Spectral Theory in Hilbert Space

108 15. SINGULAR PROBLEMS
We will give an expansion theorem generalizing the Fourier series
expansion obtained for a discrete spectrum. The first step is the following
lemma.
Lemma 15.4. Let M(λ) be as in Theorem 15.2. Then there is a
unique increasing and left-continuous matrix-valued function P with
P (0) = 0 and unique Hermitian matrices A and B ≥ 0 such that
(15.3) M(λ) = A + Bλ +
∞
−∞
( 1 t
−
t − λ t2 ) dP (t).
+ 1
Proof. If S = F (c, λ) Theorem 15.2.(5) gives
S(M(λ) − M(µ))S ∗ = (λ − µ)RλG ∗ (c, ·, µ)(c),
where the constant matrix S is invertible. Thus M(λ) is analytic in
ρ(T ), since the resolvent Rλ : L2 W → C(K) is. Furthermore, for µ = λ
non-real we obtain
1
2i Im λ (M(λ) − M ∗ 1
(λ)) = (M(λ) − M(λ))
2i Im λ
= S −1 〈G ∗ (c, ·, λ), G ∗ (c, ·, λ)〉W (S −1 ) ∗ ≥ 0.
Thus M is a ‘matrix-valued Nevanlinna function’. We now obtain
the representation (15.3) by applying Theorem 6.1 to the Nevanlinna
function m(λ, u) = u ∗ M(λ)u where u is an n × 1-matrix. Clearly the
quantities α, β and ρ in the representation (6.1) are Hermitian forms
in u, so (15.3) follows.
The function P is called the spectral matrix for T . We now define
the Hilbert space L2 P in the following way. We consider n × 1 matrixvalued
Borel functions û, so that they are measurable with respect to
all elements of dP , and for which the integral ∞
−∞û∗ (t) dP (t) û(t) <
∞. The elements of L2 P are equivalence classes of such functions, two
functions u, v being equivalent if they are equal a.e. with respect to
dP , i.e., if dP (u − v) has all elements equal to the zero measure. We
denote the scalar product in this space by 〈·, ·〉P and the norm by
·P . Note that one may write the scalar product in a somewhat more
familiar way by using the Radon-Nikodym theorem to find a measure
dµ with respect to which all the entries in dP are absolutely continuous;
one may for example let dµ be the sum of all diagonal elements in
dP . One then has dP = Ω dµ, where Ω is a non-negative matrix of
functions locally integrable with respect to dµ, and the scalar product
is 〈û, ˆv〉P = ∞
−∞ˆv∗ Ωû dµ. Alternatively, we define L2 P as the completion
of compactly supported, continuous n×1 matrix-valued functions with
respect to the norm ·P . These alternative definitions give the same
space (Exercise 15.2). The main result of this chapter is the following.