406 CHAPTER 14. ANALYSIS OF LINEAR TRANSFORMATIONS
18. In Problem 17 Ψ(t) is defined by the given series. Denote by exp(tσ (A)) the num-bers exp(tλ ) where λ ∈ σ (A) . Show exp(tσ (A)) = σ (Ψ(t)) . This is like Lemma14.3.9. Letting J be the Jordan canonical form for A, explain why
Ψ(t)≡∞
∑k=0
tkAk
k!= S
∞
∑k=0
tkJk
k!S−1
and you note that in Jk, the diagonal entries are of the form λk for λ an eigenvalue
of A. Also J = D+N where N is nilpotent and commutes with D. Argue then that
∞
∑k=0
tkJk
k!
is an upper triangular matrix which has on the diagonal the expressions eλ t whereλ ∈ σ (A) . Thus conclude
σ (Ψ(t))⊆ exp(tσ (A))
Next take etλ ∈ exp(tσ (A)) and argue it must be in σ (Ψ(t)) . You can do this asfollows:
Ψ(t)− etλ I =∞
∑k=0
tkAk
k!−
∞
∑k=0
tkλk
k!I =
∞
∑k=0
tk
k!
(Ak−λ
kI)
=
(∞
∑k=0
tk
k!
k−1
∑j=1
Ak− jλ
j
)(A−λ I)
Now you need to argue∞
∑k=0
tk
k!
k−1
∑j=1
Ak− jλ
j
converges to something in L (Rn,Rn). To do this, use the ratio test and Lemma14.3.2 after first using the triangle inequality. Since λ ∈ σ (A) , Ψ(t)−etλ I is not oneto one and so this establishes the other inclusion. You fill in the details. This theoremis a special case of theorems which go by the name “spectral mapping theorem”which was discussed in the text. However, go through it yourself.
19. Suppose Ψ(t) ∈ L (V,W ) where V,W are finite dimensional inner product spacesand t→Ψ(t) is continuous for t ∈ [a,b]: For every ε > 0 there there exists δ > 0 suchthat if |s− t|< δ then ||Ψ(t)−Ψ(s)||< ε. Show t→ (Ψ(t)v,w) is continuous. Hereit is the inner product in W. Also define what it means for t→Ψ(t)v to be continuousand show this is continuous. Do it all for differentiable in place of continuous. Nextshow t→ ||Ψ(t)|| is continuous.
20. If z(t) ∈W, a finite dimensional inner product space, what does it mean for t→ z(t)to be continuous or differentiable? If z is continuous, define∫ b
az(t)dt ∈W