ⟨ u, S^{*} v ⟩$ for all $u, v$ , and conjugate symmetry gives $⟨ S^{*} u, v ⟩ = \overline{⟨ v, S^{*} u ⟩} = \overline{⟨ S v, u ⟩} = ⟨ u, S v ⟩$ ; for a self-adjoint $S$ the two pairings with $S$ coincide (The Hilbert-space adjoint of a bounded operator, Real and complex inner-product spaces and their induced length).

[A3]

Multiplicativity on intersections and the empty-set value hold: $E (B \cap C) = E (B) E (C)$ and $E (\emptyset) = 0$ ; if $C \subseteq B$ then $E (C) = E (B \cap C) = E (B) E (C) = E (C) E (B)$ (Projection valued measure).

[A4]

Cauchy–Schwarz gives $∣ ⟨ u, v ⟩ ∣ \leq ∥ u ∥ ∥ v ∥$ (Cauchy–Schwarz: $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥ ∥ y ∥$ , with equality exactly for dependent pairs).

[A5]

The pairing is linear in the first argument and conjugate-linear in the second, and $∥ u ∥^{2} = ⟨ u, u ⟩$ (Real and complex inner-product spaces and their induced length).

[A6]

Countable Choice is the declared standing hypothesis of this block of the page (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Proof

technique · direct

Given: A measurable space $(X, Σ)$ , a complex Hilbert space $H$ , a map $E$ with orthogonal projection values, $E (\emptyset) = 0$ , $E (B \cap C) = E (B) E (C)$ , a pairwise disjoint sequence $(B_{n})$ in $Σ$ with union $B$ , vectors $x, y \in H$ , and partial sums $Q_{N} : = \sum_{n \leq N} E (B_{n})$ .

1.1

Strong implies weak: for every $N$ the difference of the two sides of the weak identity is $⟨ E (B) x - Q_{N} x, y ⟩$ , so by Cauchy–Schwarz $∣ ⟨ E (B) x, y ⟩ - \sum_{n \leq N} ⟨ E (B_{n}) x, y ⟩ ∣ \leq ∥ E (B) x - Q_{N} x ∥ ∥ y ∥$ , which tends to $0$ because $E (B) x - Q_{N} x \to 0$ in norm by hypothesis.

A1A4A5

1.2

Weak implies strong: expanding $∥ E (B) x - Q_{N} x ∥^{2} = ⟨ E (B) x - Q_{N} x, E (B) x - Q_{N} x ⟩$ and using $P^{2} = P = P^{*}$ for each projection value gives $∥ E (B) x - Q_{N} x ∥^{2} = ⟨ E (B) x, x ⟩ - 2 \sum_{n \leq N} ⟨ E (B_{n}) x, x ⟩ + \sum_{n, m \leq N} ⟨ E (B_{m} \cap B_{n}) x, x ⟩$ , because $⟨ E (B) x, E (B) x ⟩ = ⟨ E (B)^{2} x, x ⟩$ , because $⟨ E (B_{n}) x, E (B) x ⟩ = ⟨ E (B) E (B_{n}) x, x ⟩$ with $B_{n} \subseteq B$ , and because $⟨ E (B_{n}) x, E (B_{m}) x ⟩ = ⟨ E (B_{m}) E (B_{n}) x, x ⟩ = ⟨ E (B_{m} \cap B_{n}) x, x ⟩$ .

A1A2A3A5

2.1

In the last sum the off-diagonal terms are $⟨ E (\emptyset) x, x ⟩ = 0$ and the diagonal terms are $⟨ E (B_{n}) x, x ⟩$ , so $∥ E (B) x - Q_{N} x ∥^{2} = ⟨ E (B) x, x ⟩ - \sum_{n \leq N} ⟨ E (B_{n}) x, x ⟩$ , which tends to $0$ by weak countable additivity applied with $y = x$ ; hence $Q_{N} x \to E (B) x$ in norm.

step 1.2A3A5

3.1

Both implications hold for an arbitrary pairwise disjoint sequence, so weak and strong countable additivity of $E$ are equivalent.

step 1.1step 2.1A6∎

Depends on

Used by

Continuous functional calculus produces a regular PVM Lemma

Dependency tree · two levels

35 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

Theo Bühler and Dietmar Salamon, Functional Analysis, Definition 5.72 and §5.6.1, printed pp.273–277 (standard reference, not scraped)
Dana P. Williams, Lecture Notes on the Spectral Theorem, Definition 5.1 and Lemma 5.3, pp.15–16 (standard reference, not scraped)