Measure, Integration & Real Analysis, 2021a

334 Chapter 10 Linear Maps on Hilbert Spaces
10.118 Example singular values of V−V ∗
Let V denote the Volterra operator and let T = V−V ∗ . In Example 10.108,we
saw that if e k is defined by 10.112 then {e k } k∈Z is an orthonormal basis of L 2 ([0, 1])
and
2
Te k =
i(2k + 1)π e k
for each k ∈ Z, where the eigenvalue shown above corresponding to e k comes from
10.111. Now10.56 implies that
for each k ∈ Z. Hence
T ∗ e k =
10.119 T ∗ Te k =
−2
i(2k + 1)π e k
4
(2k + 1) 2 π 2 e k
for each k ∈ Z. After taking positive square roots of the eigenvalues, we see that the
equation above shows that the singular values of T are
2
π ≥ 2 π ≥ 2
3π ≥ 2
3π ≥ 2
5π ≥ 2
5π ≥···,
where the first two singular values above come from taking k = −1 and k = 0 in
10.119, the next two singular values above come from taking k = −2 and k = 1,
the next two singular values above come from taking k = −3 and k = 2, and so on.
Each singular value of T appears twice in the list of singular values above because
each eigenvalue of T ∗ T has geometric multiplicity 2.
For n ∈ Z + , the singular value s n (T) of a compact operator T tells us how well
we can approximate T by operators whose range has dimension less than n (see
Exercise 15).
The next result makes an important connection between K ∈ L 2 (μ × μ) and the
singular values of the integral operator associated with K.
10.120 sum of squares of singular values of integral operator
Suppose μ is a σ-finite measure and K ∈ L 2 (μ × μ). Then
‖K‖ 2 L 2 (μ×μ) = ∞
∑
n=1(
sn (I K ) ) 2 .
Proof
Consider a singular value decomposition
10.121 I K ( f )= ∑ s k 〈 f , e k 〉h k
k∈Ω
of the compact operator I K . Extend {e j } j∈Ω to an orthonormal basis {e j } j∈Γ of
L 2 (μ), and extend {h k } k∈Ω to an orthonormal basis {h k } k∈Γ ′ of L 2 (μ).
Measure, Integration & Real Analysis, by Sheldon Axler