Spectral Theory in Hilbert Space

70 11. STURM-LIOUVILLE EQUATIONS
function g(x, y, λ), which is in L 2 (I) as a function of y for every x ∈ I
and such that Rλu(x) = 〈u, g(x, ·, λ)〉 for any u ∈ L 2 (I). There is also
a kernel g1(x, y, λ) in L 2 (I) as a function of y for every x ∈ I such
that (Rλu) ′ (x) = 〈u, g1(x, ·, λ)〉 for any u ∈ L 2 (I).
Proof. We already noted that ρ(T ) ∋ λ ↦→ Rλ ∈ B(L 2 (I), D1) is
analytic in the uniform operator topology. Furthermore, the restriction
operator IK : D1 → C 1 (K) is bounded and independent of λ. Hence
ρ(T ) ∋ λ → IKRλ is analytic in the uniform operator topology. In
particular, for fixed λ ∈ ρ(T ) and any x ∈ I, the linear form L 2 (I) ∋
u ↦→ (IKRλu)(x) = Rλu(x) is (locally uniformly) bounded. By Riesz’
representation theorem we have Rλu(x) = 〈u, g(x, ·, λ)〉, where y ↦→
g(x, y, λ) is in L 2 (I). Similarly, since L 2 ∋ u ↦→ (Rλu) ′ (x) is a bounded
linear form for each x ∈ I the kernel g1 exists.
Among other things, Theorem 11.1 tells us that if uj → u in L 2 (I),
then Rλuj → Rλu in C 1 (K), so that Rλuj and its derivative converge
locally uniformly. This is actually true even if uj just converges weakly,
but all we need is the following weaker result.
Lemma 11.2. Suppose Rλ is the resolvent of a selfadjoint relation T
as above. Then if uj ⇀ 0 weakly in L 2 (I), it follows that both Rλuj → 0
and (Rλuj) ′ → 0 pointwise and locally boundedly.
Proof. Rλuj(x) = 〈uj, g(x, ·, λ)〉 → 0 since y ↦→ g(x, y, λ) is in
L 2 (I) for any x ∈ I. Now let K be a compact subinterval of I. A
weakly convergent sequence in L 2 (I) is bounded, so since Rλ maps
L 2 (I) boundedly into C 1 (K), it follows that Rλuj(x) is bounded independently
of j and x for x ∈ K. Similarly for the sequence of derivatives.
Corollary 11.3. If the interval I is compact, then any selfadjoint
restriction T of T1 has compact resolvent. Hence T has a complete
orthonormal sequence of eigenfunctions in L 2 (I).
Proof. Suppose uj ⇀ 0 weakly in L 2 (I). If I is compact, then
Lemma 11.2 implies that Rλuj → 0 pointwise and boundedly in I,
and hence by dominated convergence Rλuj → 0 in L 2 (I). Thus Rλ is
compact. The last statement follows from Theorem 8.3.
For a different proof, see Corollary 11.7.
If T has compact resolvent, then the generalized Fourier series of
any u ∈ L 2 (I) converges to u in L 2 (I). For functions in the domain of
T much stronger convergence is obtained.
Corollary 11.4. Suppose T has a complete orthonormal sequence
of eigenfunctions in L 2 (I). If u is in the domain of T , then the generalized
Fourier series of u, as well as the differentiated series, converges
locally uniformly in I. In particular, if I is compact, the convergence
is uniform in I.