Measure, Integration & Real Analysis, 2021a

288 Chapter 10 Linear Maps on Hilbert Spaces
Unlike linear maps from one vector space to a different vector space, operators on
the same vector space can be composed with each other and raised to powers.
10.21 Definition T k
Suppose T is an operator on a vector space V.
• For k ∈ Z + , the operator T k is defined by T k =
}
TT
{{
···T
}
.
k times
• T 0 is defined to be the identity operator I : V → V.
You should verify that powers of an operator satisfy the usual arithmetic rules:
T j T k = T j+k and (T j ) k = T jk for j, k ∈ Z + . Also, if V is a normed vector space
and T ∈B(V), then
‖T k ‖≤‖T‖ k
for every k ∈ Z + , as follows from using induction on 10.20.
Recall that if z ∈ C with |z| < 1, then the formula for the sum of a geometric
series shows that
∞
1
1 − z = ∑ z k .
k=0
The next result shows that this formula carries over to operators on Banach spaces.
10.22 operators in the open unit ball centered at the identity are invertible
If T is a bounded operator on a Banach space and ‖T‖ < 1, then I − T is
invertible and
(I − T) −1 ∞
= ∑ T k .
k=0
Proof
Suppose T is a bounded operator on a Banach space V and ‖T‖ < 1. Then
∞
∑
k=0
‖T k ‖≤
∞
∑
k=0
‖T‖ k =
1
1 −‖T‖ < ∞.
Hence 6.47 and 6.41 imply that the infinite sum ∑ ∞ k=0 Tk converges in B(V). Now
10.23 (I − T)
∞
∑
k=0
T k = lim (I − T)
n→∞
n
∑
k=0
T k = lim n→∞
(I − T n+1 )=I,
where the last equality holds because ‖T n+1 ‖≤‖T‖ n+1 and ‖T‖ < 1. Similarly,
10.24
( ∞
∑ T k) n
(I − T) = lim n→∞ ∑ T k (I − T) = lim (I − T n+1 )=I.
n→∞
k=0
k=0
Equations 10.23 and 10.24 imply that I − T is invertible and (I − T) −1 = ∑ ∞ k=0 Tk .
Measure, Integration & Real Analysis, by Sheldon Axler