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Scalar and complex measures from a pvm

Statement

Assume Countable Choice. Let $(X, Σ)$ be a measurable space, let $H$ be a complex Hilbert space, let $E$ be a projection valued measure on $(X, Σ)$ , and for $x, y \in H$ define

$E_{x} (B) : = ⟨ E (B) x, x ⟩, E_{x, y} (B) : = ⟨ E (B) x, y ⟩ (B \in Σ) .$

Then:

$E_{x}$ is a positive measure on $(X, Σ)$ with $E_{x} (X) = ∥ x ∥^{2}$ and $0 \leq E_{x} (B) \leq ∥ x ∥^{2}$ for every $B$ ;
$E_{x, y}$ is a finite complex measure on $(X, Σ)$ , the map $(x, y) \mapsto E_{x, y}$ is linear in $x$ and conjugate-linear in $y$ , and $E_{y, x} = \overline{E_{x, y}}$ , meaning $E_{y, x} (B) = \overline{E_{x, y} (B)}$ for every $B$ ;
$∣ E_{x, y} ∣ (X) \leq ∥ x ∥ ∥ y ∥$ , so $∣ E_{x, y} (B) ∣ \leq ∥ x ∥ ∥ y ∥$ for every $B$ , and the polarization identity $E_{x, y} = \frac{1}{4} \sum_{k = 0}^{3} i^{k} E_{x + i^{k} y}$ holds as an identity of complex measures, where $i^{k}$ are the fourth roots of unity $1, i, - 1, - i$ .

Facts & Assumptions

[A1]

Projection values satisfy $P^{2} = P = P^{*}$ , $⟨ P x, x ⟩ = ∥ P x ∥^{2} \geq 0$ and $∥ P x ∥ \leq ∥ x ∥$ (Projection valued measure, Hilbert projections are linear, self-adjoint and contractive).

[A2]

$E (\emptyset) = 0$ , $E (X) = I$ , $E (B \cap C) = E (B) E (C)$ , and for every pairwise disjoint sequence $(B_{n})$ with union $B$ one has $E (B) x = \lim_{N} \sum_{n \leq N} E (B_{n}) x$ in norm (Projection valued measure).

[A3]

$⟨ S u, v ⟩ = ⟨ u, S^{*} v ⟩$ for all $u, v$ , and for self-adjoint $S$ one has $⟨ S u, v ⟩ = ⟨ u, S v ⟩$ and $⟨ S u, v ⟩ = \overline{⟨ S v, u ⟩}$ (The Hilbert-space adjoint of a bounded operator, Real and complex inner-product spaces and their induced length).

[A4]

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

[A5]

A measure is a $[0, + \infty]$ -valued countably additive set function vanishing at $\emptyset$ (Measures on sigma-algebras); a complex measure is a $C$ -valued countably additive set function vanishing at $\emptyset$ (A complex measure is a finite-valued countably additive set function); its total variation is the supremum of $\sum_{n} ∣ ν (E_{n}) ∣$ over countable measurable partitions (The total variation |nu|(E) from countable measurable partitions).

[A6]

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

[A7]

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 projection valued measure $E$ on it, vectors $x, y \in H$ , and a pairwise disjoint sequence $(B_{n}) \subseteq Σ$ with union $B$ .

1.1A2A4A6

$E_{x}$ is countably additive and vanishes at $\emptyset$ : using strong additivity, $E_{x} (B) = ⟨ \lim_{N} \sum_{n \leq N} E (B_{n}) x, x ⟩ = \lim_{N} \sum_{n \leq N} ⟨ E (B_{n}) x, x ⟩ = \sum_{n} E_{x} (B_{n})$ , with the limit pulled through the linear functional $⟨ \cdot, x ⟩$ , continuous since $∣ ⟨ v, x ⟩ - ⟨ w, x ⟩ ∣ \leq ∥ v - w ∥ ∥ x ∥$ , while $E_{x} (\emptyset) = ⟨ 0, x ⟩ = 0$ and $E_{x} (X) = ⟨ x, x ⟩ = ∥ x ∥^{2}$ .

1.2A1A2A3A4A5A6

$E_{x, y}$ is a finite complex measure and the map is linear in $x$ and conjugate-linear in $y$ : countable additivity holds by the same limit argument with the continuous functional $⟨ \cdot, y ⟩$ , $E_{x, y} (\emptyset) = 0$ , and $⟨ E (B) (x + u), y ⟩ = ⟨ E (B) x, y ⟩ + ⟨ E (B) u, y ⟩$ , $⟨ E (B) (λ x), y ⟩ = λ ⟨ E (B) x, y ⟩$ , $⟨ E (B) x, λ y ⟩ = \overline{λ} ⟨ E (B) x, y ⟩$ ; finiteness follows from $∣ E_{x, y} (B) ∣ = ∣ ⟨ E (B) x, y ⟩ ∣ \leq ∥ E (B) x ∥ ∥ y ∥ \leq ∥ x ∥ ∥ y ∥$ .

1.3A1A3A6

Conjugate symmetry: $\overline{E_{x, y} (B)} = \overline{⟨ E (B) x, y ⟩} = ⟨ y, E (B) x ⟩ = ⟨ E (B) y, x ⟩ = E_{y, x} (B)$ , using that $E (B)$ is self-adjoint.

1.4A6algebra

Polarization: for fixed $B$ the form $Λ (u, v) : = ⟨ E (B) u, v ⟩$ is sesquilinear, so expanding the four terms gives $\frac{1}{4} \sum_{k = 0}^{3} i^{k} Λ (u + i^{k} v, u + i^{k} v) = Λ (u, v)$ for all $u, v$ (the $∣ u ∣^{2}$ and $∣ v ∣^{2}$ coefficients cancel and the mixed terms add to $4 Λ (u, v)$ ); reading the identity at $B$ and letting $B$ vary gives $E_{x, y} = \frac{1}{4} \sum_{k = 0}^{3} i^{k} E_{x + i^{k} y}$ .

2.1step 1.1A1A5A6

$E_{x}$ is nonnegative and bounded by its total mass: $E_{x} (B) = ⟨ E (B) x, E (B) x ⟩ = ∥ E (B) x ∥^{2} \geq 0$ and $∥ E (B) x ∥ \leq ∥ x ∥$ , so $0 \leq E_{x} (B) \leq ∥ x ∥^{2}$ ; hence $E_{x}$ is a positive measure with $E_{x} (X) = ∥ x ∥^{2}$ .

3.1step 1.1step 2.1A1A2A3A4A5algebra

Variation bound: let $(B_{j})_{j \geq 0}$ be a countable measurable partition of $X$ . For every $N$ , Cauchy--Schwarz gives $\sum_{j = 0}^{N} ∣ E_{x, y} (B_{j}) ∣ = \sum_{j = 0}^{N} ∣ ⟨ E (B_{j}) x, E (B_{j}) y ⟩ ∣ \leq (\sum_{j = 0}^{N} ∥ E (B_{j}) x ∥^{2})^{1 / 2} (\sum_{j = 0}^{N} ∥ E (B_{j}) y ∥^{2})^{1 / 2} \leq ∥ x ∥ ∥ y ∥$ . Indeed, orthogonality and strong additivity give $\sum_{j \geq 0} ∥ E (B_{j}) x ∥^{2} = \sum_{j \geq 0} ⟨ E (B_{j}) x, x ⟩ = E_{x} (X) = ∥ x ∥^{2}$ , because $E (B_{j})^{2} = E (B_{j}) = E (B_{j})^{*}$ and the $B_{j}$ partition $X$ ; the same holds with $y$ . Taking $N \to \infty$ yields $\sum_{j \geq 0} ∣ E_{x, y} (B_{j}) ∣ \leq ∥ x ∥ ∥ y ∥$ .

4.1step 3.1A5

Hence every countable partition contributes at most $∥ x ∥ ∥ y ∥$ to the defining supremum of $∣ E_{x, y} ∣ (X)$ , so $∣ E_{x, y} ∣ (X) \leq ∥ x ∥ ∥ y ∥$ and $∣ E_{x, y} (B) ∣ \leq ∣ E_{x, y} ∣ (X) \leq ∥ x ∥ ∥ y ∥$ for every $B$ , since $(B, X ∖ B, \emptyset, \dots)$ is one of those countable partitions.

5.1step 1.1step 2.1step 1.2step 1.3step 4.1step 1.4A7∎

All asserted properties of $E_{x}$ , $E_{x, y}$ hold for arbitrary $x, y$ , so the scalar pairings of a projection valued measure are a positive measure of mass $∥ x ∥^{2}$ and a family of finite complex measures of variation at most $∥ x ∥ ∥ y ∥$ , conjugate symmetric and recovered by polarization.

Scalar and complex measures from a pvm

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources