Spectral Theory in Hilbert Space

CHAPTER 14
Eigenfunction expansions
Just as in Chapter 11, we will deduce our results for the system
(13.1) from a detailed description of the resolvent. As before we will
prove that the resolvent is actually an integral operator. To see this,
first note that according to Lemma 13.6 all elements of D1 are locally
absolutely continuous, in particular they are in C(I). The set C(I) becomes
a Fréchet space if provided with the topology of locally uniform
convergence; with a little loss of elegance we may restrict ourselves to
consider C(K) for an arbitrary compact subinterval K ⊂ I. This is a
Banach space with norm uK = supx∈K|u(x)|, |·| denoting the norm
of an n × 1 matrix (Exercise 14.1). The set T1 is a closed subspace of
H ⊕ H, since T1 is a closed relation. It follows from Assumption 13.7
that the map T1 ∋ (u, v) ↦→ u ∈ C(I) is well defined, i.e., there can not
be two different locally absolutely continuous functions u in the same
L 2 W
-equivalence class satisfying (13.1) for the same v. The restriction
map IK : T1 ∋ (u, v) ↦→ u ∈ C(K) is therefore a linear map between
Banach spaces.
Proposition 14.1. For every compact subinterval K ⊂ I there
exists a constant CK such that
for any (u, v) ∈ T1.
uK ≤ CK(u, v)W
Proof. We shall show that the restriction map IK is a closed operator
if K is sufficiently large. Since IK is everywhere defined in the
Hilbert space T1 it follows by the closed graph theorem (Appendix A)
that IK is a bounded operator, which is the statement of the proposition.
Now suppose (uj, vj) → (u, v) in T1 and uj → ũ in C(K). We
must show that IK(u, v) = ũ, i.e., u = ũ pointwise in K. We have
0 ≤
K (u − uj) ∗W (u − uj) ≤ u − uj2 and by Lemma 13.4
uj(x) = F (x, λ)(uj(c) + J −1
x
F ∗ (y, λ)W (y)vj(y) dy),
so letting j → ∞ it is clear that
K (u − ũ)∗W (u − ũ) = 0 and that ũ
satisfies Jũ ′ + Qũ = W v, so Assumption 13.7 shows that u − ũ = 0
pointwise in K if K is sufficiently large. Hence IK is closed, and we
are done.
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