Measure, Integration & Real Analysis, 2021a

Section 10D Spectral Theorem for Compact Operators 337
11 For k ∈ Z, define g k ∈ L 2( (−π, π] ) and h k ∈ L 2( (−π, π] ) by
g k (t) = 1 √
2π
e it/2 e ikt and h k (t) = 1 √
2π
e ikt ;
here we are assuming that F = C.
(a) Use the conclusion of Example 10.108 to show that {g k } k∈Z is an orthonormal
basis of L 2( (−π, π] ) .
(b) Use the result in part (a) to show that {h k } k∈Z is an orthonormal basis of
L 2( (−π, π] ) .
(c) Use the result in part (b) to show that the orthonormal family in the third
bullet point of Example 8.51 is an orthonormal basis of L 2( (−π, π] ) .
12 Suppose T is a compact operator on a Hilbert space. Prove that s 1 (T) =‖T‖.
13 Suppose T is a compact operator on a Hilbert space and n ∈ Z + . Prove that
dim range T < n if and only if s n (T) =0.
14 Suppose T is a compact operator on a Hilbert space V with singular value
decomposition
∞
Tf = ∑ s k (T)〈 f , e k 〉h k
k=1
for all f ∈ V. Forn ∈ Z + , define T n : V → V by
n
T n f = ∑ s k (T)〈 f , e k 〉h k .
k=1
Prove that lim n→∞ ‖T − T n ‖ = 0.
[This exercise gives another proof, in addition to the proof suggested by Exercise
15 in Section 10C, that an operator on a Hilbert space is compact if and only if
it is the limit of bounded operators with finite-dimensional range.]
15 Suppose T is a compact operator on a Hilbert space V and n ∈ Z + . Prove that
inf{‖T − S‖ : S ∈B(V) and dim range S < n} = s n (T).
16 Suppose T is a compact operator on a Hilbert space V and n ∈ Z + . Prove that
s n (T) =inf{‖T| U ⊥‖ : U is a subspace of V with dim U < n}.
17 Suppose T is a compact operator on a Hilbert space with singular value decomposition
Tf = ∑ s k 〈 f , e k 〉h k
k∈Ω
for all f ∈ V. Prove that
for all f ∈ V.
T ∗ f = ∑ s k 〈 f , h k 〉e k
k∈Ω
Measure, Integration & Real Analysis, by Sheldon Axler