Spectral Theory in Hilbert Space

CHAPTER 7
The spectral theorem
Theorem 7.1. (Spectral theorem) Suppose T is selfadjoint. Then
there exists a unique, increasing and left-continuous family {Et}t∈R of
orthogonal projections with the following properties:
• Et commutes with T , in the sense that T Et is the closure of
EtT .
• Et → 0 as t → −∞ and Et → I (= identity on H) as t → ∞
(strong convergence).
• T = ∞
−∞ t dEt in the following sense: u ∈ D(T ) if and only if
∞
∞
−∞ t2 d〈Etu, u〉.
−∞ t2 d〈Etu, u〉 < ∞, 〈T u, v〉 = ∞
−∞ t d〈Etu, v〉 and T u 2 =
The family {Et}t∈R of projections is called the resolution of the identity
for T . The formula T = ∞
−∞t dEt can be made sense of directly by
introducing Stieltjes integrals with respect to operator-valued increasing
functions. This is a simple generalization of the scalar-valued case.
Although we then, formally, get a slightly stronger statement, it does
not appear to be any more useful than the statement above. We will
therefore omit this.
For the proof we need two lemmas, the first of which actually contains
the main step of the proof.
Lemma 7.2. For f, g ∈ H there is a unique left-continuous function
σf,g of bounded variation, with σf,g(−∞) = 0, and the following
properties:
• σf,g is Hermitian in f, g ( i.e., σf,g = σg,f and is linear in f),
and σf,f is increasing.
• dσf,g is a bounded sesquilinear form on H. In fact, we even
have ∞
−∞ |dσf,g| ≤ fg.
• 〈Rλf, g〉 = ∞ dσf,g
−∞ t−λ .
Proof. The uniqueness of σf,g follows from the Stieltjes inversion
formula, applied to F (λ) = 〈Rλf, g〉. Since 〈Rλf, g〉 is sesqui-linear in
f, g and R ∗ λ = R λ , it then follows that σf,g is Hermitian if it exists.
However, by Theorem 5.3 the function λ ↦→ 〈Rλf, f〉 is a Nevanlinna
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