Kazuaki Taira. Boundary Value Problems and Markov Processes

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2.1 Analytic Semigroups 15

converges in the uniform operator topology of the Banach space L(E,E)for

all t>0, and thus deﬁnes a bounded linear operator on E.HereL(E,E)

denotes the space of bounded linear operators on E.

Furthermore, we have the following:

Proposition 2.1. The operators U(t), deﬁned by formula (2.2), form a semi-

group on E, that is, they enjoy the semigroup property

U(t + s)=U (t) · U (s) for all t, s>0.

Proof. By Cauchy’s theorem, we may assume that

U(s)=−

2πi





μs

R(μ) dμ, s > 0.

Here Γ



is a path obtained from the path Γ by translating each point of Γ to

the right by a ﬁxed small positive distance (see Figure 2.3).



Fig. 2.3.

Then we have, by Fubini’s theorem,

U(t) · U(s)=

(2πi)





λt

μs

R(λ) R(μ) dλ dμ

(2πi)





λt

μs

R(λ) − R(μ)

λ − μ

dλ dμ

2πi



λt

R(λ)



2πi





μs

λ − μ

dμ



dλ

−

2πi





μs

R(μ)



2πi



λt

λ − μ

dλ



dμ.

We calculate the two terms in the last part.

(a) We let

f(μ)=

μs

λ − μ

,μ∈ C.

16 2 Semigroup Theory

|¹|= r

Fig. 2.4.

Then, by applying the residue theorem we obtain that (see Figure 2.4)





(1)

∩{|μ|≤r}

f(μ) dμ +





(2)

f(μ) dμ +





(3)

∩{|μ|≤r}

f(μ) dμ



π/2+ω−ε

−(π/2+ω−ε)

f(re

iθ

)rie

iθ

dθ

=2πi Res [f(μ)]

μ=λ

= −2πie

λs

However, we have, as r →∞,





(1)

∩{|μ|≤r}

f(μ) dμ −→





(1)

f(μ) dμ,





(3)

∩{|μ|≤r}

f(μ) dμ −→





(3)

f(μ) dμ,

and





π/2+ω−ε

−(π/2+ω−ε)

f(re

iθ

)rie

iθ

dθ



≤ e

−rs·sin(ω−ε)



π/2+ω−ε

−(π/2+ω−ε)

dθ



− e

iθ



−→ 0.

Therefore, we ﬁnd that

2πi





μs

λ − μ

dμ = −e

λs

(b) Similarly, since the path Γ lies to the left of the path Γ



, we ﬁnd that

2πi



λt

λ − μ

dλ =0.

2.1 Analytic Semigroups 17

Summing up, we obtain that

U(t) · U(s)=−

2πi



λ(t+s)

R(λ) dλ = U(t + s) for all t, s>0.

The proof of Proposition 2.1 is complete. 

The next theorem states that the semigroup U(t) can be extended to an

analytic semigroup in some sector containing the positive real axis.

Theorem 2.2. The semigroup U (t), deﬁned by formula (2.2), can be extended

toasemigroupU(z) which is analytic in the sector

= {z = t + is : z =0, |arg z| <ω},

and enjoys the following properties:

(a) The operators AU (z) and

(z) are bounded operators on E for each z ∈

, and satisfy the relation

(z)=AU(z) for all z ∈ Δ

. (2.3)

(b) For each 0 <ε<ω/2, there exist positive constants



(ε) and



(ε)

such that

U(z)≤



(ε) for all z ∈ Δ

2ε

, (2.4)

AU(z)≤



(ε)

|z|

for all z ∈ Δ

2ε

, (2.5)

where(seeFigure2.5)

2ε

= {z ∈ C : z =0, |arg z|≤ω − 2ε}.

2ε

U(z)x −→ x in E.

Proof. (i) The analyticity of U (z): If λ ∈ Γ

(3)

and z ∈ Δ

2ε

, that is, if we have

the formulas

λ = |λ|e

iθ

,θ= π/2+ω − ε,

z = |z|e

iϕ

, |ϕ|≤ω − 2ε,

then it follows that

λz = |λ||z|e

i(θ+ϕ)

with

π/2+ε ≤ θ + ϕ ≤ π/2+2ω − 3ε<3π/2 − 3ε.

18 2 Semigroup Theory

2ε

Fig. 2.5.

Note that

cos(θ + ϕ) ≤ cos(π/2+ε)=−sin ε.

Hence we have the inequality

λz

|≤e

−|λ||z| sin ε

for all λ ∈ Γ

(3)

and z ∈ Δ

2ε

. (2.6)

Similarly, we have the inequality

λz

|≤e

−|λ||z| sin ε

for all λ ∈ Γ

(1)

and z ∈ Δ

2ε

. (2.7)

For each small ε>0, we let

=Δ

2ε



{z ∈ C : |z|≥ε} = {z ∈ C : |z|≥ε, |arg z|≤ω − 2ε}.

Then, by combining estimates (2.1), (2.6) and (2.7) we obtain that



λz

R(λ)



≤

|λ|

−ε sin ε·|λ|

for all λ ∈ Γ

(1)



(3)

and z ∈ K

. (2.8)

On the other hand, we have the estimate



λz

R(λ)



≤ M

|z|

for all λ ∈ Γ

(2)

and z ∈ K

. (2.9)

Therefore, we ﬁnd that the integral

U(z)=−

2πi



λz

R(λ) dλ = −

2πi



k=1



(k)

λz

R(λ) dλ (2.10)

converges in the Banach space L(E, E), uniformly in z ∈ K

, for every ε>0.

This proves that the operator U(z) is analytic in the domain Δ



ε>0

By the analyticity of U(z), it follows that the operators U(z)alsoenjoy

the semigroup property

2.1 Analytic Semigroups 19

U(z + w)=U(z) · U(w) for all z, w ∈ Δ

(ii) We prove that the operators U(z) enjoy properties (a), (b) and (c).

(b) First, by using Cauchy’s theorem we obtain that

U(z)=−

2πi



λz

R(λ) dλ = −

2πi



|z|

λz

R(λ) dλ,

where Γ

|z|

is a path consisting of the following three curves (see Figure 2.6):

(1)

|z|



−i(π/2+ω−ε)

|z|

≤ r<∞



(2)

|z|



|z|

iθ

: −(π/2+ω − ε) ≤ θ ≤ π/2+ω − ε



(3)

|z|



i(π/2+ω−ε)

|z|

≤ r<∞



(1)

(2)

(3)

|z|

Fig. 2.6.

However, by estimates (2.1), (2.6) and (2.7), it follows that



λz

R(λ)



≤

|λ|

−|λ ||z| sin ε

for all λ ∈ Γ

(1)

|z|



(3)

|z|

and z ∈ Δ

2ε

Hence we have, for k =1,3,



(k)

|z|



λz

R(λ)



|dλ|≤M



∞

|z|

−ρ|z| sin ε

−1

dρ

= M



∞

− sin ε·s

−1

ds.

20 2 Semigroup Theory

We have also, for k =2,



(2)

|z|



λz

R(λ)



|dλ|≤M



π/2+ω−ε

−(π/2+ω−ε)

edθ

=2eM

(π/2+ω − ε)

≤ 2πeM

Summing up, we obtain the following estimate:

U(z)≤

2π



k=1



(k)

|z|



λz

R(λ)



|dλ|

≤

2π





∞

−1

− sin ε·s

ds +2πeM







∞

−1

− sin ε·s

ds + πe



This proves the desired estimate (2.4), with



(ε)=





∞

−1

− sin ε·s

ds + πe



To prove estimate (2.5), note that

AR(λ)=(A − λI + λI)R(λ)=I + λR(λ),

so that

AR(λ)≤1+M

for all λ ∈ Σ

Hence, by arguing just as in the proof of estimate (2.4) we obtain that





λz

AR(λ) dλ



≤ 2



∞

|z|

−ρ|z| sin ε

(1 + M

) dρ



π/2+ω−ε

−(π/2+ω−ε)

(1 + M

) e

|z|

dθ

≤ 2(1+M

)





∞

− sin ε·s

ds + πe



|z|

for all z ∈ Δ

2ε

. (2.11)

This proves that the integral



λz

AR(λ) dλ

is convergent in the Banach space L(E,E), for every z ∈ Δ

2ε

. By the closed-

ness of A, it follows that

U(z) ∈D(A) for all z ∈ Δ

2ε

2.1 Analytic Semigroups 21

and

AU(z)=−

2πi



λz

AR(λ) dλ for all z ∈ Δ

2ε

. (2.12)

Therefore, the desired estimate (2.5) follows from estimate (2.11), with



(ε)=

1+M





∞

− sin ε·s

ds + πe



We remark that formula (2.12) remains valid for all z ∈ Δ

,sinceΔ



ε>0

2ε

(a) By estimates (2.8) and (2.9), we can diﬀerentiate formula (2.10) under

theintegralsigntoobtainthat

(z)=−

2πi



λz

λR(λ) dλ for all z ∈ Δ

. (2.13)

On the other hand, it follows from formula (2.12) that

AU(z)=−

2πi



λz

AR(λ) dλ

= −

2πi



λz

(I + λR(λ)) dλ

= −

2πi



λz

λR(λ) dλ for all z ∈ Δ

, (2.14)

sincewehave,byCauchy’stheorem,



λz

dλ =0.

Therefore, the desired formula (2.3) follows immediately from formulas

(2.13) and (2.14).

be an arbitrary element of D(A). By the residue theorem,

it follows that

2πi



λz

dλ.

Hence we have the formula

U(z)x

− x

= −

2πi



λz



R(λ)+



dλ

= −

2πi



λz

R(λ)Ax

dλ.

Here we remark that



R(λ)



≤

|λ|

for all λ ∈ Γ,



λz



≤ 2e

−|λ||z| sin ε

+ e

|z|

for all z ∈ Δ

2ε

and λ ∈ Γ.

22 2 Semigroup Theory

Thus it follows from an application of the Lebesgue dominated convergence

theorem that, as z → 0, z ∈ Δ

2ε

U(z)x

− x

−→ −

2πi



R(λ)Ax

dλ.

However, we have the assertion



R(λ)Ax

dλ =0.

Indeed, by Cauchy’s theorem it follows that



R(λ)Ax

dλ = lim

r→∞



Γ ∩{|λ|≤r}

R(λ)Ax

dλ

= − lim

r→∞



R(λ)Ax

dλ

=0,

where C

is a closed path shown in Figure 2.7.

Fig. 2.7.

Summing up, we have proved that

U(z)x

−→ x

as z → 0, z ∈ Δ

2ε

for each x

∈D(A).

Since the domain D(A)isdenseinE and U (z)≤



(ε) for all z ∈ Δ

2ε

it follows that, for each x ∈ E,

U(z)x −→ x as z → 0, z ∈ Δ

2ε

The proof of Theorem 2.2 is now complete. 

2.1 Analytic Semigroups 23

Remark 2.1. Assume that the operator A satisﬁes a stronger condition than

condition (2.1):

R(λ)≤

|λ| +1

for all λ ∈ Σ

. (2.15)

Then we have the estimates

U(z)≤



(ε)e

−δ·Re z

for all z ∈ Δ

2ε

, (2.16)

AU(z)≤



(ε)

|z|

−δ·Re z

for all z ∈ Δ

2ε

, (2.17)

with some positive constant δ.

Proof. Take a real number δ such that

0 <δ<

Then we have, by estimate (2.15),



(A −λI)

−1



≤

δM

|λ| +1

≤ δM

< 1 for all λ ∈ Σ

Hence it follows that the operator (A + δI) − λI has the inverse

((A + δI) − λI)

−1

=(I + δ(A − λI)

−1

)

−1

(A −λI)

−1

and

((A + δI) − λI)

−1

≤(I + δ(A − λI)

−1

)

−1

·(A − λI)

−1



≤

|λ| +1

1 −δ(A − λI)

−1



≤

|λ| +1

1 − δM

≤



1 − δM



|λ|

This proves that the operator A+δI satisﬁes condition (2.1), so that estimates

(2.4) and (2.5) remain valid for the operator A + δI:

V (z)≤



(ε) for all z ∈ Δ

2ε

, (2.18)

(A + δI)V (z)≤



(ε)

|z|

for all z ∈ Δ

2ε

, (2.19)

where

V (z)=−

2πi



λz

(A + δI −λI)

−1

dλ.

24 2 Semigroup Theory

However, we have, by Cauchy’s theorem,

V (z)=−

2πi



λz

(A + δI − λI)

−1

dλ

= −

2πi



Γ +δ

λz

(A + δI −λI)

−1

dλ

= −

2πi



μz

δz

(A −μI)

−1

dμ = e

δz

U(z)

for all z ∈ Δ

2ε

. (2.20)

In view of formula (2.16), the desired estimates (2.16) and (2.17) follow from

estimates (2.18) and (2.19). 

2.1.2 Fractional Powers

Assume that the operator A satisﬁes a stronger condition than condition (2.1):

(1) The resolvent set of A contains the region Σ as in Figure 2.8.

Fig. 2.8.

(2) There exists a positive constant M such that the resolvent R(λ)=(A −

λI)

−1

satisﬁes the estimate

R(λ)≤

(1 + |λ|)

for all λ ∈ Σ. (2.21)

If α>0, we can deﬁne the fractional power (−A)

−α

of −A by the following

formula:

(−A)

−α

= −

2πi



(−λ)

−α

R(λ) dλ. (2.22)

Here the path Γ runs in the set Σ from ∞e

−iω

to ∞e

iω

, avoiding the positive

real axis and the origin (see Figure 2.9), and for the function (−λ)

−α