17.3. INVARIANT SUBSPACES 427
and let 0 < δ < min{δ i, i = 1, · · · ,n}. Then letting Ui ≡ Ki +B(0,δ ) , it follows that theUi are disjoint open sets. Let Γ j be a oriented cycle such that for z ∈ K j,n(Γ j,z) = 1 andfor z ∈ Ki, i ̸= j,n(Γ j,z) = 0, and if z ∈ Γ∗j , then n(Γi,z) = 0. Let Γ be the sum of theseoriented cycles. Thus n(Γ,z) = 0 if z /∈Ω≡ ∪n
i=1Ui and Ui ⊇ Γ∗i . Define
fi (λ )≡{
1 on Ui0 on U j for j ̸= i (17.8)
Thus fi is analytic on Ω. Then fi (A) ≡ Pi ≡ 12πi∫
Γfi (λ )(λ I−A)−1 dλ . By the spectral
mapping theorem, Theorem 17.2.6,
σ ( fi (A)) = fi (σ (A)) = {0,1} (17.9)
Note that for λ ∈ ρ (A) ,A(λ I−A)−1 = (λ I−A)−1 A as can be seen by multiplyingboth sides by (λ I−A) and observing that the result is A on both sides. Then since (λ I−A)is one to one, the identity follows. Now let Pk ∈ L (X ,X) be the linear transformationgiven by Pk =
12πi∫
Γk(λ I−A)−1 dλ .
From Lemma 17.1.1,
APk = A1
2πi
∫Γk
(λ I−A)−1 dλ =1
2πi
∫Γk
A(λ I−A)−1 dλ
=1
2πi
∫Γk
(λ I−A)−1 Adλ =1
2πi
(∫Γk
(λ I−A)−1 dλ
)A = PkA (17.10)
With these introductory observations, the following is the main result about invariantsubspaces. First is some notation.
Definition 17.3.1 Let X be a vector space and let Xk be a subspace. Then X =
∑nk=1 Xk means that every x ∈ X can be written in the form x = ∑
nk=1 xk,xk ∈ Xk. We write
X =n⊕
k=1
Xk if whenever 0 = ∑k xk, it follows that each xk = 0. In other words, we use the
new notation when there is a unique way to write each vector in X as a sum of vectors inthe Xk. When this uniqueness holds, the sum is called a direct sum. In case AXk ⊆ Xk, wesay that Xk is A invariant and Xk is an invariant subspace.
Theorem 17.3.2 Let σ (A) = ∪nk=1Kk where K j ∩Ki = /0, each K j being compact.
There exist Pk ∈L (X ,X) for each k = 1, · · · ,n such that
1. I = ∑nk=1 Pk
2. PiPj = 0 if i ̸= j
3. P2i = Pi for each i
4. X =n⊕
k=1
Xk where Xk = PkX and each Xk is a Banach space.
5. AXk ⊆ Xk which says that Xk is A invariant.
6. Pkx = x if x ∈ Xk. If x ∈ X j, then Pkx = 0 if k ̸= j.