Spectral Theory in Hilbert Space

6. NEVANLINNA FUNCTIONS 37
choose k and n such that ρ(rn) − ρ(rk) < ε. We then obtain
−ε ≤ lim (ρjj(x) − ρ(x)) ≤ lim (ρjj(x) − ρ(x)) ≤ ε .
j→∞
j→∞
Hence {ρjj} ∞ 1 converges pointwise to ρ, except possibly in points of
discontinuity of ρ. But there are at most countably many such discontinuities,
ρ being increasing. Hence repeating the trick of extracting
subsequences, and then using the ‘diagonal’ sequence, we get a subsequence
of the original sequence which converges everywhere in I. We
now obtain (1).
If f is the characteristic function of a compact interval whose endpoints
are points of continuity for ρ and all ρj it is obvious that (6.3)
holds. It follows that (6.3) holds if f is a stepfunction with all discontinuities
at points where ρ and all ρj are continuous. If f is continuous
and ε > 0 we may, by uniform continuity, choose such a stepfunction
g so that supI|f − g| < ε. If C is a common bound for all ρj we then
obtain |
I (f − g) dρ| < 2Cε and similarly with ρ replaced by ρj. It
follows that limj→∞|
I f dρj −
f dρ| ≤ 4Cε and since ε is arbitrary
I
positive (2) follows.
Proof of Theorem 6.3. According to Lemma 6.2 we have, for
|z| < 1,
G(Rz) = i Im G(0) + 1
2π
π
−π
e iθ + z
e iθ − z dσR(θ) ,
where σR(θ) = θ
−π Re G(Reiϕ ) dϕ. Hence σR is increasing, ≥ 0 and
bounded from above by σR(π). Now Re G is a harmonic function so it
has the mean value property, which means that σR(π) = 2π Re G(0).
This is independent of R, so by Helly’s theorem we may choose a sequence
Rj ↑ 1 such that σR converges to an increasing function σ. Use
of the second part of Helly’s theorem completes the proof.
To prove the uniqueness of the function ρ of Theorem 6.1 we need
the following simple, but important, lemma.
Lemma 6.5 (Stieltjes’ inversion formula). Let ρ be complex-valued
of locally bounded variation, and such that ∞ dρ(t)
−∞ t2 is absolutely con-
+1
vergent. Suppose F (λ) is given by (6.1). Then if y < x are points of
continuity of ρ we have
ρ(x) − ρ(y) = lim
ε↓0
1
2πi
x
y
(F (s + iε) − F (s − iε) ds
= lim
ε↓0
1
π
x
y
∞
−∞
ε dρ(t)
(t − s) 2 ds .
+ ε2