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Stone resolvent formula for spectral projections

Statement

Assume AC. Let $T$ be a bounded self-adjoint operator on a nonzero complex Hilbert space $H$ with spectral projection valued measure $E$ on the compact set $σ (T) \subseteq R$ , and let $a < b$ be real. For every Borel set $A \subseteq R$ , write $E (A) : = E (A \cap σ (T))$ . Then, with the resolvents $(T - (t \pm i ε))^{- 1}$ defined for $ε > 0$ by Spectrum and resolvent of a bounded operator and the integral of a continuous $B (H)$ -valued function understood in the Bochner sense (Bochner-integrable function),

$\frac{1}{2 π i} \int_{a}^{b} [(T - (t + i ε))^{- 1} - (T - (t - i ε))^{- 1}] d t ⟶ E ((a, b)) + \frac{E ({a}) + E ({b})}{2}$

in the strong operator topology as $ε ↓ 0$ . In particular, if $a, b \notin σ (T)$ then the limit is the spectral projection $E ((a, b))$ , and the half-masses at $a$ and $b$ appear exactly when these points are atoms of the spectrum.

Facts & Assumptions

[A1]

For $z \notin R$ the function $g_{z} (λ) : = (λ - z)^{- 1}$ is bounded and Borel on $σ (T)$ , with $∥ g_{z} ∥_{\infty} \leq ∣ Im z ∣^{- 1}$ , and its Borel calculus value satisfies $(T - z I)^{- 1} = Φ_{E} (g_{z})$ : indeed $(T - z I) Φ_{E} (g_{z}) = Φ_{E} ((λ - z) g_{z}) = Φ_{E} (1) = I$ and $Φ_{E} (g_{z}) (T - z I) = I$ by linearity and multiplicativity of the Borel calculus (Borel functional calculus for bounded normal operators, Borel functional calculus for a bounded normal operator).

[A2]

$σ (T) \subseteq R$ for self-adjoint $T$ , so the functions $g_{z}$ are defined on the spectrum (Spectrum of a self adjoint operator is real).

[A3]

A continuous function on the compact interval $[a, b]$ with values in the Banach space $B (H)$ is Bochner integrable, and a bounded linear map $Φ$ satisfies $Φ (\int_{E} f d μ) = \int_{E} Φ f d μ$ (Bochner integrability criterion, Bounded linear maps commute with Bochner integration).

[A4]

The Borel calculus is a unital star-homomorphism: $Φ_{E} (1_{B}) = E (B)$ , and uniformly bounded pointwise $E$ -almost everywhere convergence implies strong convergence (Pvm integral is a star homomorphism, Projection valued measure).

[A5]

AC is the declared choice hypothesis of this page from the construction item onward (The Axiom of Choice).

Proof

technique · direct

Given: A bounded self-adjoint operator $T$ with spectral PVM $E$ on $σ (T) \subseteq R$ , real numbers $a < b$ , and $ε > 0$ .

1.1A1A2A3

Resolvent identity: since $σ (T) \subseteq R$ , the function $g_{z}$ is bounded Borel for $z = t \pm i ε$ and (A1) identifies $(T - (t \pm i ε))^{- 1} = Φ_{E} (g_{t \pm i ε})$ , so the integrand $t \mapsto Φ_{E} (g_{t + i ε}) - Φ_{E} (g_{t - i ε})$ is a continuous $B (H)$ -valued function on the compact interval and the Bochner integral converges.

1.2A2algebra

Scalar kernel: for real $λ$ and $t$ one computes $(λ - (t + i ε))^{- 1} - (λ - (t - i ε))^{- 1} = \frac{2 i ε}{(λ - t)^{2} + ε^{2}}$ , hence the bounded Borel function $K_{ε} (λ) : = \frac{1}{2 π i} \int_{a}^{b} [(λ - (t + i ε))^{- 1} - (λ - (t - i ε))^{- 1}] d t$ equals $\frac{1}{π} \int_{a}^{b} \frac{ε d t}{(λ - t)^{2} + ε^{2}} = \frac{1}{π} [\arctan \frac{b - λ}{ε} - \arctan \frac{a - λ}{ε}]$ , with $∣ K_{ε} ∣ \leq 1$ and, for every real $λ$ , $K_{ε} (λ) \to 1_{(a, b)} (λ) + \frac{1}{2} 1_{{a, b}} (λ)$ as $ε ↓ 0$ .

2.1step 1.1step 1.2A3

The operator integral is the calculus value of the kernel: by linearity of $Φ_{E}$ and commutation of the bounded linear map $Φ_{E}$ with Bochner integration, $\frac{1}{2 π i} \int_{a}^{b} [(T - (t + i ε))^{- 1} - (T - (t - i ε))^{- 1}] d t = \frac{1}{2 π i} \int_{a}^{b} [Φ_{E} (g_{t + i ε}) - Φ_{E} (g_{t - i ε})] d t = Φ_{E} (\frac{1}{2 π i} \int_{a}^{b} [g_{t + i ε} - g_{t - i ε}] d t) = Φ_{E} (K_{ε})$ .

3.1step 2.1A4

Strong limit: $∣ K_{ε} ∣ \leq 1$ and $K_{ε} \to 1_{(a, b)} + \frac{1}{2} 1_{{a, b}}$ pointwise, so the strong-convergence clause of the calculus gives $Φ_{E} (K_{ε}) \to Φ_{E} (1_{(a, b)} + \frac{1}{2} 1_{{a, b}}) = E ((a, b)) + \frac{1}{2} (E ({a}) + E ({b}))$ in the strong operator topology.

4.1step 3.1A5∎

Therefore the resolvent expression converges strongly to $E ((a, b)) + \frac{1}{2} (E ({a}) + E ({b}))$ as $ε ↓ 0$ ; if $a, b \notin σ (T)$ the endpoint atoms vanish and the limit is the open-interval spectral projection $E ((a, b))$ .

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Sources

Gerald Teschl, Mathematical Methods in Quantum Mechanics, 2nd ed., Theorem 4.3, printed pp.113–115 (standard reference, not scraped)
Theo Bühler and Dietmar Salamon, Functional Analysis, Lemma 5.79, printed pp.285–288 (standard reference, not scraped)