Measure, Integration & Real Analysis, 2021a

338 Chapter 10 Linear Maps on Hilbert Spaces
18 Suppose that T is an operator on a finite-dimensional Hilbert space V with
dim V = n.
(a) Prove that T is invertible if and only if s n (T) ̸= 0.
(b) Suppose T is invertible and T has a singular value decomposition
for all f ∈ V. Show that
for all f ∈ V.
Tf = s 1 (T)〈 f , e 1 〉h 1 + ···+ s n (T)〈 f , e n 〉h n
T −1 f = 〈 f , h 1〉
s 1 (T) e 1 + ···+ 〈 f , h n〉
s n (T) e n
19 Suppose T is a compact operator on a Hilbert space V. Prove that
∑ ‖Te k ‖ 2 ∞ (
= ∑ sn (T) ) 2
k∈Γ
n=1
for every orthonormal basis {e k } k∈Γ of V.
20 Use the result of Example 10.124 to evaluate
∞
∑
n=1
1
n 2 .
21 Suppose T is a normal compact operator. Prove that the following are equivalent:
• range T is finite-dimensional.
• sp(T) is a finite set.
• s n (T) =0 for some n ∈ Z.
22 Find the singular values of the Volterra operator.
[Your answer, when combined with Exercise 12, should show that the norm of
the Volterra operator is
π 2 . This appearance of π can be surprising because the
definition of the Volterra operator does not involve π.]
Measure, Integration & Real Analysis, by Sheldon Axler