1700 CHAPTER 54. FUNCTIONAL ANALYSIS APPLICATIONS

as n→ ∞ because the convergence of the series requires the nth term to converge to 0.Therefore,

(λ I−A)B = B(λ I−A) = I

which shows λ I−A is both one to one and onto and the Neumann series converges to(λ I−A)−1 . This proves the lemma.

This lemma also shows that σ (A) is bounded. In fact, σ (A) is closed.

Lemma 54.1.3 r (A) is open. In fact, if λ ∈ r (A) and |µ−λ | <∣∣∣∣∣∣(λ I−A)−1

∣∣∣∣∣∣−1, then

µ ∈ r (A).

Proof: First note

(µI−A) =(

I− (λ −µ)(λ I−A)−1)(λ I−A) (54.1.1)

= (λ I−A)(

I− (λ −µ)(λ I−A)−1)

(54.1.2)

Also from the assumption about |λ −µ| ,∣∣∣∣∣∣(λ −µ)(λ I−A)−1∣∣∣∣∣∣≤ |λ −µ|

∣∣∣∣∣∣(λ I−A)−1∣∣∣∣∣∣< 1

and so by the root test,∞

∑k=0

((λ −µ)(λ I−A)−1

)k

converges to an element of L (X ,X) . As in Lemma 54.1.2,

∞

∑k=0

((λ −µ)(λ I−A)−1

)k=(

I− (λ −µ)(λ I−A)−1)−1

.

Therefore, from 54.1.1,

(µI−A)−1 = (λ I−A)−1(

I− (λ −µ)(λ I−A)−1)−1

.

This proves the lemma.

Corollary 54.1.4 σ (A) is a compact set.

Proof: Lemma 54.1.2 shows σ (A) is bounded and Lemma 54.1.3 shows it is closed.

Definition 54.1.5 The spectral radius, denoted by ρ (A) is defined by

ρ (A)≡max{|λ | : λ ∈ σ (A)} .

Since σ (A) is compact, this maximum exists. Note from Lemma 54.1.2, ρ (A)≤ ||A||.

There is a simple formula for the spectral radius.