Spectral Theory in Hilbert Space

102 14. EIGENFUNCTION EXPANSIONS
We can now show that the resolvent is an integral operator. First
note that if T is a selfadjoint realization of (13.1), i.e., a selfadjoint
restriction of T1, then setting HT = D(T ) the resolvent Rλ of the operator
part ˜ T of T is an operator on HT , defined for λ ∈ ρ( ˜ T ). We define
the resolvent set ρ(T ) = ρ( ˜ T ) and extend Rλ to all of L2 W by setting
RλH∞ = 0, and it is then clear that the resolvent has all the properties
of Theorems 5.2 and 5.3; the only difference is that the resolvent
is perhaps no longer injective. Given u ∈ L 2 W
we obtain the element1
(Rλu, λRλu+u) ∈ T1, so we may also view the resolvent as an operator
˜Rλ : L 2 W → T1. This operator is bounded since (Rλu, λRλu+u)W ≤
((1 + |λ|)Rλ + 1)uW . Hence ˜ Rλ ≤ (1 + |λ|)Rλ + 1, where Rλ
is the norm of Rλ as an operator on HT . It is also clear that the analyticity
of Rλ implies the analyticity of ˜ Rλ. We obtain the following
theorem.
Theorem 14.2. Suppose I is an arbitrary interval, and that T is a
selfadjoint realization in L2 W of the system (13.1), satisfying Assumption
13.7. Then the resolvent Rλ of T may be viewed as a bounded
to C(K), for any compact subinterval K of I,
linear map from L2 W
which depends analytically on λ ∈ ρ(T ), in the uniform operator topology.
Furthermore, there exists Green’s function G(x, y, λ), an n × n
matrix-valued function, such that Rλu(x) = 〈u, G∗ (x, ·, λ)〉W for any
u ∈ L2 W . The columns of y ↦→ G∗ (x, y, λ) are in HT = D(T ) for any
x ∈ I.
Proof. We already noted that ρ(T ) ∋ λ ↦→ ˜ Rλ ∈ B(L 2 W , T1) is
analytic in the uniform operator topology. Furthermore, the restriction
operator IK : T1 → C(K) is bounded and independent of λ. Hence
ρ(T ) ∋ λ → IK ˜ Rλ is analytic in the uniform operator topology. In
particular, for fixed λ ∈ ρ(T ) and any x ∈ I, the components of the
linear map L 2 W ∋ u ↦→ (IK ˜ Rλu)(x) = Rλu(x) are bounded linear forms.
By Riesz’ representation theorem we have Rλu(x) = 〈u, G ∗ (x, ·, λ)〉W ,
where the columns of y ↦→ G ∗ (x, y, λ) are in L 2 W . Since Rλu = 0 for
u ∈ H∞ it follows that the columns of G ∗ (x, ·, λ) are actually in HT
for each x ∈ I.
Among other things, Theorem 14.2 tells us that if uj → u in L 2 W ,
then Rλuj → Rλu in C(K), so that Rλuj converges locally uniformly.
This is actually true even if uj just converges weakly, but we only need
is the following weaker result.
Lemma 14.3. Suppose Rλ is the resolvent of a selfadjoint relation
T as above. Then if uj ⇀ 0 weakly in L 2 W , it follows that Rλuj → 0
pointwise and locally boundedly.
1 u = uT + u∞ with uT ∈ HT and (0, u∞) ∈ T and ˜ T RλuT = ( ˜ T − λ)RλuT +
λRλuT = uT + λRλu. Thus (Rλu, λRλu + u) = (Rλu, λRλu + uT ) + (0, u∞) ∈ T ⊂
T1.