Spectral Theory in Hilbert Space

42 7. THE SPECTRAL THEOREM
as
The resolvent relation Rλ − Rµ = (λ − µ)RλRµ may be expressed
∞
−∞
1
t − λ
d〈Etf, g〉
t − µ =
∞
−∞
d〈EtRµf, g〉
t − λ
(check this!), so the uniqueness of the Stieltjes transform shows that
〈EtRµf, g〉 = t d〈Esf,g〉
. But
−∞ s−µ
〈EtRµf, g〉 = 〈Rµf, Etg〉 =
∞
−∞
d〈Esf, Etg〉
.
s − µ
So, again by uniqueness, 〈Esf, Etg〉 = 〈Euf, g〉 where u = min(s, t),
i.e., EtEs = Emin(s,t). For s = t this shows that Et is a projection, and
if t > s we get 0 ≤ (Et − Es) ∗ (Et − Es) = (Et − Es) 2 = Et − Es so that
{Et}t∈R is an increasing family of orthogonal projections.
Now suppose f ∈ D(T ) and v = T f. For any non-real λ we then
have f = Rλ(v−λf) or Rλv = f +λRλf. Since 1+λ/(t−λ) = t/(t−λ)
we therefore obtain
∞
−∞
dσv,g(t)
t − λ =
∞
−∞
t dσf,g
t − λ
so that σv,g(t) = t
−∞ s dσf,g(s). In particular, 〈T f, g〉 = ∞
−∞t d〈Etf, g〉.
We also get σv,v(t) = t
−∞ s dσf,v(s) = t
−∞ s2 dσf,f(s), so that T f2 =
∞
−∞s2d〈Esf, f〉.
Next we prove that any u ∈ H for which ∞
−∞s2 d〈Esu, u〉 < ∞ is
in D(T ). To see this, note that
|d〈Esu, v〉| ≤ d〈Esu, u〉 d〈Esv, v〉
∆
∆
if ∆ is a finite union of intervals. This follows just as in the proof of
Lemma 7.2. Now let ∆k = {s | 2 k−1 < |s| ≤ 2 k }, k ∈ Z. Then
s d〈Esu, v〉
≤ 2k |d〈Esu, v〉|
∆k
∆k
≤ 2 k
d〈Esu, u〉 d〈Esv, v〉 ≤ 2
∆k
∆k
∆k
∆
s2
d〈Esu, u〉
∆k
d〈Esv, v〉.
If now ∞
−∞s2 d〈Esu, u〉 < ∞ we obtain from this by adding over all k
and using Cauchy-Schwarz’ inequality for sums that ∞
−∞s d〈Esu,
v〉 ≤
∞
2 −∞s2d〈Esu, u〉v so that the anti-linear form v ↦→ ∞
−∞s d〈Esu, v〉