Kenneth Kuttler

17.2. FUNCTIONS OF LINEAR TRANSFORMATIONS 425

Now using the Fubini theorem, Theorem 14.4.9,

=∫

f (λ )(λ I−A)−1∫

Γ̂

g(µ)1

µ−λdµdλ +

∫Γ̂

g(µ)(µI−A)−1∫

f (λ )1

λ −µdλdµ

The first term is 0 from Cauchy’s integral formula, Theorem 14.11.1 applied to the indi-vidual simple closed curves whose union is Γ because the winding number n

(Γ̂,λ

)= 0.

Thus the inside integral vanishes. From the Cauchy integral formula, the second term is2πi

∫Γ̂

g(µ)(µI−A)−1 f (µ)dµ and so

f (A)g(A) =1

2πi

∫Γ̂

f (µ)g(µ)(µI−A)−1 dµ =1

2πi

∫Γ

f (λ )g(λ )(λ I−A)−1 dλ ■

Now consider the case that f (λ ) = λ . Is f (A), defined in terms of integrals as above,equal to A? If so, then from the theorem just shown, λ

n used in the integral formula doeslead to An. Thus one considers 1

2πi∫

Γλ (λ I−A)−1 dλ where Γ is a cycle such that n(Γ,z)=

1 if z ∈ σ (A), n(Γ,z) is an integer if z ∈Ω, and n(Γ,z) = 0 if z /∈Ω,Γ∗∩σ (A) = /0.Let Ω̂≡ σ (A)C and let γ̂R be a large circle of radius R > ∥A∥ oriented clockwise, which

includes σ (A)∪Γ∗on its inside.Consider the following picture in which σ (A) is the union of the two compact sets,

K1,K2 which are contained in the closed curves shown and Γ is the union of the orientedcycles Γi. A similar picture would apply if there were more than two Ki. All that is ofinterest here is that there is a cycle Γ oriented such that for all z ∈ σ (A) ,n(Γ,z) = 1,n(Γ,z) is an integer if z∈ Ω̂, and γ̂R is a large circle oriented clockwise as shown, R > ∥A∥.

γ̂R

Γ1

Γ2

Then in this case, f (λ ) = λ is analytic everywhere and Ω̂ ≡ σ (A)C . Let γR ≡ −γ̂R.Thus, by Corollary 17.2.3, 1

2πi∫

Γf (λ )(λ I−A)−1 dλ + 1

2πi∫

γ̂Rf (λ )(λ I−A)−1 dλ = 0 and

so, f (A) ≡ 12πi∫

Γf (λ )(λ I−A)−1 dλ = 1

2πi∫

γRf (λ )(λ I−A)−1 dλ . The integrand in the

integral on the right is λ ∑∞k=0

λk+1 for λ ∈ γ∗R and convergence is uniform on γ∗R. Then all

terms vanish except the one when k = 1 because all the other terms have primitives. Theuniform convergence implies that the integral of the sum is the sum of the integrals andthere is only one which survives. Therefore,

f (A)≡ 12πi

∫γR

Aλ

dλ = A

It follows from Theorem 17.2.5 that if f (λ ) = λn, then f (A) = An. This shows that it is

not unreasonable to make this definition. Similar reasoning yields

f (A) = I if f (λ ) = λ0 = 1. (17.7)

17.2. FUNCTIONS OF LINEAR TRANSFORMATIONS 425Now using the Fubini theorem, Theorem 14.4.9,= [ rayar—ay [9(u) pandas [ey —ay" [ Fa) araThe first term is 0 from Cauchy’s integral formula, Theorem 14.11.1 applied to the indi-vidual simple closed curves whose union is I’ because the winding number n (fA) =0.Thus the inside integral vanishes. From the Cauchy integral formula, the second term is2mi frg (ML) (uI—A) | f (HL) du and soFAA) = 55 [ Fw) (u) (ula) an = 5 [Pa @(Q)(Qr-ay "aaNow consider the case that f(A) =A. Is f(A), defined in terms of integrals as above,equal to A? If so, then from the theorem just shown, A” used in the integral formula doeslead to A”. Thus one considers 55; fA (AI —A)~' dA where is acycle such that n (I,z) =1 if z€ o(A), n(T,z) is an integer if z€ Q, andn(I,z) =O if z€ Q,T*No(A) =9.Let © = (A) and let Yp be a large circle of radius R > ||A|| oriented clockwise, whichincludes o (A) UI*on its inside.Consider the following picture in which o (A) is the union of the two compact sets,K,, Ko which are contained in the closed curves shown and I is the union of the orientedcycles T;. A similar picture would apply if there were more than two Kj. All that is ofinterest here is that there is a cycle I’ oriented such that for all z € o(A),n(I,z) = 1,n(I,z) is an integer if z € ©, and ¥p is a large circle oriented clockwise as shown, R > ||A||.Then in this case, f(A )= A is analytic everywhere and Q = o (A). Let yp = —p.Thus, by Corollary 17.2.3, 5} ai Inf (A )(AI—A)"! dats Lig f (A) (Al—A)'da =Oandso, f(A) = orn Ip f(A) (AT—A)~ ‘da= ani Jy, f (A) (AI—A)~' dd. The integrand in theintegral on the right is A VP 5 ia for A € Yp and convergence is uniform on Yp. Then allterms vanish except the one when k = | because all the other terms have primitives. Theuniform convergence implies that the integral of the sum is the sum of the integrals andthere is only one which survives. Therefore,1 AA)=—]|] ~dA=AFC ) 20i Yr AIt follows from Theorem 17.2.5 that if f(A) =A", then f(A) =A”. This shows that it isnot unreasonable to make this definition. Similar reasoning yieldsf(A) =1if f(A) =A° =1. (17.7)