Spectral Theory in Hilbert Space

8. COMPACTNESS 47
A natural question is now: How do I, in a concrete case, recognize
that an operator is compact? One class of compact operators which
are sometimes easy to recognize, are the Hilbert-Schmidt operators.
Definition 8.5. A : H → H is called a Hilbert-Schmidt operator if
for some complete orthonormal sequence e1, e2, . . . we have Aej 2 <
∞. The number ||A|| = Aej 2 is called the Hilbert-Schmidt norm
of A.
Lemma 8.6. ||A|| is independent of the particular complete orthonormal
sequence used in the definition, it is a norm, ||A|| = |||A ∗ ||, and
any Hilbert-Schmidt operator is compact. The set of Hilbert-Schmidt
operators on H is a Hilbert space in the Hilbert-Schmidt norm.
Proof. It is clear that ||| · || is a norm. Now suppose {ej} ∞ 1 and
{fj} ∞ 1 are arbitrary complete orthonormal sequences. Using Parseval’s
formula twice it follows that
jAej2 =
j,k |〈Aej, fk〉| 2 =
j,k |〈ej, A∗fk〉| 2 =
kA∗fk2 . Thus the Hilbert-Schmidt norm has
the claimed properties. To see that A is compact, suppose uj ⇀ 0 and
let ε > 0. Choose N so large that ∞
N A∗ ej 2 < ε and let C be a
bound for the sequence {uj} ∞ 1 . By Parseval’s formula we then have
Auk 2 = |〈Auk, ej〉| 2 = |〈uk, A ∗ ej〉| 2 . We obtain
Auk 2 ≤
N
1
|〈uk, A ∗ ej〉| 2 + C 2 ε → C 2 ε
as k → ∞ since |〈uk, A ∗ ej〉| ≤ CA ∗ ej . It follows that Auk → 0
so that A is compact. We leave the proof of the last statement as an
exercise for the reader (Exercise 8.4).
It is usual to consider a differential operator defined in some domain
Ω ⊂ R n as an operator in the space L 2 (Ω, w) where w > 0 is
measurable
and the scalar product in the space is given by 〈u, v〉 =
u(x)v(x)w(x) dx. In all reasonable cases the resolvent of such an
Ω
operator can be realized as an integral operator, i.e., an operator of
the form
(8.1) Au(x) = g(x, y)u(y)w(y) dy for x ∈ Ω.
Ω
The function g, defined in Ω ×Ω, is called the integral kernel of the operator
A. The integral kernel of the resolvent of a differential operator
is usually called Green’s function for the operator.
Theorem 8.7. Assume g(x, y) is measurable as a function of both
its variables and that y ↦→ g(x, y) is in L 2 (Ω, w) for a.a. x ∈ Ω. Then
the operator A of (8.1) is a Hilbert-Schmidt operator in L 2 (Ω, w) if and