Spectral Theory in Hilbert Space

48 8. COMPACTNESS
only if g ∈ L 2 (Ω, w) ⊗ L 2 (Ω, w), i.e., if and only if
Ω×Ω
|g(x, y)| 2 w(x)w(y) dx dy < ∞.
Proof. Let {ej} ∞ 1 be a complete orthonormal sequence in the
space L 2 (Ω, w). For fixed x ∈ Ω we may view Aej(x) as the j:th
Fourier coefficient of g(x, ·) so Parseval’s formula gives |Aej(x)| 2 =
Ω |g(x, y)|2 w(y) dy for a.a x ∈ Ω. By monotone convergence the product
of this function by w is in L 1 (Ω) if and only if the Hilbert-Schmidt
norm of A is finite. The theorem now follows by an application of
Tonelli’s theorem (i.e., a positive, measurable function is integrable
over Ω × Ω if and only if the iterated integral is finite).
Example 8.8. Consider the operator T in L 2 (−π, π) with domain
D(T ), consisting of those absolutely continuous functions u with deriva-
tive in L 2 (−π, π) for which u(π) = u(−π), and given by T u = −i du
dx (cf.
Example 4.8). This operator is self-adjoint and its resolvent is given
by Rλu(x) = π
g(x, y, λ)u(y) dy where Green’s function g(x, y, λ) is
−π
given by
⎧
⎪⎨ −
g(x, y, λ) =
⎪⎩
e−iλπ
2 sin λπ eiλ(x−y) y < x,
− eiλπ
2 sin λπ eiλ(x−y) y > x.
The reader should verify this! Since |g(x, y, λ)| 2 dx dy < ∞ for noninteger
λ the resolvent is a Hilbert-Schmidt operator, so it is compact.
Now consider the operator of Example 4.6. Green’s function is now
only defined for non-real λ and given by
Im λ i |Im λ| (8.2) g(x, y, λ) =
eiλ(x−y) if (x − y) Im λ > 0,
0 otherwise.
The reader should verify this as well! In this case there is no value of λ
for which g(·, ·, λ) ∈ L 2 (R 2 ) so the resolvent is not a Hilbert-Schmidt
operator.