partialdiﬀerentialequationsand …grsam/hs16/pdesblo/sg_031.pdf · 2016-12-23 · the fact that...

Partial Differential Equations andSemigroups of Bounded Linear Operators

(Currently under construction!)

Arnulf Jentzen

December 23, 2016

2

PrefaceThese lecture notes are based on the lecture notes “Stochastic Partial DifferentialEquations: Analysis and Numerical Approximations”. These lecture notes are faraway from being complete and remain under construction. In particular, these lecturenotes do not yet contain a suitable comparison of the presented material with existingresults, arguments, and notions in the literature. Furthermore, these lecture notesdo not contain a number of proofs, arguments, and intuitions. For most of thisadditional material, the reader is referred to the lectures of the course “401-4475-66LPartial Differential Equations and Semigroups of Bounded Linear Operators” in thespring semester 2016. Special thanks are due to Sonja Cox, Ryan Kurniawan, PrimozPusnik, Diyora Salimova, and Timo Welti for their very helpful advice and their veryuseful assistance. The students of the courses “401-4606-00L Numerical Analysis ofStochastic Partial Differential Equations” are gratefully acknowledged for pointingout a number of misprints to me.

Zürich, September 2016

Arnulf Jentzen

3

ExercisesSolutions to the exercises can be turned in the designated mailbox in the anteroomHG G 53.x.

Exercise Exercises Deadlinesheet1 Exercises 1.3.5, 1.9.3, 1.9.4, and 1.9.5. 04.11.2016, 15:15 AM2 Exercises 2.1.18, 2.1.19, 2.1.24, and 2.2.6. 25.11.2016, 15:15 AM3 Exercises 2.3.7, 4.1.10, and 4.1.18. 23.12.2016, 15:15 AM

Contents

1 Semigroups of bounded linear operators 91.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.2 Definition of a semigroup of bounded linear operators . . . . . . . . . 91.3 Types of semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.4 The generator of a semigroup . . . . . . . . . . . . . . . . . . . . . . 111.5 Global a priori bounds for semigroups . . . . . . . . . . . . . . . . . . 121.6 Strongly continuous semigroups . . . . . . . . . . . . . . . . . . . . . 12

1.6.1 A priori bounds for strongly continuous semigroups . . . . . . 121.6.1.1 The Baire category theorem on complete metric spaces 131.6.1.2 The uniform boundedness principle . . . . . . . . . . 171.6.1.3 Local a priori bounds . . . . . . . . . . . . . . . . . 181.6.1.4 Global a priori bounds . . . . . . . . . . . . . . . . . 19

1.6.2 Existence of solutions of linear ordinary differential equationsin Banach spaces . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.6.3 Pointwise convergence in the space of bounded linear operators 211.6.4 Domains of generators of strongly continuous semigroups . . . 221.6.5 Generators of strongly continuous semigroups . . . . . . . . . 231.6.6 A generalization of matrix exponentials to infinite dimensions 271.6.7 A characterization of strongly continuous semigroups . . . . . 27

1.7 Uniformly continuous semigroups . . . . . . . . . . . . . . . . . . . . 281.7.1 Matrix exponential in Banach spaces . . . . . . . . . . . . . . 291.7.2 Continuous invertibility of bounded linear operators in Banach

spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321.7.3 Generators of uniformly continuous semigroup . . . . . . . . . 341.7.4 A characterization result for uniformly continuous semigroups 351.7.5 An a priori bound for uniformly continuous semigroups . . . . 36

1.8 The Hille-Yosida theorem . . . . . . . . . . . . . . . . . . . . . . . . 371.8.1 Yosida approximations . . . . . . . . . . . . . . . . . . . . . . 37

5

6 CONTENTS

1.8.2 Scalar shifts of generators of strongly continuous semigroups . 381.8.3 Convergence of linear-implicit Euler approximations . . . . . . 401.8.4 Properties of Yosida approximations . . . . . . . . . . . . . . 40

1.8.4.1 Convergence of Yosida approximations . . . . . . . . 401.8.4.2 Semigroups generated by Yosida approximations . . 41

1.8.5 A characterization for generators of strongly continuous con-traction semigroups . . . . . . . . . . . . . . . . . . . . . . . . 42

1.9 Diagonal linear operators on Hilbert spaces . . . . . . . . . . . . . . . 491.10 Semigroups generated by diagonal linear operators . . . . . . . . . . . 50

1.10.1 A characterization for strongly continuous semigroups gener-ated by diagonal linear operators . . . . . . . . . . . . . . . . 55

1.10.2 Contraction semigroups generated by diagonal linear operators 561.10.3 Smoothing effect of the semigroup . . . . . . . . . . . . . . . . 571.10.4 Semigroup generated by the Laplace operator . . . . . . . . . 59

2 Nonlinear functions and nonlinear spaces 612.1 Continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

2.1.1 Topological spaces . . . . . . . . . . . . . . . . . . . . . . . . 612.1.1.1 Topological spaces induced by distance-type functions 612.1.1.2 Semi-metric spaces . . . . . . . . . . . . . . . . . . . 63

2.1.2 Continuity properties of functions . . . . . . . . . . . . . . . . 652.1.2.1 Uniform continuity . . . . . . . . . . . . . . . . . . . 652.1.2.2 Hölder continuity . . . . . . . . . . . . . . . . . . . . 65

2.1.3 Modulus of continuity . . . . . . . . . . . . . . . . . . . . . . 672.1.3.1 Properties of the modulus of continuity . . . . . . . . 672.1.3.2 Convergence of the modulus of continuity . . . . . . 69

2.1.4 Extensions of uniformly continuous functions . . . . . . . . . . 692.2 Measurable functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

2.2.1 Nonlinear characterization of the Borel sigma-algebra . . . . . 732.2.2 Pointwise limits of measurable functions . . . . . . . . . . . . 74

2.3 Strongly measurable functions . . . . . . . . . . . . . . . . . . . . . . 752.3.1 Simple functions . . . . . . . . . . . . . . . . . . . . . . . . . 752.3.2 Separability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 762.3.3 Strongly measurable functions . . . . . . . . . . . . . . . . . . 782.3.4 Pointwise approximations of strongly measurable functions . . 782.3.5 Sums of strongly measurable functions . . . . . . . . . . . . . 81

CONTENTS 7

3 The Bochner integral 833.1 Sets of integrable functions . . . . . . . . . . . . . . . . . . . . . . . . 83

3.1.1 Lp-sets of measurable functions for p P r0,8q . . . . . . . . . . 833.1.2 Lp-spaces of strongly measurable functions for p P r0,8q . . . 84

3.2 Existence and uniqueness of the Bochner integral . . . . . . . . . . . 863.3 Definition of the Bochner integral . . . . . . . . . . . . . . . . . . . . 87

4 Nonlinear partial differential equations 894.1 Diagonal linear operators on Hilbert spaces . . . . . . . . . . . . . . . 89

4.1.1 Closedness of diagonal linear operators . . . . . . . . . . . . . 894.1.2 Laplace operators on bounded domains . . . . . . . . . . . . . 90

4.1.2.1 Laplace operators with Dirichlet boundary conditions 904.1.2.2 Laplace operators with Neumann boundary conditions 914.1.2.3 Laplace operators with periodic boundary conditions 92

4.1.3 Spectral decomposition for a diagonal linear operator . . . . . 934.1.4 Fractional powers of a diagonal linear operator . . . . . . . . . 974.1.5 Domain Hilbert space associated to a diagonal linear operator 984.1.6 Completion of metric spaces . . . . . . . . . . . . . . . . . . . 100

4.1.6.1 An extension of a metric space . . . . . . . . . . . . 1004.1.6.2 An artificial completion . . . . . . . . . . . . . . . . 1014.1.6.3 The natural completion . . . . . . . . . . . . . . . . 1034.1.6.4 Further completions . . . . . . . . . . . . . . . . . . 104

4.1.7 Interpolation spaces associated to a diagonal linear operator . 1044.2 The Sobolev space W 1,2

0 pp0, 1q,Rq “ H10 pp0, 1q,Rq . . . . . . . . . . . 105

4.2.1 Weak derivatives in W 1,20 pp0, 1q,Rq “ H1

0 pp0, 1q,Rq . . . . . . . 1054.2.2 A special case of the Sobolev embedding theorem . . . . . . . 108

4.3 Nonlinear evolution equations . . . . . . . . . . . . . . . . . . . . . . 1104.3.1 Complete function spaces . . . . . . . . . . . . . . . . . . . . 1104.3.2 Measurability properties . . . . . . . . . . . . . . . . . . . . . 1124.3.3 Local existence of mild solutions . . . . . . . . . . . . . . . . . 115

8 CONTENTS

Chapter 1

Semigroups of bounded linearoperators

In this chapter we mostly follow the presentations in Pazy [5].

1.1 PreliminariesDefinition 1.1.1 (Power set). Let A be a set. Then we denote by PpAq the powerset of A (the set of all subsets of A).

Definition 1.1.2 (Set of functions). Let A and B be sets. Then we denote byMpA,Bq the set of all functions from A to B.

1.2 Definition of a semigroup of bounded linear op-erators

Definition 1.2.1 (Semigroups of bounded linear operators). Let K P tR,Cu andlet pV, ¨V q be a normed K-vector space. Then we say that S is a semigroup on V(we say that S is a semigroup of bounded linear operators on V , we say that S isa semigroup) if and only if S P Mpr0,8q, LpV qq is a function from r0,8q to LpV qwhich satisfies for all t1, t2 P r0,8q that

S0 “ IdV and St1St2 “ St1`t2l jh n

semigroup property

. (1.1)

9

10 CHAPTER 1. SEMIGROUPS OF BOUNDED LINEAR OPERATORS

1.3 Types of semigroupsDefinition 1.3.1 (Contraction semigroups). Let K P tR,Cu, let pV, ¨V q be anormed K-vector space, and let S : r0,8q Ñ LpV q be a semigroup. Then we saythat S is contractive if and only if it holds that

suptPr0,8q

StLpV q ď 1. (1.2)

Definition 1.3.2 (Strongly continuous semigroups). Let K P tR,Cu, let pV, ¨V qbe a normed K-vector space, and let S : r0,8q Ñ LpV q be a semigroup. Then we saythat S is strongly continuous if and only if it holds for every v P V that the function

r0,8q Q t ÞÑ Stv P V (1.3)

is continuous.

Definition 1.3.3 (Uniformly continuous semigroups). Let K P tR,Cu, let pV, ¨V qbe a normed K-vector space, and let S : r0,8q Ñ LpV q be a semigroup. Then we saythat S is uniformly continuous if and only if the function

r0,8q Q t ÞÑ St P LpV q (1.4)

is continuous.

Example 1.3.4 (Matrix exponential). Let d P N and let A P Rdˆd be an arbitrarydˆ d-matrix. Then the function

r0,8q Q t ÞÑ eAt P Rdˆd (1.5)

is a uniformly continuous semigroup.

Clearly, it holds that every uniformly continuous semigroup is also strongly con-tinuous. However, not every strongly continuous semigroup is uniformly continuoustoo. This is the subject of the next exercise.

Exercise 1.3.5. Give an example of an R-Banach space pV, ¨V q and a stronglycontinuous semigroup S : r0,8q Ñ LpV q so that S is not a uniformly continuoussemigroup. Prove that your function S does indeed fulfill the desired properties.

1.4. THE GENERATOR OF A SEMIGROUP 11

1.4 The generator of a semigroupDefinition 1.4.1 (Generator). Let K P tR,Cu, let pV, ¨V q be a normed K-vectorspace, and let S : r0,8q Ñ LpV q be a semigroup. Then we denote by GS : DpGSq ĎV Ñ V the function with the property that

DpGSq “"

v P V :

ˆ„

Stv ´ v

t

converges as p0,8q Q tŒ 0

*

(1.6)

and with the property that for all v P DpGSq it holds that

GSv “ limtŒ0

„

Stv ´ v

t

(1.7)

and we call GS the generator of S (we call GS the infinitesmal generator of S).

In the next notion we label all linear operators that are generators of stronglycontinuous semigroups.

Definition 1.4.2 (Generator of a strongly continuous semigroup). Let K P tR,Cuand let pV, ¨V q be a normed K-vector space. Then we say that A is the generator ofa strongly continuous semigroup on V (we say that A is the generator of a stronglycontinuous semigroup) if and only if there exists a strongly continuous semigroupS : r0,8q Ñ LpV q of bounded linear operators on V such that

GS “ A. (1.8)

We complete this section with a simple exercise which aims to illustrate and relatethe different concepts introduced above.

Exercise 1.4.3. Let K P tR,Cu, let pV, ¨V q be a normed K-vector space with#V ą 1, and let S : r0,8q Ñ LpV q be the function which satisfies for all t P r0,8qthat

St “

#

IdV : t “ 0

0 : t ą 0. (1.9)

(i) Is S a semigroup? Prove that your answer is correct.

(ii) Is S a strongly continuous semigroup? Prove that your answer is correct.

(iii) Is S a uniformly continuous semigroup? Prove that your answer is correct.

(iv) Is S a contractive semigroup? Prove that your answer is correct.

(v) Specify DpGSq and GS.


1.5 Global a priori bounds for semigroupsIn the next result, Proposition 1.5.1, we present a global a priori bound for semigroupsof bounded linear operators.

Proposition 1.5.1 (Global a priori bound). Let K P tR,Cu, let pV, ¨V q be anormed K-vector space, and let S : r0,8q Ñ LpV q be a semigroup. Then it holds forall t P r0,8q, ε P p0,8q that

supsPr0,ts SsLpV q ď“

supsPr0,εs SsLpV q‰

¨ et“

lnpSε1εLpV qq

‰`

. (1.10)

Proof of Proposition 1.5.1. Note that for all t P r0,8q, ε P p0,8q, n P N0Xptε´1, tεsit holds that

StLpV q “›

›Snε`ptńεq›

›

LpV q“›

›SnεSptńεq›

›

LpV qď SnεLpV q

›

›Sptńεq›

›

LpV q

“ rSεsnLpV q

›

›Sptńεq›

›

LpV qď Sε

nLpV q

›

›Sptńεq›

›

LpV q

ď max

1, SεnLpV q

( “


““

supsPr0,εs SsLpV q‰ “

max

1, SεLpV q(‰n

ď“

supsPr0,εs SsLpV q‰ “

max

1, SεLpV q(‰tε

““


max!

e0, exp´

t ln´

Sε1εLpV q

¯¯)

ď“


exp´

tmax

0, ln`

Sε1εLpV q

˘(

¯

.

(1.11)

This completes the proof of Proposition 1.5.1.

1.6 Strongly continuous semigroups

1.6.1 A priori bounds for strongly continuous semigroups

In Corollary 1.6.8 below we present a global priori bound for strongly continuoussemigroups. The proof of Corollary 1.6.8 uses the local a priori bound in Lemma 1.6.7below. The proof of Lemma 1.6.7, in turn, exploits the uniform boundedness princi-ple. This is the subject of the next result.

1.6. STRONGLY CONTINUOUS SEMIGROUPS 13

1.6.1.1 The Baire category theorem on complete metric spaces

Lemma 1.6.1 (A set contains an open ball). Let pE, dEq be a metric space and letA P PpEq. Then EzA ‰ E if and only if there exist ε P p0,8q, x P E such that

ty P E : dEpx, yq ă εu Ď A. (1.12)

Proof of Lemma 1.6.1. Observe that

EzA “ E

ô EzA is dense in Eô @x P E : @ ε P p0,8q : D y P pEzAq : dEpx, yq ă ε

ô @x P E : @ ε P p0,8q : pEzAq X ty P E : dEpx, yq ă εu ‰ H.

(1.13)

This implies that

EzA ‰ E

ô Dx P E : D ε P p0,8q : pEzAq X ty P E : dEpx, yq ă εu “ H

ô Dx P E : D ε P p0,8q : ty P E : dEpx, yq ă εu Ď A.

(1.14)

The proof of Lemma 1.6.1 is thus completed.

Corollary 1.6.2 (A characterization for density). Let pE, dEq be a metric space andlet A P PpEq. Then the following three statements are equivalent:

(i) It holds that A “ E.

(ii) It holds for all ε P p0,8q, x P E that AX ty P E : dEpx, yq ă εu ‰ H.

(iii) It holds for all non-empty open sets O P PpEq that AXO ‰ H.

Proof of Corollary 1.6.2. Note that Item (ii) and Item (iii) are equivalent. Moreover,observe that Lemma 1.6.1 shows that Item (i) and Item (ii) are equivalent. The proofof Corollary 1.6.2 is thus completed.

Lemma 1.6.3. Let pE, dEq be a metric space, let U P PpEq, let V P PpEq be anopen set, and assume that U “ V “ E. Then U X V “ E.


Proof of Lemma 1.6.3. First of all, observe that the assumption that V is an openset proves that for every open set W Ď E it holds that

V XW (1.15)

is an open set. Next observe that Corollary 1.6.2 proves that for every non-emptyopen set W Ď E and every set E P PpEq with E “ E it holds that

pE XW q ‰ H. (1.16)

This and the assumption that V “ E assure that for every non-empty open setW Ď E it holds that

pV XW q ‰ H. (1.17)

Combining this with (1.15) ensures that for every non-empty open set W Ď E itholds that

V XW (1.18)

is a non-empty open set. The assumption that U “ E together with (1.16) henceimplies that for all non-empty open sets W Ď E it holds that

pU X V q XW “ U X pV XW q ‰ H. (1.19)

Combining this with Corollary 1.6.2 completes the proof of Lemma 1.6.3.

Theorem 1.6.4 (Baire category theorem for complete metric spaces). Let pE, dEq bea complete metric space and let An Ď E, n P N, be closed subsets of E which satisfyEz rYnPNAns ‰ E. Then there exists a natural number n P N such that EzAn ‰ E.

Proof of Theorem 1.6.4. Throughout this proof let Nn P PpNq, n P N0 Y t8u, bethe sets which satisfy for all n P N0 Y t8u that

Nn “

k P pNX r0, nsq : EzAk “ E(

(1.20)

and let U Ď E be a non-empty open set (why does such a set exist?). We intendto prove that N8 ‰ N. We may thus assume w.l.o.g. that N8 is not a finite set.The fact that for every n P N8 it holds that EzAn is an open set, the fact that@n P N8 : EzAn “ E, and Lemma 1.6.3 assure that for all n P N it holds that

XkPNnrEzAks “ E. (1.21)

Hence, we obtain that for all n P N and all non-empty open sets B P PpEq it holdsthat

´

`

XkPNn rEzAks˘

XB¯

‰ H. (1.22)


Combining this with the fact that every n P N it holds that

XkPNnrEzAks (1.23)

is an open set ensures that for every n P N and every non-empty open set B P PpRqit holds that

`

XkPNn rEzAks˘

XB (1.24)

is a non-empty open set. This assures that for every n P N and every non-emptyopen set B P PpRq there exist w P E and ε P p0,8q which satisfy

tv P E : dEpw, vq ă εu Ď´

`

XkPNn rEzAks˘

XB¯

. (1.25)

This implies the existence of functions e “ penqnPN : N Ñ E and r “ prnqnPN : N Ñp0,8q which satisfy for all n P N that rn ă 1

2nand

tv P E : dEpen, vq ă 2rnu

Ď

´

pXkPNn rEzAksq X”

U X`

XkPNn´1

v P E : dEpek, vq ă rk(˘

ı¯

. (1.26)

Hence, we obtain that for all n P N it holds that

tv P E : dEpen, vq ă rnu Ď tv P E : dEpen, vq ă 2rnu

Ď`

XkPNn´1

v P E : dEpek, vq ă rk(˘

.(1.27)

Therefore, we get that for all n P N, k P Nn´1 it holds that

tv P E : dEpen, vq ă rnu Ď

v P E : dEpek, vq ă rk(

. (1.28)

This assures that for all m,n P N8 with m ď n it holds that

tv P E : dEpen, vq ă rnu Ď tv P E : dEpem, vq ă rmu. (1.29)

In particular, we obtain that for all m,n P N8 with m ď n it holds that

en P tv P E : dEpem, vq ă rmu. (1.30)

Therefore, we get that for all m,n P N8 with m ď n it holds that

dEpem, enq ă rm. (1.31)


This together with the fact that lim supmÑ8 rm “ 0 shows that en P E, n P N8,is a Cauchy sequence in pE, dEq. The completeness of pE, dEq hence establishes theexistence of an e P E which satisfies

lim supN8QnÑ8

dEpe, enq “ 0. (1.32)

Combining this with (1.31) implies that for all m P N8 it holds that

dEpem, eq “ lim supN8QnÑ8

dEpem, enq ď lim supN8QnÑ8

rm “ rm. (1.33)

This and (1.26) assure that for all n P N8 it holds that

e P tv P E : dEpen, vq ď rnu Ď tv P E : dEpen, vq ă 2rnu Ď´

pXkPNn rEzAksq X U¯

.

(1.34)Therefore, we get that

e P”

XnPN8

´

pXkPNn rEzAksq X U¯ı

“

´

pXkPN8 rEzAksq X U¯

. (1.35)

This establishes, in particular, that´

pXkPN8 rEzAksq X U¯

‰ H. (1.36)

As U was an arbitrary non-empty open set, we obtain from Corollary 1.6.2 that

XkPN8 rEzAks “ E. (1.37)

This together with the fact that XkPNrEzAks “ EzrYkPNAks ‰ E ensures that N8 ‰

N. Hence, we obtain thatrNzN8s ‰ H. (1.38)

The proof of Theorem 1.6.4 is completed.

Corollary 1.6.5 (Baire category theorem for complete metric spaces – on the densityof the countable intersection of dense open sets). Let pE, dEq be a complete metricspace and let On Ď E, n P N, be open and dense subsets of E. Then it holds thatXnPNOn is a dense subset of E.


1.6.1.2 The uniform boundedness principle

Theorem 1.6.6 (Uniform boundedness principle). Let K P tR,Cu, let pU, Üq bea K-Banach space, let pV, ¨V q be a normed K-vector space, and let A Ď LpU, V q bea non-empty set which satisfies for all u P Uzt0u that

supAPA

AuV ă 8. (1.39)

Then

supAPA

ALpU,V q “ sup

ˆ"

supAPA

”

AuVuU

ı

: u P Uzt0u

*

Y t0u

˙

ă 8. (1.40)

Proof of Theorem 1.6.6. Throughout this proof let Un Ď U , n P N, be the sets whichsatisfy for all n P N that

Un “

"

u P U : supAPA

AuV ď n

*

. (1.41)

Note that the triangle inequality shows that for all A P A, n,K P N, v P U ,pukqkPN Ď Un with lim supkÑ8 v ´ ukU “ 0 it holds that

AvV ď Apv ´ uKqV ` AuKV ď ALpU,V q v ´ uKU ` n. (1.42)

This implies that for all A P A, n P N, v P U , pukqkPN Ď Un with lim supkÑ8 v úkU “ 0 it holds that

AvV “ lim supKÑ8

AvV ď lim supKÑ8

“

ALpU,V q v ´ uKU ` n‰

“ n. (1.43)

This establishes that for all n P N, v P U , pukqkPN Ď Un with lim supkÑ8 vúkU “ 0it holds that

supAPA

AvV ď n. (1.44)

Hence, we obtain that for all n P N, v P U , pukqkPN Ď Un with lim supkÑ8 vúkU “0 it holds that v P Un. This proves that for every n P N it holds that Un is a closedset. In addition, note that assumption (1.39) ensures that

YnPNUn “ U. (1.45)

This shows thatUzrYnPNUns “ H “ H ‰ U. (1.46)


Theorem 1.6.4 and the fact that for every n P N it holds that Un is a closed settherefore show that there exists a natural number n P N such that

UzUn ‰ U. (1.47)

Combining this with Lemma 1.6.1 implies that there exist v P Un and ε P p0,8q suchthat

tu P U : u´ vU ă 2εu Ď Un. (1.48)

This shows that for all A P A, u P U with uU ď 1 it holds that

AuV “1εApεuqV “

1εApv ` εuq ´ AvV

ď 1εrApv ` εuqV ` AvV s ď

1εpn` nq “ 2n

ε.

(1.49)

Hence, we obtain that

supAPA

ALpU,V q “ supAPA

supuPU,uUď1

AuV ď2nεă 8. (1.50)

The proof of Theorem 1.6.6 is thus completed.

1.6.1.3 Local a priori bounds

Lemma 1.6.7 (Local a priori bound). Let K P tR,Cu, let pV, ¨V q be a K-Banachspace, and let S : r0,8q Ñ LpV q be a semigroup which satisfies for all v P V thatlim suptŒ0 Stv ´ vV “ 0. Then

lim suptŒ0

StLpV q “ limtŒ0

supsPr0,ts

SsLpV q ă 8. (1.51)

Proof of Lemma 1.6.7. We prove Lemma 1.6.7 by a contradiction. More specifically,we assume in the following that

limtŒ0

supsPr0,ts

SsLpV q “ 8. (1.52)

This and the fact that S0LpV q “ 1 ă 8 imply that for all t P p0,8q it holds that

supsPp0,ts

SsLpV q “ 8. (1.53)

Hence, there exists a strictly decreasing sequence tn P p0,8q, n P N, with limnÑ8 tn “0 and with the property that for all n P N it holds that

StnLpV q ě n. (1.54)


This ensures thatsupnPN

StnLpV q “ 8. (1.55)

Theorem 1.6.6 hence implies that there exists a vector v P V such that

supnPN

StnvV “ 8. (1.56)

Combining this and the fact that @n P N : StnvV ă 8 implies that

lim supnÑ8

StnvV “ 8. (1.57)

This and the assumption that @w P V : lim suptŒ0 Stw ´ wV show that

8 ą vV “›

›

›limnÑ8

rStnvs›

›

›

V“ lim

nÑ8StnvV “ lim sup

nÑ8StnvV “ 8. (1.58)

This contradiction completes the proof of Lemma 1.6.7.

1.6.1.4 Global a priori bounds

The next result, Corollary 1.6.8, proves a stronger version of Lemma 1.6.7. Observethat Lemma 1.6.7 and Corollary 1.6.8 apply to strongly continuous semigroups onBanach spaces.

Corollary 1.6.8 (Global a priori bound). Let K P tR,Cu, let pV, ¨V q be a K-Banach space, and let S : r0,8q Ñ LpV q be a semigroup which satisfies for all v P Vthat lim suptŒ0 Stv ´ vV “ 0. Then it holds for all t P r0,8q, ε P p0,8q that

supsPr0,ts SsLpV q ď“


¨ et“

lnpSε1εLpV qq

‰`

ă 8. (1.59)

Corollary 1.6.8 is an immediate consequence of Proposition 1.5.1 and Lemma 1.6.7above.

1.6.2 Existence of solutions of linear ordinary differential equa-tions in Banach spaces

Lemma 1.6.9 (Invariance of the domain of the generator). Let K P tR,Cu, letpV, ¨V q be a normed K-vector space, and let S : r0,8q Ñ LpV q be a semigroup.Then it holds for all t P r0,8q, v P DpGSq that

St`

DpGSq˘

Ď DpGSq and GSStv “ StGSv. (1.60)


Proof of Lemma 1.6.9. Observe that for all t P r0,8q, v P DpGSq it holds that

limsŒ0

„

Ss rStvs ´ rStvs

s

“ limsŒ0

„

St

„

Ssv ´ v

s

“ St

„

limsŒ0

„

Ssv ´ v

s

“ StGSv.

(1.61)This completes the proof of Lemma 1.6.9.

Lemma 1.6.10. Let K P tR,Cu, let pV, ¨V q be a K-Banach space, let S : r0,8q ÑLpV q be a strongly continuous semigroup, and let v P DpGSq. Then

(i) it holds that the function r0,8q Q t ÞÑ Stv P V is continuously differentiableand

(ii) it holds for all t P r0,8q that

ddtrStvs “ GSStv “ StGSv. (1.62)

Proof of Lemma 1.6.10. Observe that for all s, t P r0,8q with s ‰ t it holds that›

›

›

›

Ssv ´ Stv

s´ t´ StGSv

›

›

›

›

V

“

›

›

›

›

Smints,tu

„

Ss´mints,tuv ´ St´mints,tuv

s´ t

´ StGSv›

›

›

›

V

ď

›

›

›

›

Sminps,tq

„

Smaxts,tu´mints,tuv ´ v

maxts, tu ´mints, tu´ GSv

›

›

›

›

V

`›

›

“

Smints,tu ´ St‰

GSv›

›

V

ď›

›Sminps,tq

›

›

LpV q

›

›

›

›



›

›

›

›

V

`›

›

“

Smints,tu ´ St‰

GSv›

›

V.

(1.63)

Corollary 1.6.8 and the fact that S is strongly continuous hence imply that for allt P r0,8q it holds that

lim supr0,8qzttuQsÑt

›

›

›

›

Ssv ´ Stv

s´ t´ StGSv

›

›

›

›

V

ď

«

supsPr0,t`1s

SsLpV q

ff«

lim supr0,8qzttuQsÑt

›

›

›

›



›

›

›

›

V

ff

` lim supr0,8qzttuQsÑt

›

›

“

Smints,tu ´ St‰

GSv›

›

V“ 0.

(1.64)

This and Lemma 1.6.9 complete the proof of Lemma 1.6.10.


1.6.3 Pointwise convergence in the space of bounded linearoperators

Lemma 1.6.11 (A characterization of pointwise convergence in the space of boundedlinear operators). Let K P tR,Cu, let pV, ¨V q be a K-Banach space, and let pSnqnPN0

Ď LpV q. Then @ v P V : lim supnÑ8 Snv ´ S0vV “ 0 if and only if it holds for allnon-empty compact sets K Ď V that lim supnÑ8 supvPK Snv ´ S0vV “ 0.

Proof of Lemma 1.6.11. The proof of the “ð” direction in the statement of Lemma1.6.11 is clear. It thus remains to prove the “ñ” direction in the statement ofLemma 1.6.11. To this end we assume that for all v P V it holds that

lim supnÑ8

Snv ´ S0vV “ 0 (1.65)

and we assume that there exists a non-empty compact set K Ď V such that

lim supnÑ8

supvPK

Snv ´ S0vV ą 0. (1.66)

In the next step we note that there exists a sequence pvnqnPN Ď K such that for alln P N it holds that

Snvn ´ S0vnV “ supvPK

Snv ´ S0vV . (1.67)

The compactness of K ensures that there exist a w P K and a strictly increasingsequence pnkqkPN Ď N such that

lim supkÑ8

vnk ´ wV “ 0. (1.68)

Next note that (1.65) ensures that

lim supkÑ8

Snkw ´ S0wV “ 0. (1.69)


This, (1.67), (1.68), and Theorem 1.6.6 imply that

0 “ lim supkÑ8

Snkw ´ S0wV

“ lim supkÑ8

Snkpw ´ vnkq ` pSnk ´ S0q vnk ` S0 pvnk ´ wqV

ě lim supkÑ8

“

pSnk ´ S0q vnkV ´ Snkpw ´ vnkqV ´ S0 pvnk ´ wqV‰

ě lim supkÑ8

pSnk ´ S0q vnkV ´ lim supkÑ8

Snkpw ´ vnkqV ´ lim supkÑ8

S0 pvnk ´ wqV

ě lim supkÑ8

supvPK

pSnk ´ S0q vV ´

„

supkPN

SnkLpV q

lim supkÑ8

w ´ vnkV

“ lim supkÑ8

supvPK

pSnk ´ S0q vV ą 0.

(1.70)

This condradiction completes the proof of Lemma 1.6.11.

1.6.4 Domains of generators of strongly continuous semigroups

In this subsection we prove that the generator of a strongly continuous semigroup isdensely defined ; see Corollary 1.6.13 below. In the proof of Corollary 1.6.13 we usethe following result, Lemma 1.6.12. Lemma 1.6.12 and its proof can, e.g., be foundas Theorem 2.4 (b) in Pazy [5] and Corollary 1.6.13 and its proof can, e.g., be foundas Corollary 2.5 in Pazy [5].

Lemma 1.6.12 (Fundamental theorem of calculus for strongly continuous semi-groups). LetK P tR,Cu, t P r0,8q, let pV, ¨V q be aK-Banach space, let S : r0,8q ÑLpV q be a strongly continuous semigroup, and let v P V . Then it holds that

ż t

0

Ssv ds P DpGSq and GSˆż t

0

Ssv ds

˙

“ Stv ´ v. (1.71)

Proof of Lemma 1.6.12. Throughout this proof we assume w.l.o.g. that t P p0,8q.Next we observe that for all u P p0, tq it holds that

rSu ´ IdV s

u

„ż t

0

Ssv ds

“1

u

ż t

0

rSu`sv ´ Ssvs ds “1

u

ż t`u

t

Ssv ds´1

u

ż u

0

Ssv ds.

(1.72)


Continuity of the function r0,8q Q s ÞÑ Ssv P V hence proves thatşt

0Ssv ds P DpGSq

and that

GSˆż t

0

Ssv ds

˙

“ limuŒ0

„

rSu ´ IdV s

u

„ż t

0

Ssv ds

“ Stv ´ S0v “ Stv ´ v. (1.73)


We are now ready to prove that the generator of a strongly continuous semigroupis densely defined.

Corollary 1.6.13. Let K P tR,Cu, let pV, ¨V q be a K-Banach space, and letS : r0,8q Ñ LpV q be a strongly continuous semigroup. Then it holds that DpGSq isdense in V .

Proof of Corollary 1.6.13. Let v P V be arbitrary. The assumption that S is astrongly continuous semigroup together with the fundamental theorem of calculusensures that

limtŒ0

ˆ

1

t

ż t

0

Ssv ds

˙

“ S0v “ v. (1.74)

In addition, Lemma 1.6.12 proves that for all t P p0,8q it holds that

1

t

ż t

0

Ssv ds P DpGSq. (1.75)

This and (1.74) imply that v P DpGSq. The proof of Corollary 1.6.13 is thus com-pleted.

1.6.5 Generators of strongly continuous semigroups

In this section we show that a strongly continuous semigroup is uniquely determinedby its generator; see Proposition 1.6.15 below. In Proposition 1.6.15 we use theassumption that the graph of one function is a subset of the graph of another function.To develop a better understanding for this assumption, we first note the followingremark.

Remark 1.6.14. Let A1, A2, B be sets and let f1 : A1 Ñ B and f2 : A2 Ñ B befunctions. Then it holds that Graphpf1q Ď Graphpf2q if and only if (A1 Ď A2 andf2|A1 “ f1).


We are now ready to show that a strongly continuous semigroup is uniquelydetermined by its generator.

Proposition 1.6.15 (The generator determines the semigroup). Let K P tR,Cu, letpV, ¨V q be a K-Banach space, and let S, S : r0,8q Ñ LpV q be strongly continuoussemigroups with GraphpGSq Ď GraphpGSq. Then it holds that S “ S and GS “ GS.

Proof of Proposition 1.6.15. Let v P DpGSq Ď DpGSq, t P p0,8q and let η : r0, ts Ñ Vbe the function which satisfies for all s P r0, ts that

ηpsq “ St´s Ss v. (1.76)

Next note that for all s, u P r0, ts with s ‰ u it holds that

›

›

›

›

ηpuq ´ ηpsq

u´ s

›

›

›

›

V

“

›

›

›

›

›

Stú Su v ´ St´s Ss v

u´ s

›

›

›

›

›

V

“

›

›

›

›

›

Stú

„

Su v ´ Ss v

u´ s

`

“

Stú ´ St´s‰

Ssv

u´ s

›

›

›

›

›

V

“

›

›

›

›

›

ˆ

”

Stú ´ St´s

ı

„

Su v ´ Ss v

u´ s

` St´s

„

Su v ´ Ss v

u´ s

˙

`

“

Stú ´ St´s‰

Ssv

u´ s

›

›

›

›

›

V

.

(1.77)

The triangle inequality hence implies that for all s, u P r0, ts with s ‰ u it holds that

›

›

›

›

ηpuq ´ ηpsq

u´ s

›

›

›

›

V

“

›

›

›

›

›

”

Stú ´ St´s

ı

„

Su v ´ Ss v

u´ s

` St´s

„

Su v ´ Ss v

u´ s

´

“

Stú ´ St´s‰

Ssv

pt´ uq ´ pt´ sq

›

›

›

›

›

V

ď

›

›

›

›

”

Stú ´ St´s

ı

„

Su v ´ Ss v

u´ s

›

›

›

›

V

`

›

›

›

›

›

St´s

„

Su v ´ Ss v

u´ s

´

“

Stú ´ St´s‰

Ssv

pt´ uq ´ pt´ sq

›

›

›

›

›

V

.

(1.78)

This implies that for all s P r0, ts, n P N and all pumqmPN P MpN, r0, tsztsuq with


lim supmÑ8 |um ´ s| “ 0 it holds that›

›

›

›

ηpunq ´ ηpsq

un ´ s

›

›

›

›

V

ď

›

›

›

›

›

St´s

„

Sunv ´ Ssv

un ´ s

´

“

Stún ´ St´s‰

Ssv

pt´ unq ´ pt´ sq

›

›

›

›

›

V

` sup!

›

›Stúnw ´ St´sw›

›

V: w P tGSSsvu Y

!

Sumv´Ssvum´s

: m P N

))

.

(1.79)

Next observe that Lemma 1.6.10 and Lemma 1.6.9 prove that for all s P r0, ts it holdsthat

limpr0,tsztsuqQuÑs

«

“

Stú ´ St´s‰

Ssv

pt´ uq ´ pt´ sq

ff

“ GSSt´sSsv “ St´sGSSsv “ St´sGSSsv (1.80)

andlim

pr0,tsztsuqQuÑs

„

Suv ´ Ssv

u´ s

“ GSSsv. (1.81)

Putting (1.80)–(1.81) into (1.79) proves that for all s P r0, ts and all punqnPN Ď

r0, tsztsu with lim supnÑ8 |un ´ s| “ 0 it holds that

lim supnÑ8

›

›

›

›

ηpunq ´ ηpsq

un ´ s

›

›

›

›

V

ď lim supnÑ8

›

›

›

›

›

St´s

„

Sunv ´ Ssv

un ´ s

´

“

Stún ´ St´s‰

Ssv

pt´ unq ´ pt´ sq

›

›

›

›

›

V

` lim supnÑ8

sup!

›


›

V: w P tGSSsvu Y

!

Sumv´Ssvum´s

: m P N

))

“›

›St´sGSSsv ´ St´sGSSsv›

›

V

` lim supnÑ8

sup!

›


›

V: w P tGSSsvu Y

!

Sumv´Ssvum´s

: m P N

))

“ lim supnÑ8

sup!

›


›

V: w P tGSSsvu Y

!

Sumv´Ssvum´s

: m P N

))

.

(1.82)

Next observe that the assumption that S is a strongly continuous semigroup ensuresthat for all s P r0, ts, w P V and all punqnPN Ď r0, tsztsu with lim supnÑ8 |un´ s| “ 0it holds that

lim supnÑ8

›


›

V“ 0. (1.83)


Moreover, note that Lemma 1.6.10 shows that for all s P r0, ts and all punqnPN Ďr0, tsztsu with lim supnÑ8 |un ´ s| “ 0 it holds that

tGSSsvu Y!

Sumv´Ssvum´s

: m P N

)

(1.84)

is a compact set. Combining this and Lemma 1.6.11 with (1.83) and (1.82) provesthat η is differentiable and that for all s P r0, ts it holds that η1psq “ 0. Thefundamental theorem of calculus hence implies that

Stv “ ηp0q “ ηptq “ Stv. (1.85)

As v P DpGSq was arbitrary, we obtain that St|DpGSq “ St|DpGSq. Corollary 1.6.13hence proves that St “ St. This completes the proof of Proposition 1.6.15.

Lemma 1.6.16 (Closedness of generators of strongly continuous semigroups). LetK P tR,Cu, let pV, ¨V q be a K-Banach space, and let S : r0,8q Ñ LpV q be astrongly continuous semigroup. Then it holds that GS is a closed linear operator.

Proof of Lemma 1.6.16. Throughout this proof let x, y P V and let pvnqnPN Ď DpGSqbe a sequence which satisfies that

lim supnÑ8

x´ vnV “ lim supnÑ8

y ´ GSvnV “ 0. (1.86)

Next note that Lemma 1.6.10 and the fundamental theorem of calculus show thatfor all t P r0,8q, n P N it holds that

Stvn ´ vn “

ż t

0

SsGSvn ds. (1.87)

Furthermore, observe that (1.86) and Corollary 1.6.8 imply that for all t P r0,8q itholds that

lim supnÑ8

›

›

›

›

ż t

0

Ssy ds´

ż t

0

SsGSvn ds›

›

›

›

V

ď lim supnÑ8

ż t

0

Ssy ´ SsGSvnV ds

ď t“

supsPr0,ts SsLpV q‰ “

lim supnÑ8 y ´ GSvnV‰

“ 0.

(1.88)

This, (1.86), and (1.87) prove that for all t P r0,8q it holds that

Stx´ x “

ż t

0

Ssy ds. (1.89)


Again the fundamental theorem of calculus hence shows that

lim suptŒ0

›

›

›

›

Stx´ x

t´ y

›

›

›

›

V

“ lim suptŒ0

›

›

›

›

ˆ

1

t

ż t

0

Ssy ds

˙

´ y

›

›

›

›

V

“ 0. (1.90)

This ensures that x P DpGSq and GSx “ y. The proof of Lemma 1.6.16 is thuscompleted.

1.6.6 A generalization of matrix exponentials to infinite di-mensions

Definition 1.4.2 and Proposition 1.6.15 ensure that the next concept, Definition 1.6.17,makes sense.

Definition 1.6.17 (Generalized matrix exponential). Let K P tR,Cu, t P r0,8q, letpV, ¨V q be aK-Banach space, and let A : DpAq Ď V Ñ V be a generator of a stronglycontinuous semigroup. Then we denote by etA P LpV q the linear operator whichsatisfies for every strongly continuous semigroup S : r0,8q Ñ LpV q with GS “ Athat

etA “ St. (1.91)

1.6.7 A characterization of strongly continuous semigroups

Lemma 1.6.18 (Characterization of strongly continuous semigroups). Let K P

tR,Cu, let pV, ¨V q be a K-Banach space, and let S : r0,8q Ñ LpV q be a semi-group on V . Then it holds that S is strongly continuous if and only if it holds for allv P V that

lim suptŒ0

Stv ´ vV “ 0. (1.92)

Proof of Lemma 1.6.18. It is clear that a strongly continuous semigroup satisfies con-dition (1.92). In the following we thus assume that S : r0,8q Ñ LpV q is a semigroupwhich fulfills for all v P V that

lim suptŒ0

Stv ´ vV “ 0. (1.93)


Corollary 1.6.8 hence implies that for all t P r0,8q it holds that

lim supr0,8qQsÑt

Ssv ´ StvV “ lim supr0,8qQsÑt

›

›Smints,tu

`

Ss´mints,tuv ´ St´mints,tuv˘›

›

V

“ lim supr0,8qQsÑt

›

›Smints,tu

`

S|t´s|v ´ v˘›

›

V

ď lim supr0,8qQsÑt

”

›

›Sminps,tq

›

›

LpV q

›

›S|t´s|v ´ v›

›

V

ı

ď

«

supuPr0,t`1s

SuLpV q

ff«

lim supr0,8qQsÑt

›

›S|t´s|v ´ v›

›

V

ff

“ 0.

(1.94)


1.7 Uniformly continuous semigroupsLemma 1.7.1. Let K P tR,Cu, let pV, ¨V q be a normed K-vector space, and letS : r0,8q Ñ LpV q be a semigroup which satisfies that

lim suptŒ0

St ´ S0LpV q “ 0. (1.95)

Then it holds for all t P r0,8q that supsPr0,ts SsLpV q ă 8.

Proof of Lemma 1.7.1. The assumption that lim suptŒ0 St ´ S0LpV q “ 0 ensuresthat there exists a real number ε P p0,8q such that

supsPr0,εs

SsLpV q ă 8. (1.96)

Combining this with Proposition 1.5.1 completes the proof of Lemma 1.7.1.

Lemma 1.7.2. Let K P tR,Cu, let pV, ¨V q be a normed K-vector space, and letS : r0,8q Ñ LpV q be a semigroup. Then S is uniformly continuous if and only iflim suptŒ0 St ´ S0LpV q “ 0.

Proof of Lemma 1.7.2. Clearly, it holds that if S is uniformly continuous, then itholds that limtŒ0 St ´ S0LpV q “ 0. It thus remains to prove that the conditionlimtŒ0 St ´ S0LpV q “ 0 ensures that S is uniformly continuous. We thus assume inthe following that

lim suptŒ0

St ´ S0LpV q “ 0. (1.97)

1.7. UNIFORMLY CONTINUOUS SEMIGROUPS 29

Lemma 1.7.1 hence implies that for all t P r0,8q it holds that

supsPr0,ts

SsLpV q ă 8. (1.98)

This and (1.97) show that for all t P r0,8q it holds that

lim supr0,8qQsÑt

Ss ´ StLpV q “ lim supr0,8qQsÑt

›

›Smints,tu

“

Srmaxts,tu´mints,tus ´ S0

‰›

›

LpV q

ď

«

lim supr0,8qQsÑt

›

›Srmaxps,tq´minps,tqs ´ S0

›

›

LpV q

ff«

supsPr0,t`1s

SsLpV q

ff

“ 0.(1.99)


1.7.1 Matrix exponential in Banach spaces

The next result, Lemma 1.7.3, demonstrates one way how uniformly continuoussemigroups can be constructed.

Lemma 1.7.3 (Matrix exponential in Banach spaces). Let K P tR,Cu, let pV, ¨V qbe a K-Banach space, and let A P LpV q. Then

(i) it holds that A is a generator of a strongly continuous semigroup,

(ii) it holds that petAqtPr0,8q Ď LpV q is a uniformly continuous semigroup,

(iii) it holds for all t P r0,8q that›

›etA›

›

LpV qďř8

n“0

›

›

ptAqn

n!

›

›

LpV qď et ALpV q ă 8, and

(iv) it holds for all t P r0,8q that

etA “8ÿ

n“0

ptAqn

n!. (1.100)

Proof of Lemma 1.7.3. First of all, observe that8ÿ

n“0

›

›

›

›

ptAqn

n!

›

›

›

›

LpV q

ď

8ÿ

n“0

tn AnLpV qn!

“ etALpV q ă 8. (1.101)

Next let S : r0,8q Ñ LpV q be the function which satisfies for all t P r0,8q that

St “8ÿ

n“0

pAtqn

n!. (1.102)


In addition, note that for all t1, t2 P r0,8q it holds that

St1St2 “

«

8ÿ

n“0

pAt1qn

n!

ff«

8ÿ

n“0

pAt2qn

n!

ff

“

8ÿ

n,m“0

An`m pt1qnpt2q

m

n!m!

“

8ÿ

k“0

ÿ

n,mPN0,n`m“k

Ak pt1qnpt2q

m

n!m!“

8ÿ

k“0

Ak

k!

«

kÿ

n“0

k!

n! pk ´ nq!¨ pt1q

n¨ pt2q

pk´nq

ff

“

8ÿ

k“0

Ak

k!rt1 ` t2s

k“ St1`t2 .

(1.103)

This shows that S is a semigroup. Moreover, observe that for all t P r0,8q it holdsthat

St ´ S0LpV q “

›

›

›

›

›

8ÿ

n“1

pAtqn

n!

›

›

›

›

›

LpV q

ď t ALpV q

›

›

›

›

›

8ÿ

n“1

pAtqpn´1q

n!

›

›

›

›

›

LpV q

“ t ALpV q

›

›

›

›

›

8ÿ

n“0

pAtqn

pn` 1q!

›

›

›

›

›

LpV q

ď t ALpV q

«

8ÿ

n“0

AtnLpV qpn` 1q!

ff

ď t ALpV q etALpV q .

(1.104)

This together with Lemma 1.7.2 proves that S is uniformly continuous. Furthermore,note that for all t P p0,8q it holds that

›

›

›

›

St ´ S0

t´ A

›

›

›

›

LpV q

“

›

›

›

›

›

›

”

ř8

n“1ptAqn

n!

ı

t´ A

›

›

›

›

›

›

LpV q

“

›

›

›

›

›

A

«

8ÿ

n“1

ptAqpn´1q

n!

ff

´ A

›

›

›

›

›

LpV q

“

›

›

›

›

›

A

«

8ÿ

n“0

ptAqn

pn` 1q!

ff

´ A

›

›

›

›

›

LpV q

“

›

›

›

›

›

A

«

8ÿ

n“1

ptAqn

pn` 1q!

ff›

›

›

›

›

LpV q

“ t

›

›

›

›

›

A2

«

8ÿ

n“1

ptAqpn´1q

pn` 1q!

ff›

›

›

›

›

LpV q

“ t

›

›

›

›

›

A2

«

8ÿ

n“0

ptAqn

pn` 2q!

ff›

›

›

›

›

LpV q

ď t A2LpV q e

tALpV q .

(1.105)

Therefore, we obtain thatGS “ A. (1.106)

This, in turn, establishes Item (i). Proposition 1.6.15, (1.106), and the fact that Sis a uniformly continuous semigroup hence prove Item (ii) and Item (iv). Next note


that Item (iii) follows from Item (iv) and (1.101). The proof of Lemma 1.7.3 is thuscompleted.

Lemma 1.7.4 (Groups of bounded linear operators generated by bounded opera-tors). Let K P tR,Cu, let pV, ¨V q be a K-Banach space, let A P LpV q, and letS : RÑ LpV q be the function which satisfies for all t P R that

St “8ÿ

n“0

tnAn

n!. (1.107)

Then

(i) it holds for all t1, t2 P R that St1St2 “ St1`t2,

(ii) it holds that R Q t ÞÑ St P LpV q is infinitely often differentiable, and

(iii) it holds for all n P N0, t P R that Spnqt “ AnSt “ StAn.

Proof of Lemma 1.7.4. First of all, note that for all t1, t2 P R it holds that

St1St2 “

«

8ÿ

n“0

pAt1qn

n!

ff«

8ÿ

n“0

pAt2qn

n!

ff

“

8ÿ

n,m“0

An`m pt1qnpt2q

m

n!m!

“

8ÿ

k“0

ÿ

n,mPN0,n`m“k

Ak pt1qnpt2q

m

n!m!“

8ÿ

k“0

Ak

k!

«

kÿ

n“0

k!

n! pk ´ nq!¨ pt1q

n¨ pt2q

pk´nq

ff

“

8ÿ

k“0

Ak

k!rt1 ` t2s

k“ St1`t2 .

(1.108)

This proves Item (i). It thus remains to prove Item (ii) and Item (iii). For this note


that Item (i) ensures that for all t P R it holds that

lim supRzttuQhÑ0

«

St`h ´ St ´ hAStLpV q|h|

ff

ď lim supRzttuQhÑ0

«

Sh ´ IdV ´hALpV q StLpV q|h|

ff

“

«

lim supRzttuQhÑ0

›

›

ř8

n“2hnAn

n!

›

›

LpV q

|h|

ff

StLpV q ď

«

lim supRzttuQhÑ0

8ÿ

n“2

|h|n´1 AnLpV qn!

ff

StLpV q

“

«

lim supRzttuQhÑ0

8ÿ

n“1

|h|n AnLpV qpn` 1q!

ff

ALpV q StLpV q

“ lim supRzttuQhÑ0

˜«

8ÿ

n“0

|h|n AnLpV qpn` 2q!

ff

|h|

¸

A2LpV q StLpV q

ď lim supRzttuQhÑ0

`

|h| ehALpV q˘

A2LpV q StLpV q “ 0.

(1.109)

Hence, we obtain that for all t P R it holds that S 1t “ ASt “ StA. Induction henceestablishes Item (ii) and Item (iii). The proof of Lemma 1.7.4 is thus completed.

1.7.2 Continuous invertibility of bounded linear operators inBanach spaces

Lemma 1.7.5 (Geometric series in Banach spaces and inversion of bounded linearoperators). Let K P tR,Cu, let pV, ¨V q be a K-Banach space, and let A P LpV q bea bounded linear operator with IdV ÁLpV q ă 1. Then it holds

(i) that A is bijective,

(ii) that A´1 P LpV q,

(iii) thatř8

n“0 rIdV ÁsnLpV q ă 8, and

(iv) that

A´1“

8ÿ

n“0

rIdV Ásn . (1.110)


Proof of Lemma 1.7.5. Throughout this proof let Q P LpV q and Sn P LpV q, n P N0,be the bounded linear operators which satisfy for all n P N that

Q “ IdV ´A and Sn “nÿ

k“0

Qk. (1.111)

Note that the assumption that QLpV q ă 1 ensures that

8ÿ

k“0

›

›Qk›

›

LpV qď

8ÿ

k“0

QkLpV q “1

“

1´ QLpV q‰ ă 8. (1.112)

This implies that Sn, n P N0, is a Cauchy-sequence in LpV q. Completeness of LpV qhence shows that Sn, n P N0, converges in LpV q. Next we claim that for all n P N0

it holds thatASn “ IdV ´Q

n`1. (1.113)

We show (1.113) by induction on n P N0. For this observe that

AS0 “ A IdV “ A “ IdV ´Q. (1.114)

This proves the base case n “ 0 in (1.113). Next note that for all n P N withASn´1 “ IdV ´Q

n it holds that

ASn “ A

«

nÿ

k“0

Qk

ff

“ A

«

IdV `nÿ

k“1

Qk

ff

“ A

«

IdV `Q

«

n´1ÿ

k“0

Qk

ffff

“ A rIdV `QSn´1s “ A`QASn´1 “ A`Q rIdV ´Qns

“ A`Q´Qn`1“ IdV ´Q

n`1.

(1.115)

This proves (1.113) by induction on n P N0. Next note that (1.113) and the factthat AQ “ QA imply that for all n P N0 it holds that

ASn “ SnA “ IdV ´Qn`1. (1.116)

This and the fact that pSnqnPN0 Ď LpV q converges shows that

A”

limnÑ8

Sn

ı

l jh n

PLpV q

“

”

limnÑ8

Sn

ı

l jh n

PLpV q

A “ IdV ´ limnÑ8

Qn“ IdV . (1.117)

This implies that A is bijective and that A´1 “ limnÑ8 Sn P LpV q. The proof ofLemma 1.7.5 is thus completed.


1.7.3 Generators of uniformly continuous semigroup

Lemma 1.7.6 (The generator of a uniformly continuous semigroup). Let K P

tR,Cu, let pV, ¨V q be a K-Banach space, and let S : r0,8q Ñ LpV q be a uniformlycontinuous semigroup. Then GS P LpV q.

Proof of Lemma 1.7.6. The assumption that S is uniformly continuous assures thatfor all t P r0,8q it holds that

lim supp0,8qQsÑ0

›

›

›

›

1

s

ż t`s

t

Su du´ St

›

›

›

›

LpV q

“ lim supp0,8qQsÑ0

›

›

›

›

1

s

ż t`s

t

rSu ´ Sts du

›

›

›

›

LpV q

ď lim supp0,8qQsÑ0

„

1

s

ż t`s

t

Su ´ StLpV q du

ď lim supp0,8qQsÑ0

«

supuPrt,t`ss

Su ´ StLpV q

ff

ď StLpV q

«

lim supp0,8qQsÑ0

«

supuPr0,ss

Su ´ S0LpV q

ffff

“ StLpV q

«

lim supp0,8qQsÑ0

Ss ´ S0LpV q

ff

“ 0.

(1.118)

This implies that there exists a real number ε P p0,8q such that›

›

›

›

1

ε

ż ε

0

Ss ds´ S0

›

›

›

›

LpV q

ă 1. (1.119)

Lemma 1.7.5 hence shows thatşε

0Ss ds P LpV q is bijective and that

„ż ε

0

Ss ds

´1

P LpV q. (1.120)

Therefore, we obtain that for all t P p0, εq it holds that

St ´ IdVt

“

„

St ´ S0

t

„ż ε

0

Ss ds

„ż ε

0

Ss ds

´1

“

„

şε

0rSt`s ´ Sss ds

t

„ż ε

0

Ss ds

´1

“

«

şt`ε

tSs ds´

şε

0Ss ds

t

ff

„ż ε

0

Ss ds

´1

“

«

şε`t

εSs ds´

şt

0Ss ds

t

ff

„ż ε

0

Ss ds

´1

“

„

1

t

ż ε`t

ε

Ss ds´1

t

ż t

0

Ss ds

„ż ε

0

Ss ds

´1

.

(1.121)


This together with (1.118) shows that

lim supp0,8qQtÑ0

›

›

›

›

›

St ´ IdVt

´ rSε ´ S0s

„ż ε

0

Ss ds

´1›

›

›

›

›

LpV q

“ 0. (1.122)

Hence, we obtain that GS P LpV q and

GS “ rSε ´ S0s

„ż ε

0

Ss ds

´1

. (1.123)


1.7.4 A characterization result for uniformly continuous semi-groups

Theorem 1.7.7 (Characterization of uniformly continuous semigroups). Let K P

tR,Cu, let pV, ¨V q be a K-Banach space, and let S : r0,8q Ñ LpV q be a semigroup.Then the following three statements are equivalent:

(i) It holds that S is uniformly continuous.

(ii) It holds that lim suptŒ0 St ´ S0LpV q “ 0.

(iii) It holds that GS P LpV q.

Proof of Theorem 1.7.7. Lemma 1.7.2 implies that Item (i) and Item (ii) are equiva-lent. Moreover, Lemma 1.7.6 ensures that Item (i) implies Item (iii). It thus remainsto prove that (iii) implies (i). We thus assume in the following that GS P LpV q. Nextlet S : r0,8q Ñ LpV q be the function which satisfies for all t P r0,8q that

St “8ÿ

n“0

ptGSqn

n!. (1.124)

Observe that Lemma 1.7.3 shows that S is a uniformly continuous semigroup andthat

GS “ GS. (1.125)

In particular, we obtain that S is a strongly continuous semigroup. Next note thatthe assumption that GS P LpV q ensures that for all v P V it holds that

lim supp0,8qQtÑ0

„

Stv ´ v ´ tGSvVt

“ lim supp0,8qQtÑ0

›

›

›

›

Stv ´ v

t´ GSv

›

›

›

›

V

“ 0. (1.126)


This implies that for all v P V it holds that

lim supp0,8qQtÑ0

Stv ´ v ´ tGSvV “ lim supp0,8qQtÑ0

„

t ¨Stv ´ v ´ tGSvV

t

“ 0. (1.127)

Hence, we obtain that for all v P V it holds that

lim supp0,8qQtÑ0

Stv ´ vV “ lim supp0,8qQtÑ0

Stv ´ v ´ tGSv ` tGSvV

ď lim supp0,8qQtÑ0

rStv ´ v ´ tGSvV ` t GSvV s

ď lim supp0,8qQtÑ0

Stv ´ v ´ tGSvV ` lim supp0,8qQtÑ0

rt GSvV s

“ lim supp0,8qQtÑ0

Stv ´ v ´ tGSvV “ 0.

(1.128)

This allows us to apply Lemma 1.6.18 to obtain that S is a strongly continuoussemigroup. This, the fact that S is also a stongly continuous semigroup, and (1.125)enables us to apply Proposition 1.6.15 to obtain that

S “ S. (1.129)

This and the fact that S is uniformly continuous complete the proof of Theorem 1.7.7is thus completed.

1.7.5 An a priori bound for uniformly continuous semigroups

Combining Lemma 1.7.3 and Theorem 1.7.7 immediately results in the followingestimate.

Proposition 1.7.8 (A priori bounds for uniformly continuous semigroups). Let K PtR,Cu, let pV, ¨V q be a K-Banach space, and let S : r0,8q Ñ LpV q be a uniformlycontinuous semigroup. Then it holds for all t P r0,8q that GS P LpV q and

supsPr0,ts

SsLpV q ď et GSLpV q ă 8. (1.130)

1.8. THE HILLE-YOSIDA THEOREM 37

1.8 The Hille-Yosida theorem

1.8.1 Yosida approximations

Definition 1.8.1 (Resolvent set of a linear operator). Let K P tR,Cu, let pV, ¨V qbe a normed K-vector space, and let A : DpAq Ď V Ñ V be a linear operator. Thenwe denote by ρpAq the set given by

ρpAq “

λ P K :`

λ´ A : DpAq Ñ V is bijective and rV Q v ÞÑ pλ´ Aq´1v P V s P LpV q˘(

(1.131)

and we call ρpAq the resolvent set of A.

Definition 1.8.2 (Yosida approximations of a linear operator). Let K P tR,Cu,let pV, ¨V q be a normed K-vector space, and let A : DpAq Ď V Ñ V be a linearoperator. Then we say that A is the family of Yosida approximations of A if andonly if A P MpρpAq, LpV qq is a function from ρpAq to LpV q which satisfies for allλ P ρpAq, v P V that

Aλv “ λ2pλ´ Aq´1v ´ λv. (1.132)

Remark 1.8.3 (Representations of the Yosida approximations). Let K P tR,Cu,let pV, ¨V q be a normed K-vector space, let A : DpAq Ď V Ñ V be a linear operator,and let pAλqλPρpAq Ď LpV q be the family of Yosida approximations of A. Then itholds for all λ P ρpAqzt0u, v P V that

Aλv “ Ap1´ 1λ ¨ Aq´1v “ λApλ´ Aq´1v. (1.133)

Remark 1.8.4 (An intuition for the Yosida approximations). Let K P tR,Cu, letpV, ¨V q be a normed K-vector space, and let A : DpAq Ď V Ñ V be a linear operatorwhich satisfies that p0,8q Ď ρpAq, and let pAλqλPρpAq Ď LpV q be the family of Yosidaapproximations of A. Then it holds for all v P DpAq and all sufficiently large n P Nthat

Anv “ ApIdV ´1nAq´1v « Av (1.134)


Remark 1.8.5 (Boundedness of implicit Euler approximations for linear differentialequations). Let K P tR,Cu, let pV, ¨V q be a K-Banach space, and let A : DpAq ĎV Ñ V be a linear operator such that p0,8q Ď ρpAq and suphPp0,8q p1´ hAq

´1LpV q ď

1, and let pY hn qnPN0 Ď V , h P p0,8q, satisfy for all h P p0,8q, n P N0 that

Y hn`1 “ p1´ hAq

´1Y hn . (1.135)

Then it holds for all h P p0,8q, n P N0 that

Y hn`1V ď

„

suprPp0,8q

›

›p1´ rAq´1›

›

LpV q

Y hn V ď Y

hn V . (1.136)

1.8.2 Scalar shifts of generators of strongly continuous semi-groups

Lemma 1.8.6 (Scalar shifts of generators of strongly continuous semigroups). LetK P tR,Cu, λ P K, let pV, ¨V q be a K-Banach space, and let A : DpAq Ď V Ñ Vbe a generator of a strongly continuous semigroup. Then

(i) it holds that A ´ λ : DpAq Ď V Ñ V is a generator of a strongly continuoussemigroup and

(ii) it holds for all t P r0,8q that etpA´λq “ e´λtetA.

Proof of Lemma 1.8.6. Throughout this proof let S : r0,8q Ñ LpV q be the functionwhich satisfies for all t P r0,8q that

St “ e´λtetA. (1.137)

Next note that for all t1, t2 P r0,8q it holds that S0 “ IdV and

St1St2 “ e´λt1e´λt2et1Aet2A “ e´λpt1`t2qept1`t2qA “ St1`t2 . (1.138)

Moreover, observe that the fact that r0,8q Q t ÞÑ etA P LpV q is a strongly continuoussemigroup ensures that for every v P V it holds that the function

r0,8q Q t ÞÑ Stv P V (1.139)


is continuous. This and (1.138) show that S is a strongly continuous semigroup. Inaddition, note that for all v P DpAq it holds that

lim suptŒ0

›

›

›

›

Stv ´ v

t´ pA´ λqv

›

›

›

›

V

“ lim suptŒ0

›

›

1tpe´λtetAv ´ vq ´ pA´ λqv

›

›

V

“ lim suptŒ0

›

›

1tpe´λtetAv ´ vq ` λetAv ´ Av ` λpv ´ etAvq

›

›

V

“ lim suptŒ0

›

›

1te´λtetAv ´ 1

tetAv ` λetAv ` 1

tetAv ´ 1

tv ´ Av ` λpv ´ etAvq

›

›

V

“ lim suptŒ0

›

›

“

1tpe´λt ´ 1q ` λ

‰

etAv ` 1tpetAv ´ vq ´ Av ` λpv ´ etAvq

›

›

V

ď

„

lim suptŒ0

ˇ

ˇ

1tpe´λt ´ 1q ` λ

ˇ

ˇ

„

lim suptŒ0

etAvV

`

„

lim suptŒ0

›

›

1tpetAv ´ vq ´ Av

›

›

V

` |λ|

„

lim suptŒ0

v ´ etAvV

“ 0.

(1.140)

This implies that DpAq Ď DpGSq. Furthermore, observe that for all v P DpGSq itholds that

lim suptŒ0

›

›

›

›

etAv ´ v

t´ pGS ` λqv

›

›

›

›

V

“ lim suptŒ0

›

›

1tpetAv ´ vq ´ λetAv ´ GSv ` λpetAv ´ vq

›

›

V

“ lim suptŒ0

›

›

1tetAv ´ 1

te´λtetAv ´ λetAv ` 1

te´λtetAv ´ 1

tv ´ GSv ` λpetAv ´ vq

›

›

V

“ lim suptŒ0

›

›

“

1tp1´ e´λtq ´ λ

‰

etAv ` 1tpe´λtetAv ´ vq ´ GSv ` λpetAv ´ vq

›

›

V

ď

„

lim suptŒ0

ˇ

ˇ

1tp1´ e´λtq ´ λ

ˇ

ˇ

„

lim suptŒ0

etAvV

`

„

lim suptŒ0

›

›

1tpStv ´ vq ´ GSv

›

›

V

` |λ|

„

lim suptŒ0

etAv ´ vV

“ 0.

(1.141)

This proves that DpAq Ě DpGSq. Combining (1.140) and (1.141) hence establishesthat for all v P DpAq it holds that

DpAq “ DpGSq and GSv “ pA´ λqv. (1.142)



1.8.3 Convergence of linear-implicit Euler approximations

Lemma 1.8.7. LetK P tR,Cu, let pV, ¨V q be aK-Banach space, and let A : DpAq Ď

V Ñ V be a linear operator which satisfies DpAq “ V , p0,8q Ď ρpAq, and

lim suphŒ0

›

›p1´ hAq´1›

›

LpV q“ lim sup

λÑ8

›

›p1´ 1λ ¨ Aq´1›

›

LpV qď 1. (1.143)

Then it holds for all v P V that

lim suphŒ0

p1´ hAq´1v ´ vV “ lim supλÑ8

p1´ 1λ ¨ Aq´1v ´ vV “ 0. (1.144)

Proof of Lemma 1.8.7. Note that it holds for all v P DpAq that

lim supλÑ8

p1´ 1λ ¨ Aq´1v ´ vV “ lim supλÑ8

“

1λAp1´ 1λ ¨ Aq´1vV

‰

“ lim supλÑ8

“

1λp1´ 1λ ¨ Aq´1AvV

‰

ď lim supλÑ8

“

p1´ 1λ ¨ Aq´1LpV q

1λAvV

‰

ď

„

lim supλÑ8

p1´ 1λ ¨ Aq´1LpV q

„

lim supλÑ8

1λAvV

ď lim supλÑ8

“

1λAvV

‰

“ 0.

(1.145)

This implies that for all x P V and all pvnqnPN Ď DpAq with lim supnÑ8 x´vnV “ 0it holds that

lim supλÑ8

p1´ 1λ ¨ Aq´1x´ xV

“ lim supnÑ8

lim supλÑ8

p1´ 1λ ¨ Aq´1px´ vnq ` p1´ 1λ ¨ Aq´1vn ´ vn ` vn ´ xV

ď 2 ¨ lim supnÑ8

x´ vn ` lim supnÑ8

lim supλÑ8

p1´ 1λ ¨ Aq´1vn ´ vnV “ 0. (1.146)


1.8.4 Properties of Yosida approximations

1.8.4.1 Convergence of Yosida approximations

Corollary 1.8.8 (Approximation property of Yosida approximations). Let K P

tR,Cu, let pV, ¨V q be a K-Banach space, let A : DpAq Ď V Ñ V be a linear operatorwhich satisfies DpAq “ V , p0,8q Ď ρpAq, and lim suphŒ0 p1´ hAq

´1LpV q ď 1, andlet pAλqλPρpAq Ď LpV q be the family of Yosida approximations of A. Then it holdsfor all v P DpAq that

lim supλÑ8

Aλv ´ AvV “ 0. (1.147)


Proof of Corollary 1.8.8. Lemma 1.8.7 and Remark 1.8.3 prove that for all v P DpAqit holds that

lim supλÑ8

Aλv ´ AvV “ lim supλÑ8

Ap1´ 1λ ¨ Aq´1v ´ AvV

“ lim supλÑ8

p1´ 1λ ¨ Aq´1Av ´ AvV “ 0.(1.148)

The proof of Corollary 1.8.8 is thus completed.

1.8.4.2 Semigroups generated by Yosida approximations

Definition 1.8.9 (Contraction semigroup). Let K P tR,Cu and let pV, ¨V q be anormed K-vector space. Then we say that S is a contraction semigroup on V (wesay that S is a contraction semigroup, we say that S is a semigroup of contractionson V , we say that S is a semigroup of contractions) if and only if it holds

(i) that S is a semigroup of bounded linear operators on V and

(ii) that S is contractive.

Lemma 1.8.10 (Semigroups generated by Yosida approximations). Let K P tR,Cu,let pV, ¨V q be a K-Banach space, let A : DpAq Ď V Ñ V be a linear operatorwhich satisfies p0,8q Ď ρpAq and suphPp0,8q p1 ´ hAq´1LpV q “ supλPp0,8q p1 ´ 1λ ¨

Aq´1LpV q ď 1, and let pAλqλPρpAq Ď LpV q be the family of Yosida approximations ofA. Then it holds for all λ, µ P p0,8q, t P r0,8q, v P V that pesAλqsPr0,8q Ď LpV q isa uniformly continuous contraction semigroup and that

etAλv ´ etAµvV ď t Aλv ´ AµvV . (1.149)

Proof of Lemma 1.8.10. First of all, note that the fact that

@λ P p0,8q : Aλ P LpV q (1.150)

and Lemma 1.7.3 show that for every λ P p0,8q it holds that etAλ P LpV q, t Pr0,8q, is a uniformly continuous semigroup. In addition, observe that Lemma 1.8.6,Item (iii) in Lemma 1.7.3, and the fact that @λ P p0,8q : p1 ´ 1λ ¨ Aq´1LpV q ď 1ensure that for all λ P p0,8q, t P r0,8q it holds that

etAλLpV q “ e´λtetrλ2pλÁq´1s

LpV q ď e´λtet λ2pλÁq´1LpV q

“ e´λteλt p1´1λÄq´1LpV q ď e´λteλt “ 1.

(1.151)


This proves that petAλqtPr0,8q Ď LpV q is a contraction semigroup. Next note thatItem (iv) in Lemma 1.7.3, Lemma 1.7.4, and the product rule ensure that for everyλ, µ P p0,8q, t P r0,8q, v P V it holds that the function

r0, 1s Q s ÞÑ etsAλetp1´sqAµv P V (1.152)

is continuously differentiable. The product rule, Lemma 1.7.4, the fact that @λ, µ Pp0,8q : AλAµ “ AµAλ, and Item (iv) in Lemma 1.7.3 hence show that for all λ, µ Pp0,8q, t P r0,8q, s P r0, 1s, v P V it holds that

ddsretsAλetp1´sqAµvs “ tAλe

tsAλetp1´sqAµv ´ etsAλtAµetp1´sqAµv

“ tetsAλetp1´sqAµpAλ ´ Aµqv.(1.153)

This, (1.152), and (1.151) imply that for all λ, µ P p0,8q, t P r0,8q, v P V it holdsthat

etAλv ´ etAµvV “

›

›

›

›

ż 1

0

ddsretsAλetp1´sqAµvs ds

›

›

›

›

V

ď t

ż 1

0

etsAλetp1´sqAµpAλ ´ AµqvV ds ď t Aλv ´ AµvV .

(1.154)


1.8.5 A characterization for generators of strongly continuouscontraction semigroups

Theorem 1.8.11 (Hille-Yosida). Let K P tR,Cu, let pV, ¨V q be a K-Banach space,and let A : DpAq Ď V Ñ V be a linear operator. Then it holds that A is a generatorof a strongly continuous contraction semigroup if and only if it holds

(i) that DpAqV“ V ,

(ii) that A is a closed linear operator,

(iii) that p0,8q Ď ρpAq, and

(iv) that suphPp0,8q p1´ hAq´1LpV q ď 1.


Proof of Theorem 1.8.11. We first prove the “ñ” direction in the statement of The-orem 1.8.11. To this end assume that A is a generator of a strongly continuouscontraction semigroup. Corollary 1.6.13 and Lemma 1.6.16 prove that DpAq is densein V and that A is a closed linear operator. Furthermore, observe that Lemma 1.8.6and the fact that

supsPr0,8q

esALpV q ď 1 (1.155)

shows that for all λ P p0,8q, t1 P r0,8q, t2 P rt1,8q, v P V it holds that›

›

›

›

ż t2

0

espA´λqv ds´

ż t1

0

espA´λqv ds

›

›

›

›

V

“

›

›

›

›

ż t2

t1

espA´λqv ds

›

›

›

›

V

“

›

›

›

›

ż t2

t1

e´λsesAv ds

›

›

›

›

V

ď

«

supsPrt1,t2s

esALpV q

ff

ż t2

t1

e´λsvV ds

ď vV

ż t2

t1

e´λs ds “ 1λpe´λt1 ´ e´λt2qvV ď

1λe´λt1vV .

(1.156)

This proves that for all λ P p0,8q, v P V it holds thatşn

0espA´λqv ds P V , n P N, is

a Cauchy sequence in pV, ¨V q. In particular, inequality (1.156) ensures that for allλ P p0,8q, v P V it holds that

›

›

›

›

limnÑ8

ż n

0

espA´λqv ds

›

›

›

›

V

“ limnÑ8

›

›

›

›

ż n

0

espA´λqv ds

›

›

›

›

V

ď 1λvV . (1.157)

Next we introduce some additional notation. Let Rλ : V Ñ V , λ P p0,8q, be thefunctions which satisfy for all λ P p0,8q, v P V that

Rλv “ limnÑ8

ż n

0

espA´λqv ds. (1.158)

In the following we establish that for all λ P p0,8q it holds

(a) that RλpV q Ď DpAq,

(b) that Rλ P LpV q,

(c) that λ´ A : Dpλ´ Aq “ DpAq Ď V Ñ V is bijective,

(d) that pλ´ Aq´1 “`

V Q v ÞÑ Rλv P DpAq˘

, and

(e) that p0,8q Ď ρpAq.


In this sense the functions Rλ : V Ñ V , λ P p0,8q, are candidates for the resolventoperators. We first observe that (1.157) proves that for every λ P p0,8q it holds that

Rλ P LpV q and RλLpV q ď1λ. (1.159)

Next note that Lemma 1.8.6 and the fact that

supsPr0,8q

esALpV q ď 1 (1.160)

imply that for every λ P p0,8q, v P V it holds that A´λ is a generator of a stronglycontinuous contraction semigroup and

lim supnÑ8

enpA´λqvV “ lim supnÑ8

eńλenAvV ď lim supnÑ8

“

eńλenALpV qvV‰

ď lim supnÑ8

“

e´λnvV‰

“ 0.(1.161)

Lemma 1.6.12 hence ensures that for all λ P p0,8q, v P V , n P N it holds thatn

∫0espA´λqv ds P DpA´ λq “ DpAq “ Dpλ´ Aq (1.162)

and

lim supnÑ8

›

›

›

›

pλ´ Aq

ˆż n

0

espA´λqv ds

˙

´ v

›

›

›

›

V

“ lim supnÑ8

›

›

›

›

pA´ λq

ˆż n

0

espA´λqv ds

˙

` v

›

›

›

›

V

“ lim supnÑ8

penpA´λqv ´ vq ` vV “ lim supnÑ8

enpA´λqvV “ 0.

(1.163)

Next observe that the fact that for every λ P p0,8q it holds that A´λ : DpAq Ď V ÑV is a generator of a strongly continuous semigroup and Lemma 1.6.16 ensure thatfor every λ P p0,8q it holds that pA´λq : DpAq Ď V Ñ V is a closed linear operator.This, in turn, assures that for every λ P p0,8q it holds that pλÁq : DpAq Ď V Ñ Vis a closed linear operator. Combining this with (1.162)–(1.163) proves that for allλ P p0,8q, v P V it holds that

Rλv P Dpλ´ Aq “ DpAq (1.164)

and

pλ´ AqRλv “ pλ´ Aq

ˆ

limnÑ8

ż n

0

espA´λqv ds

˙

“ limnÑ8

„

pλ´ Aq

ˆż n

0

espA´λqv ds

˙

“ v.

(1.165)


In the next step we observe that Lemma 1.6.10, the fundamental theorem of calculus,and (1.161) show that for all λ P p0,8q, v P DpAq it holds that

Rλpλ´ Aqv “ limnÑ8

ż n

0

espA´λqpλ´ Aqv ds “ ´ limnÑ8

ż n

0

espA´λqpA´ λqv ds

“ ´ limnÑ8

ż n

0

“

dds

`

espA´λqv˘‰

ds “ ´”

limnÑ8

penpA´λqv ´ vqı

“ limnÑ8

pv ´ enpA´λqvq “ v ´ limnÑ8

“

enpA´λqv‰

“ v.

(1.166)

This, (1.164), and (1.165) prove that for all λ P p0,8q it holds that

DpAq Q v ÞÑ pλ´ Aqv P V (1.167)

is bijective and that

pλ´ Aq´1“`

V Q v ÞÑ Rλv P DpAq˘

. (1.168)

Combining this with (1.159) ensures that p0,8q Ď ρpAq and

suphPp0,8q

›

›p1´ hAq´1›

›

LpV q“ sup

λPp0,8q

›

›p1´ 1λ ¨ Aq´1›

›

LpV q

“ supλPp0,8q

›

›λpλ´ Aq´1›

›

LpV q“ sup

λPp0,8q

”

λ›

›pλ´ Aq´1›

›

LpV q

ı

“ supλPp0,8q

”

λ RλLpV q

ı

ď supλPp0,8q

“

λ ¨ 1λ

‰

“ 1.

(1.169)

The proof of the “ñ” direction in the statement of Theorem 1.8.11 is therebycompleted. It thus remains to prove the “ð” direction in the statement of Theo-rem 1.8.11. To this end let pAλqλPρpAq Ď LpV q be the family of Yosida approximationsof A and assume for the rest of this proof that DpAq is dense in V , that A is a closedlinear operator, that p0,8q Ď ρpAq, and that

suphPp0,8q

›

›p1´ hAq´1›

›

LpV qď 1. (1.170)

Lemma 1.8.10 implies that for all λ P p0,8q it holds that

r0,8q Q t ÞÑ etAλ P LpV q (1.171)

is a uniformly continuous contraction semigroup. This, again Lemma 1.8.10, andCorollary 1.8.8 prove that for all t P r0,8q, v P V , pxnqnPN Ď DpAq with lim supnÑ8 v´


xnV “ 0 it holds that

lim supNÑ8

supλ,µPrN,8q

supsPr0,ts

esAλv ´ esAµvV

“ lim supnÑ8

lim supNÑ8

supλ,µPrN,8q

supsPr0,ts

”

›

›esAλv ´ esAλxn ` esAλxn ´ e

sAµxn

` esAµxn ´ esAµv

›

›

V

ı

ď lim supnÑ8

lim supNÑ8

supλ,µPrN,8q

supsPr0,ts

”

`

esAλLpV q ` esAµLpV q

˘

v ´ xnV

` esAλxn ´ esAµxnV

ı

ď lim supnÑ8

lim supNÑ8

supλ,µPrN,8q

supsPr0,ts

“

2 v ´ xnV ` esAλxn ´ e

sAµxnV‰

ď lim supnÑ8

lim supNÑ8

supλ,µPrN,8q

supsPr0,ts

“

2 v ´ xnV ` s Aλxn ´ AµxnV‰

ď 2 ¨ lim supnÑ8

v ´ xnV ` t ¨ lim supnÑ8

lim supNÑ8

supλ,µPrN,8q

Aλxn ´ AµxnV “ 0.

(1.172)

Combining this with the assumption that DpAq is dense in V shows that for allt P r0,8q, v P V it holds that

etAnv P V, n P N, (1.173)

is a Cauchy sequence in pV, ¨V q. In addition, we observe that for all t P r0,8q,v P V it holds that

›

›limnÑ8 etAnv

›

›

V“ lim

nÑ8etAnvV ď vV lim sup

nÑ8etAnLpV q ď vV . (1.174)

Next we note that (1.173) and (1.174) ensure that there exists a unique functionS : r0,8q Ñ LpV q which satisfies for all t P r0,8q, v P V that

Stv “ limnÑ8

etAnv. (1.175)

Inequality (1.174) implies that

suptPr0,8q

StLpV q ď 1. (1.176)


In addition, note that S0 “ IdV and that it holds for all t1, t2 P r0,8q, v P V that

St1St2v ´ St1`t2vV

“

›

›

›

”

limnÑ8

et1An´

limmÑ8

et2Amv¯ı

´

”

limnÑ8

et1Anet2Anvı›

›

›

V

“ limnÑ8

›

›

›et1An

´”

limmÑ8

et2Amvı

´ et2Anv¯›

›

›

V

ď lim supnÑ8

´

›

›et1An›

›

LpV q

›

›

›

”

limmÑ8

et2Amvı

´ et2Anv›

›

›

V

¯

ď lim supnÑ8

›

›

›

”

limmÑ8

et2Amvı

´ et2Anv›

›

›

V“ lim

nÑ8

›

›

›

”

limmÑ8

et2Amvı

´ et2Anv›

›

›

V

“

›

›

›

”

limmÑ8

et2Amvı

´

”

limnÑ8

et2Anvı›

›

›

V“ 0.

(1.177)

Moreover, (1.172) implies that for all t P r0,8q, v P V it holds that

lim supnÑ8

supsPr0,ts

esAnv ´ SsvV “ lim supnÑ8

supsPr0,ts

limmÑ8

esAnv ´ esAmvV

ď lim supnÑ8

lim supmÑ8

supsPr0,ts

esAnv ´ esAmvV

ď lim supNÑ8

supn,mPtN,N`1,...u

supsPr0,ts

esAnv ´ esAmvV “ 0.

(1.178)

This ensures that for all v P V it holds that

lim suptŒ0

Stv ´ vV “ limtŒ0

supsPr0,ts

Ssv ´ vV

“ lim supnÑ8

limtŒ0

supsPr0,ts

Ssv ´ esAnv ` esAnv ´ vV

ď lim supnÑ8

supsPr0,1s

Ssv ´ esAnvV ` lim sup

nÑ8lim suptŒ0

etAnv ´ vV “ 0.

(1.179)

Combining this with (1.176) shows that S is a strongly continuous contraction semi-group. Next observe that (1.171), (1.178), and Corollary 1.8.8 show that for all


t P r0,8q, v P DpAq it holds that

lim supnÑ8

›

›

›

›

ż t

0

esAnAnv ds´

ż t

0

SsAv ds

›

›

›

›

V

“ lim supnÑ8

›

›

›

›

ż t

0

“

esAnAnv ´ SsAv‰

ds

›

›

›

›

V

ď lim supnÑ8

ż t

0

esAnAnv ´ SsAvV ds ď t ¨ lim supnÑ8

supsPr0,ts

esAnAnv ´ SsAvV

“ t ¨ lim supnÑ8

supsPr0,ts

esAnAnv ´ esAnAv ` esAnAv ´ SsAvV

ď t ¨ lim supnÑ8

Anv ´ AvV ` t ¨ lim supnÑ8

supsPr0,ts

esAnAv ´ SsAvV “ 0.

(1.180)

The fundamental theorem of calculus and Lemma 1.6.10 hence ensure for all v P DpAqthat

lim suptŒ0

›

›

›

›

Stv ´ v

t´ Av

›

›

›

›

V

“ lim suptŒ0

›

›

1t

`

limnÑ8retAnv ´ vs

˘

´ Av›

›

V

“ lim suptŒ0

›

›

›

›

1

t

ˆ

limnÑ8

ż t

0

“

ddsesAnv

‰

ds

˙

´ Av

›

›

›

›

V

“ lim suptŒ0

›

›

›

›

1

t

ˆ

limnÑ8

ż t

0

esAnAnv ds

˙

´ Av

›

›

›

›

V

“ lim suptŒ0

›

›

›

›

1

t

ˆż t

0

SsAv ds

˙

´ Av

›

›

›

›

V

“ 0.

(1.181)

Therefore, we obtain that for all v P DpAq it holds that

v P DpGSq and GSv “ Av. (1.182)

Next note that the fact that p0,8q Ď ρpAq shows that p1 ´ Aq : DpAq Ď V Ñ V isbijective. This and (1.182) ensure that

p1´ GSq`

DpAq˘

“ p1´ Aq`

DpAq˘

“ V. (1.183)

This implies that for all w P V there exists a v P DpAq such that

v ´ GSv “ p1´ GSqv “ w. (1.184)

Hence, we obtain that for all u P pDpGSqzDpAqq there exists a v P DpAq such that

v ´ GSv “ p1´ GSqv “ p1´ GSqu. (1.185)

1.9. DIAGONAL LINEAR OPERATORS ON HILBERT SPACES 49

In the next step we observe that the “ñ” direction in the statement of Theorem 1.8.11proves that p0,8q Ď ρpGSq. Hence, we obtain that

p1´ GSq : DpGSq Ď V Ñ V (1.186)

is bijective. Combining this with (1.185) shows that for all u P pDpGSqzDpAqq thereexists a v P DpAq such that v “ u. Hence, we obtain that

pDpGSqzDpAqq “ H. (1.187)

This and (1.182) assure thatDpGSq “ DpAq. (1.188)

Combining this with (1.182) establishes that GS “ A. The proof of Theorem 1.8.11is thus completed.

1.9 Diagonal linear operators on Hilbert spacesDefinition 1.9.1 (Diagonal linear operators on Hilbert spaces). Let K P tR,Cu andlet pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space. We say that A is a diagonal linear operatoron pH, 〈¨, ¨〉H , ¨Hq (we say that A is a diagonal linear operator on H, we say that Ais a diagonal linear operator) if and only if there exist H P PpHq and λ P MpH,Kqsuch

(i) that H is an orthonormal basis of H,

(ii) that A PMptv P H :ř

hPH |λhxh, vyH |2 ă 8u, Hq, and

(iii) that it holds for all v P DpAq that

Av “ÿ

hPH

λh 〈h, v〉H h. (1.189)

Definition 1.9.2 (Densely defined). Let K P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, and let A : DpAq Ď H Ñ H be a linear operator. Then we say that Ais a densely defined if and only if DpAq

H“ H.

Exercise 1.9.3 (Diagonal operators are densely defined). Let K P tR,Cu, letpH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, and let A : DpAq Ď H Ñ H be a diagonallinear operator. Prove that A is densely defined.


Exercise 1.9.4 (Symmetry of diagonal linear operators). Let pH, 〈¨, ¨〉H , ¨Hq be anR-Hilbert space and let A : DpAq Ď H Ñ H be a diagonal linear operator. Prove thatA is symmetric.

Exercise 1.9.5 (The point spectrum of a diagonal linear operator). Let K P tR,Cu,let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, let A : DpAq Ď H Ñ H be a linear operator,let H Ď H be an orthonormal basis of H, and let λ : HÑ K be a function such thatfor all v P DpAq it holds that

DpAq “

#

v P H :ÿ

hPH

|λh 〈h, v〉H |2ă 8

+

(1.190)

and Av “ř

hPH λh 〈h, v〉H h. Prove that σP pAq “ impλq.

1.10 Semigroups generated by diagonal linear oper-ators

Lemma 1.10.1. It holds for all z P Czt0u thatˇ

ˇ

ez´1z

ˇ

ˇ ď?

2` emaxtRepzq,0u. (1.191)

Proof of Lemma 1.10.1. First, observe that for all a, b P R with a2 ` b2 ą 0 it holdsthat

ˇ

ˇea`ib ´ 1ˇ

ˇ

?a2 ` b2

ď

ˇ

ˇea`ib ´ eibˇ

ˇ`ˇ

ˇeib ´ 1ˇ

ˇ

?a2 ` b2

“|ea ´ 1|?a2 ` b2

`

ˇ

ˇeib ´ 1ˇ

ˇ

?a2 ` b2

“emaxta,0u

ˇ

ˇea´maxta,0u ´ e´maxta,0uˇ

ˇ

?a2 ` b2

`

«

|cospbq ´ 1|2 ` |sinpbq|2

a2 ` b2

ff12

“emaxta,0u

ˇ

ˇe´|a| ´ 1ˇ

ˇ

?a2 ` b2

`

«

|cospbq ´ 1|2 ` |sinpbq|2

a2 ` b2

ff12

.

(1.192)

The fact that

@ b P R : |sinpbq| “ |sinp|b|q| “

ˇ

ˇ

ˇ

ˇ

ˇ

ż |b|

0

cospuq du

ˇ

ˇ

ˇ

ˇ

ˇ

ď

ż |b|

0

|cospuq| du ď |b|, (1.193)

1.10. SEMIGROUPS GENERATED BY DIAGONAL LINEAR OPERATORS 51

the fact that

@ b P R : |1´ cospbq| “ 1´ cospbq “ 1´ cosp|b|q “ ´ rcosp|b|q ´ cosp0qs

“

ż |b|

0

sinpuq du ď

ż |b|

0

|sinpuq| du ď |b|,(1.194)

and the fact that

@x P p´8, 0s : |ex ´ 1| “ˇ

ˇe´|x| ´ e´0ˇ

ˇ “

ż |x|

0

e´u du ď |x| (1.195)

hence show that for all a, b P R with a2 ` b2 ą 0 it holds thatˇ

ˇea`ib ´ 1ˇ

ˇ

?a2 ` b2

ď|a| emaxta,0u

?a2 ` b2

`

„

b2 ` b2

a2 ` b2

12

“|a| emaxta,0u `

?2 |b|

?a2 ` b2

ď?

2` emaxta,0u.

(1.196)


Theorem 1.10.2 (Semigroups generated by diagonal operators). Let K P tR,Cu,let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, let B Ď H be an orthonormal basis, letλ : B Ñ K be a function which satisfies supptRepλbq : b P Bu Y t0uq ă 8, and letA : DpAq Ď H Ñ H be the linear operator which satisfies

DpAq “

#

v P H :ÿ

bPB

|λb 〈b, v〉H |2ă 8

+

(1.197)

and @ v P DpAq : Av “ř

bPB λb 〈b, v〉H b. Then

(i) it holds that A is a generator of a strongly continuous semigroup,

(ii) it holds for all v P H, t P r0,8q thatř

bPB |eλbtxb, vyH |

2 ă 8 and

eAtv “ÿ

bPB

eλbt 〈b, v〉H b, (1.198)

and

(iii) it holds for all t P r0,8q that eAt P LpHq is a diagonal linear operator.


Proof of Theorem 1.10.2. Throughout this proof assume w.l.o.g. that B ‰ H, letC Ď C be the set given by

C “

"

z P Czt0u : Repzq ď max

"

supbPB

Repλbq, 0

**

, (1.199)

and let S : r0,8q Ñ LpHq be the function which satisfies for all v P H, t P r0,8qthat

Stpvq “ÿ

bPB

eλbt 〈b, v〉H b. (1.200)

Note that the assumption that supbPBRepλbq ă 8 ensures that for all v P H, t Pr0,8q it holds that

ÿ

bPB

|eλbtxb, vyH |2“

ÿ

bPB

e2tRepλbq|xb, vyH |2ď

„

supbPB

e2tRepλbq

«

ÿ

bPB

|xb, vyH |2

ff

ă 8.

(1.201)Next observe that for all t1, t2 P r0,8q, v P H it holds that

St1pSt2pvqq “ St1

˜

ÿ

bPB

eλbt2 〈b, v〉H b

¸

“ÿ

bPB

eλbt2 〈b, v〉H St1pbq

“ÿ

bPB

eλbt2 〈b, v〉H

«

ÿ

cPB

eλct1 〈c, b〉H c

ff

“ÿ

bPB

eλbt2 〈b, v〉H“

eλbt1b‰

“ÿ

bPB

eλbpt1`t2q 〈b, v〉H b “ St1`t2pvq.

(1.202)

This establishes that the function S is a semigroup. Moreover, observe that Lebesgue’stheorem of dominated convergence proves that for all v P H it holds that

lim suptŒ0

Stv ´ v2H “ lim sup

tŒ0

›

›

›

›

›

ÿ

bPB

“

eλbt ´ 1‰

〈b, v〉H b

›

›

›

›

›

2

H

“ lim suptŒ0

«

ÿ

bPB

›

›

“

eλbt ´ 1‰

〈b, v〉H b›

›

2

H

ff

“ lim suptŒ0

«

ÿ

bPB

ˇ

ˇeλbt ´ 1ˇ

ˇ

2|〈b, v〉H |

2

ff

“ lim suptŒ0

ż

B

ˇ

ˇeλbt ´ 1ˇ

ˇ

2|〈b, v〉H |

2 #Bpdbq

“

ż

B

lim suptŒ0

´

ˇ

ˇeλbt ´ 1ˇ

ˇ

2|〈b, v〉H |

2¯

#Bpdbq “ 0.

(1.203)


Combining this with Lemma 1.6.18 proves that S is a strongly continuous semigroup.In the next step we observe that Lemma 1.10.1 ensures that

sup

˜#

ˇ

ˇ

ˇ

ˇ

reλbt ´ 1´ λbts

λbt

ˇ

ˇ

ˇ

ˇ

2

: b P λ´1pCzt0uq, t P p0, 1s

+

Y t0u

¸

ď supzPC

ˇ

ˇ

ˇ

ˇ

ez ´ 1´ z

z

ˇ

ˇ

ˇ

ˇ

2

ď supzPC

„ˇ

ˇ

ˇ

ˇ

ez ´ 1

z

ˇ

ˇ

ˇ

ˇ

` 1

2

ď 2` 2 supzPC

ˇ

ˇ

ˇ

ˇ

ez ´ 1

z

ˇ

ˇ

ˇ

ˇ

2

ď 2` 2 supzPC

”?2` emaxtRepzq,0u

ı2

“ 2` 2”?

2` esupzPC maxtRepzq,0uı2

“ 2` 2”?

2` emaxtsupbPBRepλbq,0uı2

ă 8.

(1.204)

This and Lebesgue’s theorem of dominated convergence show that for all v P DpAqit holds that

lim suptŒ0

›

›

›

›

Stv ´ v

t´ Av

›

›

›

›

2

H

“ lim suptŒ0

˜

Stv ´ v ´ tAv2H

t2

¸

“ lim suptŒ0

˜

1

t2

ÿ

bPB

›

›

“

eλbt ´ 1´ λbt‰

〈b, v〉H b›

›

2

H

¸

“ lim suptŒ0

¨

˚

˝

ÿ

bPB,λb‰0

ˇ

ˇ

ˇ

ˇ

ˇ

“


λbt

ˇ

ˇ

ˇ

ˇ

ˇ

2

|λb 〈b, v〉H |2

˛

‹

‚

“

ż

λ´1pCzt0uq

lim suptŒ0

ˇ

ˇ

ˇ

ˇ

ˇ

“


λbt

ˇ

ˇ

ˇ

ˇ

ˇ

2

|λb 〈b, v〉H |2 #Bpdbq

“ÿ

bPλ´1pCzt0uq

lim suptŒ0

ˇ

ˇ

ˇ

ˇ

ˇ

8ÿ

k“2

rλbtsk´1

k!

ˇ

ˇ

ˇ

ˇ

ˇ

2

|λb 〈b, v〉H |2

ďÿ

bPB

|λb 〈b, v〉H |2 lim sup

tŒ0

«

|λbt|8ÿ

k“0

|λbt|k

k!

ff2

“ÿ

bPB

|λb 〈b, v〉H |2

„

lim suptŒ0

|λbt|2 e2|λbt|

“ 0.

(1.205)

Hence, we obtain that DpAq Ď DpGSq and GS|DpAq “ A. Next note that the fact that


@ z P Czt0u :

ˇ

ˇ

ˇ

ˇ

ez ´ 1

z´ 1

ˇ

ˇ

ˇ

ˇ

“

ˇ

ˇ

ˇ

ˇ

ˇ

«

8ÿ

k“1

zk´1

k!

ff

´ 1

ˇ

ˇ

ˇ

ˇ

ˇ

“

ˇ

ˇ

ˇ

ˇ

ˇ

«

8ÿ

k“0

zk

pk ` 1q!

ff

´ 1

ˇ

ˇ

ˇ

ˇ

ˇ

“

ˇ

ˇ

ˇ

ˇ

ˇ

8ÿ

k“1

zk

pk ` 1q!

ˇ

ˇ

ˇ

ˇ

ˇ

ď

8ÿ

k“1

|z|k

pk ` 1q!“ |z|

«

8ÿ

k“0

|z|k

pk ` 2q!

ff

ď |z| e|z|

(1.206)

ensures that for all v P DpGSq it holds that

ÿ

bPB

|λb 〈b, v〉H |2“

ÿ

bPB,λb‰0

|λb 〈b, v〉H |2“

ÿ

bPB,λb‰0

lim inftŒ0

ˇ

ˇ

ˇ

ˇ

eλbt ´ 1

λbt¨ λb ¨ 〈b, v〉H

ˇ

ˇ

ˇ

ˇ

2

“

ż

λ´1pCzt0uq

lim inftŒ0

ˇ

ˇ

ˇ

ˇ

eλbt ´ 1


ˇ

ˇ

ˇ

ˇ

2

#Bpdbq

ď lim inftŒ0

ż

λ´1pCzt0uq

ˇ

ˇ

ˇ

ˇ

eλbt ´ 1


ˇ

ˇ

ˇ

ˇ

2

#Bpdbq

“ lim inftŒ0

˜

1

t2

ÿ

bPB

ˇ

ˇ

“

eλbt ´ 1‰

〈b, v〉Hˇ

ˇ

2

¸

“ lim inftŒ0

˜

1

t2

ÿ

bPB

›

›

“

eλbt ´ 1‰

〈b, v〉H b›

›

2

H

¸

“ lim inftŒ0

ˆ

1

t2pSt ´ IdHqv

2H

˙

ď lim suptŒ0

˜

„

Stv ´ vHt

2¸

ď lim suptŒ0

˜

„

Stv ´ v ´ tGSvHt

` GSvH

2¸

“ GSv2H ă 8.

(1.207)

This establishes that DpGSq Ď DpAq. This together with the facts that DpAq ĎDpGSq and GS|DpAq “ A assures that GS “ A. The proof of Theorem 1.10.2 is thuscompleted.


1.10.1 A characterization for strongly continuous semigroupsgenerated by diagonal linear operators

Proposition 1.10.3 (Semigroups generated by diagonal operators). Let K P tR,Cu,let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, and let A : DpAq Ď H Ñ H be a diagonallinear operator. Then it holds that

supptRepλq : λ P σP pAqu Y t0uq ă 8 (1.208)

if and only if it holds that A is a generator of a strongly continuous semigroup.

Proof of Proposition 1.10.3. First, observe that Theorem 1.10.2 shows that the con-dition

supptRepλq : λ P σP pAqu Y t0uq ă 8 (1.209)

implies that A is a generator of a strongly continuous semigroup. In remainder ofthis proof we thus assume that A is the generator of a strongly continuous semigroupS : r0,8q Ñ LpHq and we assume w.l.o.g. that H ‰ t0u. Note that the assumptionthat A is a diagonal linear operator ensures that there exists an orthonormal basisB Ď H of H and a function λ : BÑ K such that it holds that

DpAq “

#

v P H :ÿ

bPB

|λb 〈b, v〉H |2ă 8

+

(1.210)

and@ v P DpAq : Av “

ÿ

bPB

λb 〈b, v〉H b. (1.211)

Next observe that the fact that GS “ A and Lemma 1.6.10 imply that for all b P B,t P r0,8q, v P H it holds

(i) that the functionr0,8q Q s ÞÑ Ssb P H (1.212)

is continuously differentiable,

(ii) that 〈v, S0pbq〉H “ 〈v, b〉H , and

(iii) thatddt〈v, Stpbq〉H “ 〈v,GSStpbq〉H “ λb 〈v, Stpbq〉H . (1.213)


This shows that for all b P B, v P H, t P r0,8q it holds that

〈v, Stb〉H “ eλbt 〈v, b〉H . (1.214)

Hence, we obtain that for all b P B, t P r0,8q it holds that

Stb “ eλbtb. (1.215)

This, in turn, ensures that

8 ą S1LpHq ě supbPB

S1bH “ supbPB

ˇ

ˇeλbˇ

ˇ “ supbPB

ˇ

ˇeRepλbqˇ

ˇ “ esupbPBRepλbq. (1.216)

This implies that supbPBRepλbq ă 8. The proof of Proposition 1.10.3 is thus com-pleted.

1.10.2 Contraction semigroups generated by diagonal linearoperators

Corollary 1.10.4 (Contraction semigroups generated by diagonal operators). LetK P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, and let A : DpAq Ď H Ñ Hbe a diagonal linear operator. Then it holds that A is a generator of a stronglycontinuous contraction semigroup if and only if it holds that

σP pAq Ď tz P C : Repzq ď 0u . (1.217)

Proof of Corollary 1.10.5. Throughout this proof we assume w.l.o.g. that H ‰ t0u.Next observe that

suptPr0,8q

›

›etA›

›

LpHq“ sup

tPr0,8q

supλPσP pAq

ˇ

ˇetλˇ

ˇ “ suptPr0,8q

supλPσP pAq

ˇ

ˇetRepλq¨ eit Impλq

ˇ

ˇ

“ suptPr0,8q

supλPσP pAq

etRepλq“ max

#

1, suptPr0,8q

supλPσP pAq

etRepλq

+

“ max

#

1, suptPr0,8q

etrsupλPσP pAqRepλqs

+

“

#

8 : supλPσP pAqRepλq ą 0

1 : supλPσP pAqRepλq ď 0.

(1.218)

This completes the proof of Corollary 1.10.4.


In the next result, Corollary 1.10.5, we specialise Corollary 1.10.4 to the caseof where the underlying field K is the field of real numbers. Corollary 1.10.5 is animmediate consequence from Corollary 1.10.4.

Corollary 1.10.5 (Contraction semigroups generated by diagonal operators). LetpH, 〈¨, ¨〉H , ¨Hq be an R-Hilbert space, and let A : DpAq Ď H Ñ H be a diagonal lin-ear operator. Then it holds that A is a generator of a strongly continuous contractionsemigroup if and only if it holds that

σP pAq Ď p´8, 0s. (1.219)

1.10.3 Smoothing effect of the semigroup

Proposition 1.10.6. Let K P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, letA : DpAq Ď H Ñ H be a symmetric linear operator with infpσP pAqq ą 0, let B Ď Hbe an orthonormal basis, let λ : BÑ K be a function which satisfies

DpAq “

#

w P H :ÿ

bPB

|λb 〈b, w〉H |2ă 8

+

(1.220)

and @ v P DpAq : Av “ř

bPB λb 〈b, v〉H b, and let pHr, 〈¨, ¨〉Hr , ¨Hrq, r P R, be afamily of interpolation spaces associated to A. Then

(i) it holds that B Ď pXrPRHrq,

(ii) it holds for all r P R that spanpBqHr“ Hr, and

(iii) it holds for all r P R that A´rpBq “

bpλbqr

P H : b P B(

is an orthonormal basisof Hr.

Proof of Proposition 1.10.6. Observe that Proposition 1.10.6 follows immediately fromDefinition 4.1.13, Definition 4.1.14, and Definition 4.1.25.

In the next result, Theorem 1.10.7, we establish a smoothing effect for stronglycontinuous semigroups generated by diagonal linear operators. We recall Remark 2.1.25for the formulation of Theorem 1.10.7.


Theorem 1.10.7 (Smoothing effect of semigroups generated by diagonal operators).Let K P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, let A : DpAq Ď H Ñ Hbe a symmetric linear operator with suppσP pAqq ă 0, let B Ď H be an orthonormalbasis, let λ : BÑ K be a function which satisfies

DpAq “

#

v P H :ÿ

bPB

|λb 〈b, v〉H |2ă 8

+

(1.221)


ÿ

bPB

λb 〈b, v〉H b, (1.222)

and let pHr, 〈¨, ¨〉Hr , ¨Hrq, r P R, be a family of interpolation spaces associated to´A. Then

(i) it holds that for all r P r0,8q that

suptPr0,8q

›

›p´tAqreAt›

›

LpHqď

”r

e

ır

ď rr ă 8, (1.223)

(ii) it holds for all t P p0,8q, r P p´8, 0q, v P H that eAtvH ď |r||r| tr vHr ă 8,

(iii) it holds for all t P p0,8q, r P R that

eAtpHrq Ď pXsPRHsq (1.224)

(cf. Proposition 2.1.23 and Item (ii)),

(iv) it holds for all t P r0,8q, r P p´8, 0q, v P H that etAvHr ď vHr , and

(v) it holds for all t P r0,8q, v P pYrPRHrq that

eAtv “ÿ

bPB

eλbt 〈b, v〉H b (1.225)

(cf. Proposition 2.1.23 and Item (iv)).

Proof of Theorem 1.10.7. Observe that Proposition 1.10.6 implies that for all r Pr0,8q it holds that

suptPr0,8q

›

›p´tAqreAt›

›

LpHq“ sup

tPr0,8q

supbPB

ˇ

ˇp´tλbqreλbt

ˇ

ˇ ď supxPp0,8q

„

xr

ex

ď

”r

e

ır

ă 8.

(1.226)The proof of Theorem 1.10.7 is thus completed.


Lemma 1.10.8. Let K P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, and letA : DpAq Ď H Ñ H be a symmetric diagonal linear operator with suppσP pAqq ă 0.Then

suprPr0,1s

suptPp0,8q

›

›p´tAq´r`

eAt ´ IdH˘›

›

LpHqď 1. (1.227)

Proof of Lemma 1.10.8. Observe that for all t P p0,8q, r P r0, 1s it holds that›

›p´tAq´r`

eAt ´ IdH˘›

›

LpHq“ sup

λPσP ptAqq

ˇ

ˇp´λq´r`

eλ ´ 1˘ˇ

ˇ

ď supxPp0,8q

„

p1´ e´xq

xr

ď 1.(1.228)


Exercise 1.10.9. Let T P p0,8q, r P r0, 1q, K P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be aK-Hilbert space, let A : DpAq Ď H Ñ H be a symmetric diagonal linear operatorwith suppσP pAqq ă 0, and let e : r0, T s Ñ H be a continuous function which satisfiesfor all t P r0, T s that

eptq “ ep0q `

ż t

0

p´Aqr ept´sqA epsq ds. (1.229)

Prove thatsuptPr0,T s

eptqH ď ep0qH ¨ E1´r

“

T 1´r‰

. (1.230)

1.10.4 Semigroup generated by the Laplace operator

Example 1.10.10 (Heat equation with Dirichlet boundary conditions). Let A : DpAq ĎL2pBorelp0,1q; |¨|Rq Ñ L2pBorelp0,1q; |¨|Rq be the Laplace operator with Dirichlet bound-ary conditions on L2pBorelp0,1q; |¨|Rq and let v : p0, 1q Ñ R be a twice continuouslydifferentiable function with vp0`q “ vp1´q “ 0. Then

(i) it holds that supλPσP pAq λ “ ´π2 ă 8,


(ii) it holds that A is the generator of a strongly continuous semigroup (cf. Theo-rem 1.10.2 and Item (i)), and

(iii) it holds that the function u : r0,8q ˆ p0, 1q Ñ R with @ pt, xq P r0,8q ˆp0, 1q : upt, xq “ peAtvqpxq satisfies for all pt, xq P p0,8qˆp0, 1q that u|p0,8qˆp0,1q PC2pp0,8q ˆ p0, 1q,Rq and

B

Btupt, xq “ B2

Bx2upt, xq, up0, xq “ vpxq, upt, 0`q “ upt, 1´q “ 0 (1.231)

(cf. Theorem 1.10.7, Item (ii), and Lemma 1.6.10).

Exercise 1.10.11 (Laplacian on L2pBorelp0,1q; |¨|Rq). Let A : DpAq Ď L2pBorelp0,1q;|¨|Rq Ñ L2pBorelp0,1q; |¨|Rq be the linear operator which satisfies for all v P H2pp0, 1q,Rq

that DpAq “ H2pp0, 1q,Rq and Av “ v2. Is A a generator of a strongly continuoussemigroup?

Example 1.10.12 (Laplacian on L2pBorelp0,1q; |¨|Rq). Let A : DpAq Ď L2pBorelp0,1q; |¨|RqÑ L2pBorelp0,1q; |¨|Rq be the linear operator which satisfies for all v P H2pp0, 1q,Rqthat DpAq “ H2pp0, 1q,Rq and Av “ v2. Then

(i) it holds that rp1qxPp0,1qsBorelp0,1q,BpRq P Kernp0´ Aq “ KernpAq,

(ii) it holds that 0 P σP pAq,

(iii) it holds for all n P N, x P p0, 1q that d2

dx2 sinpnπxq “ ń2π2 sinpnπxq,

(iv) it holds for all n P N that rpsinpnπxqqxPp0,1qsBorelp0,1q,BpRq P Kernpń2π2 ´ Aq,

(v) it holds for all n P N that ń2π2 P σP pAq,

(vi) it holds thatş1

01 ¨ sinpπxq dx “ r´ cospπxqsx“1

x“0 “ 1´ cospπq ‰ 0,

(vii) it does not hold that for all v P Kernp0 ´ Aq, w P Kernp´π2 ´ Aq it holds thatş1

0vpxqwpxq dx “ 0, and

(viii) it holds that A is not a diagonal linear operator (cf. Proposition 4.1.8 andDefinition 1.9.1).

Chapter 2

Nonlinear functions and nonlinearspaces

Most of this chapter is based on Da Prato & Zabczyk [3] and Prévôt & Röckner [6].

2.1 Continuous functions

2.1.1 Topological spaces

2.1.1.1 Topological spaces induced by distance-type functions

Proposition 2.1.1 (Topology induced by a function). Let E be a set, let T Ď R bea set, and let d : E ˆ E Ñ T be a function. Then it holds that the set

"

A P PpEq :´

@ v P A :“

D ε P p0,8q : tu P E : dpv, uq ă εu Ď A‰

¯

*

(2.1)

is a topology on E.

Proof of Proposition 2.1.1. Throughout this proof let E Ď PpEq be the set given by

E “"

A P PpEq :´

@ v P A :“


¯

*

. (2.2)

First, we observe thatH, E P E . Next we note that for all A Ď E and all v P rYAPAAsthere exists a set A P A and a real number ε P p0,8q such that v P A and such thattu P E : dpv, uq ă εu Ď A. In particular, this implies that for all A Ď E and all

61

62 CHAPTER 2. NONLINEAR FUNCTIONS AND NONLINEAR SPACES

v P rYAPAAs there exists a real number ε P p0,8q such that tu P E : dpv, uq ă εu ĎrYAPAAs. Hence, we obtain that for all A Ď E it holds that

rYAPAAs P E . (2.3)

In the next step we observe that for all A,B P E and all v P pAXBq there existsreal numbers εA, εB P p0,8q such that

tu P E : dpv, uq ă εAu Ď A and tu P E : dpv, uq ă εBu Ď B. (2.4)

Hence, we obtain that for all A,B P E and all v P pAXBq there exists real numbersεA, εB P p0,8q such that

u P E : dpv, uq ă mintεA, εBu(

Ď pAXBq . (2.5)

This proves that for all A,B P E it holds that pAXBq P E . The proof of Proposi-tion 2.1.1 is thus completed.

Proposition 2.1.1 above ensures that the designation in the next definition isreasonable.

Definition 2.1.2 (Topology induced by a function). Let E be a set, let T Ď R be aset, and let d : E ˆ E Ñ T be a function. Then we denote by τpdq Ď PpEq the setgiven by

τpdq “

"

A P PpEq :´

@ v P A :“


¯

*

(2.6)

and we call τpdq the topology induced by d.

Lemma 2.1.3 (Balls are open). Let E be a set, let T Ď R be a set, let d : E ˆE ÑT be a function which satisfies @x, y, z P E : dpx, zq ď dpx, yq ` dpy, zq, and letε P p0,8q, v P E. Then it holds that

tu P E : dpv, uq ă εu P τpdq. (2.7)

Proof of Lemma 2.1.3. First, observe that for all x P tu P E : dpv, uq ă εu, y P tu PE : dpx, uq ă ε´ dpv, xqu it holds that

dpv, yq ď dpv, xq ` dpx, yq ă dpv, xq ` rε´ dpv, xqs “ ε. (2.8)

2.1. CONTINUOUS FUNCTIONS 63

This ensures that for all x P tu P E : dpv, uq ă εu it holds that

tu P E : dpx, uq ă ε´ dpv, xqu Ď tu P E : dpv, uq ă εu (2.9)

Hence, we obtain that for all x P tu P E : dpv, uq ă εu there exists a real numberδ P p0,8q such that

tu P E : dpx, uq ă δu Ď tu P E : dpv, uq ă εu . (2.10)

This completes the proof of Lemma 2.1.3.

Proposition 2.1.4 (Convergence in the induced topology). Let E be a set, letd : E ˆ E Ñ r0,8q be a function which satisfies @x P E : dpx, xq “ 0 and @x, y, z PE : dpx, zq ď dpx, yq ` dpy, zq, and let e : N0 Ñ E be a function. Then it holds that

lim supnÑ8

d`

ep0q, epnq˘

“ 0 (2.11)

if and only if for every A P τpdq with ep0q P A there exists a natural number N P N

such that for all n P tN,N ` 1, . . . u it holds that epnq P A.

Proof of Proposition 2.1.4. First of all, recall that lim supnÑ8 d`

ep0q, epnq˘

“ 0 ifand only if

@ ε P p0,8q : DN P N : @n P tN,N ` 1, . . . u : d`

ep0q, epnq˘

ă ε. (2.12)

Hence, we obtain that lim supnÑ8 d`

ep0q, epnq˘

“ 0 if and only if for all ε P p0,8qthere exists a natural number N P N such that for all n P tN,N ` 1, . . . u it holdsthat epnq P tu P E : dpep0q, uq ă εu. This and Lemma 2.1.3 complete the proof ofProposition 2.1.4.

2.1.1.2 Semi-metric spaces

Definition 2.1.5 (Semi-metric). We say that d is a semi-metric on E if and only ifit holds that d PMpE ˆE, r0,8qq is a function from E ˆE to r0,8q which satisfiesfor all x, y, z P E that

(i) dpx, xq “ 0,

(ii) dpx, yq “ dpy, xq, and

(iii) dpx, zq ď dpx, yq ` dpy, zq.


Definition 2.1.6 (Semi-metric). We say that d is a semi-metric if and only if thereexists a set E such that d is a semi-metric on E.

Definition 2.1.7 (Semi-metric spaces). We say that E is a semi-metric space if andonly if there exist d and E such

(i) that d is a semi-metric on E and

(ii) that E “ pE, dq.

Definition 2.1.8 (Globally bounded sets). We say that A is d-globally bounded (wesay that A is globally bounded) if and only if there exists an E such that it holds

(i) that d is a semi-metric on E,

(ii) that A Ď E, and

(iii) that sup`

t0u Y tdpa, bq : a, b P Au˘

ă 8.

Lemma 2.1.9 (Globally bounded sets). Let pE, dq be a semi-metric space and letA Ď E be a non-empty set. Then the following three statements are equivalent:

(i) It holds that A is d-globally bounded.

(ii) It holds that @ e P E : supaPA dpa, eq ă 8.

(iii) It holds that D e P E : supaPA dpa, eq ă 8.

Proof of Lemma 2.1.9. First, observe that for all e P E it holds that

supaPA

dpa, eq ď supa,bPA

dpa, bq. (2.13)

This ensures that ((i) ñ (ii)). Next note that the fact that E ‰ H shows that ((ii)ñ (iii)). It thus remains to prove that ((iii)ñ (i)). For this observe that the triangleinequality ensures that for all e P E it holds that

supa,bPA

dpa, bq ď supa,bPA

rdpa, eq ` dpe, bqs “ 2

„

supaPA

dpa, eq

. (2.14)

This establishes that ((iii) ñ (i)). The proof of Lemma 2.1.9 is thus completed.


Definition 2.1.10 (Globally bounded functions). We say that f is a d-globallybounded function (we say that f is a globally bounded function) if and only if itholds

(i) that f is a function,

(ii) that d is a semi-metric on codompfq, and

(iii) that impfq is a d-globally bounded set.

2.1.2 Continuity properties of functions

2.1.2.1 Uniform continuity

Definition 2.1.11 (Uniform continuity). We say that f is d/δ-uniformly continuous(we say that f is uniformly continuous) if and only if it holds


(ii) that d is a semi-metric on dompfq,

(iii) that δ is a semi-metric on codompfq, and

(iv) that @ ε P p0,8q : D δ P p0,8q : @x, y P dompfq :`

pdpx, yq ă δq ñ pδpfpxq, fpyqq ăεq˘

.

2.1.2.2 Hölder continuity

Definition 2.1.12 (α-Hölder continuity). We say that f is d/δ-α-Hölder continuous(we say that f is α-Hölder continuous) if and only if it holds


(ii) that d is a semi-metric on dompfq,

(iii) that δ is a semi-metric on codompfq,

(iv) that α P p0,8q, and

(v) that DC P R : @x, y P dompfq : δpfpxq, fpyqq ď C |dpx, yq|α.


Definition 2.1.13 (Hölder continuity). We say that f is d/δ-Hölder continuous(we say that f is Hölder continuous) if and only if there exists an α such that f isd/δ-α-Hölder continuous.

Definition 2.1.14 (Hölder constants). Let pE, dq and pF, δq be semi-metric spacesand let α P p0,8q. Then we denote by

|¨|Cαd,δ: MpE,F q Ñ r0,8s (2.15)

the function which satisfies for all f PMpE,F q that

|f |Cαd,δ“ sup

ˆ

t0u Y

"

δpfpxq, fpyqq

|dpx, yq|αP p0,8s : px, y P E, δpfpxq, fpyqq ą 0q

*˙

.

(2.16)

Lemma 2.1.15 (Pointwise Lipschitz constants). Let pV, ¨V q and pW, ¨W q be normedR-vector spaces, let U Ď V be an open set, let v P U , and let f : U Ñ W be a functionwhich is Fréchet differentiable at v. Then

lim suphŒ0

„

fpv ` hq ´ fpvqVhV

“ limεŒ0

suphPV zt0u,hV ďε

„

fpv ` hq ´ fpvqVhV

“ f 1pvqLpV,W q .

(2.17)

Proof of Lemma 2.1.15. Observe that the triangle inequality ensures that

lim suphŒ0

„

fpv ` hq ´ fpvqVhV

“ lim suphŒ0

„

fpv ` hq ´ fpvq ´ f 1pvqh` f 1pvqhVhV

ď lim suphŒ0

„

fpv ` hq ´ fpvq ´ f 1pvqhVhV

`f 1pvqhVhV

ď lim suphŒ0

„

fpv ` hq ´ fpvq ´ f 1pvqhVhV

` f 1pvqLpV,W q

“ f 1pvqLpV,W q .

(2.18)



2.1.3 Modulus of continuity

Definition 2.1.16 (Modulus of continuity). Let pE, dq and pF, δq be semi-metricspaces and let f : E Ñ F be a function. Then we denote by

wd,δf : r0,8s Ñ r0,8s (2.19)

the function which satisfies for all h P r0,8s that

wd,δf phq “ sup´

t0u Y!

δ`

fpxq, fpyq˘

P r0,8q :“

x, y P E with dpx, yq ď h‰

)¯

(2.20)

and we call wd,δf the d/δ-modulus of continuity of f (we call wd,δf the modulus ofcontinuity of f).

2.1.3.1 Properties of the modulus of continuity

Lemma 2.1.17 (Properties of the modulus of continuity). Let pE, dq and pF, δq besemi-metric spaces and let f : E Ñ F be a function. Then

(i) it holds that wd,δf is non-decreasing,

(ii) it holds that f is d/δ-uniformly continuous if and only if limhŒ0wd,δf phq “ 0,

(iii) it holds that f is a δ-globally bounded function if and only if wd,δf p8q ă 8,

(iv) it holds for all x, y P E that δ`

fpxq, fpyq˘

ď wd,δf`

dpx, yq˘

, and

(v) it holds for all r P p0,8q that |f |Crd,δ “ suphPp0,8q`

h´rwd,δf phq˘

.

Proof of Lemma 2.1.17. First, observe that (i), (ii), and (iv) are an immediate con-sequence of the definition of wdE ,dFf . Next note that (iii) follows directly fromLemma 2.1.9. It thus remains to prove (v). For this observe that (iv) ensuresthat for all r P p0,8q, x, y P E with dpx, yq ą 0 ă δpfpxq, fpyqq it holds that

δpfpxq, fpyqq

|dpx, yq|rďwd,δf pdpx, yqq

|dpx, yq|rď sup

hPp0,8q

˜

wd,δf phq

hr

¸

. (2.21)

Next note that for all r P p0,8q, x, y P E with dpx, yq “ 0 ă δpfpxq, fpyqq it holdsthat

suphPp0,8q

˜

wd,δf phq

hr

¸

ě suphPp0,8q

ˆ

δpfpxq, fpyqq

hr

˙

“ 8. (2.22)


Combining this with (2.21) ensures that for all r P p0,8q it holds that |f |Crd,δ ď

suphPp0,8qph´rwd,δf phqq. Moreover, note that the fact that

@ r P p0,8q, x, y P E : δpfpxq, fpyqq ď |f |Crd,δ|dpx, yq|r (2.23)

proves that

suphPp0,8q

˜

wd,δf phq

hr

¸

ď suphPp0,8q

˜

|f |Crd,δhr

hr

¸

“ |f |Crd,δ. (2.24)


Exercise 2.1.18. Prove or disprove the following statement: For all semi-metricspaces pE, dq and pF, δq and all functions f : E Ñ F it holds that f is d/δ-uniformlycontinuous if and only if

limhŒ0

wd,δf phq “ wd,δf p0q. (2.25)

Exercise 2.1.19. Give an example of a metric space pE, dq such that for all h Pr0,8s it holds that

wd,didEphq “

#

0 : h P r0, 1q

1 : h P r1,8s. (2.26)

Prove that your metric space has the desired properties.

Lemma 2.1.20 (Uniformly continuous functions). Let pE, dq and pF, δq be semi-metric spaces, let f : E Ñ F be a uniformly continuous function, and let penqnPN Ď Ebe a Cauchy sequence in pE, dq. Then it holds that fpenq P F , n P N, is a Cauchysequence in pF, δq.

Proof of Lemma 2.1.20. The assumption that penqnPN is a Cauchy sequence in pE, dqand the assumption that f is uniformly continuous imply that

lim supNÑ8

supn,mPtN,N`1,... u

δpfpenq, fpemqq ď lim supNÑ8

supn,mPtN,N`1,... u

wd,δf`

dpen, emq˘

ď lim supNÑ8

wd,δf`

supn,mPtN,N`1,... u dpen, emq˘

“ 0.(2.27)

This shows that fpenq P F , n P N, is a Cauchy sequence. The proof of Lemma 2.1.20is thus completed.


2.1.3.2 Convergence of the modulus of continuity

The next result, Lemma 2.1.21, provides an upper bound for the modulus of conti-nuity of the point limit of a sequence of functions. Lemma 2.1.21 is, e.g., related toItem (i) in Corollary 2.10 in Cox et al. [2].

Lemma 2.1.21 (Convergence of the modulus of continuity). Let pE, dq and pF, δqbe semi-metric spaces and let fn : E Ñ F , n P N0, be functions which satisfy @ e PE : lim supnÑ8 δpf0peq, fnpeqq “ 0. Then

(i) it holds for all h P r0,8s that wd,δf0phq ď lim supnÑ8w

d,δfnphq and

(ii) it holds for all r P p0,8q that |f0|Crd,δď lim supnÑ8 |fn|Crd,δ

.

Proof of Lemma 2.1.21. Observe that for all h P r0,8s it holds that

wd,δf0phq “ suppt0u Y tδpf0pxq, f0pyqq P r0,8q : rx, y P E with dEpx, yq ď hsuq

ď sup

˜

t0u Y

#

lim supnÑ8“

δpf0pxq, fnpxqq ` δpfNpxq, fNpyqq

` δpfnpyq, f0pyqq‰

P r0,8q : rx, y P E with dpx, yq ď hs

+¸

ď sup

ˆ

t0u Y

"

lim supnÑ8

δpfnpxq, fnpyqq P r0,8q : rx, y P E with dpx, yq ď hs

*˙

ď lim supnÑ8wd,δfnphq.

(2.28)


2.1.4 Extensions of uniformly continuous functions

Lemma 2.1.22 (Uniqueness of continuous extensions). Let pE, dq be a semi-metricspace, let pE , δq be a metric space, let A P PpEq, and let f, g P CpA, Eq be functionswhich satisfy

f |A “ g|A. (2.29)

Then it holds that f “ g.

Proof of Lemma 2.1.22. Throughout this proof assume w.l.o.g. that A ‰ H, let v PA, and let xn P A, n P N, be a sequence which satisfies

lim supnÑ8

dpxn, vq “ 0. (2.30)


The assumption that f, g P CpA, Eq hence ensures that

lim supnÑ8

δpfpxnq, fpvqq “ lim supnÑ8

δpgpxnq, gpvqq “ 0. (2.31)

The triangle inequality and the assumption that f |A “ g|A therefore show that

δpfpvq, gpvqq “ lim supnÑ8

δpfpvq, gpvqq

ď lim supnÑ8

rδpfpvq, fpxnqq ` δpfpxnq, gpxnqq ` δpgpxnq, gpvqqs

ď

„

lim supnÑ8

δpfpvq, fpxnqq

`

„

lim supnÑ8

δpfpxnq, gpxnqq

`

„

lim supnÑ8

δpgpxnq, gpvqq

“

„

lim supnÑ8

δpfpvq, fpxnqq

`

„

lim supnÑ8

δpgpxnq, gpvqq

“ 0.

(2.32)

This and the fact that pE , δq is a metric space establishes that

fpvq “ gpvq. (2.33)


Proposition 2.1.23 (Extension of uniformly continuous functions). Let pE, dq bea semi-metric space, let pF, δq be a complete metric space, let A P PpEq, and letf : AÑ F be a d|AÂ/δ-uniformly continuous function. Then

(i) there exists a unique f P CpA, F q with the property that f |A “ f ,

(ii) it holds for all h P r0,8s that

wd|AÂ,δf phq ď w

d|AÂ,δ

fphq ď lim

εŒ0wd|AÂ,δf ph` εq, (2.34)

and

(iii) it holds that f is dAEÂE/δ-uniformly continuous.

Proof of Proposition 2.1.23. The uniqueness of f is an immediate consequence ofLemma 2.1.22. It thus remains to prove the existence of a function f with thedesired properties. For this observe that for all x P A, penqnPN Ď A, penqnPN Ď Awith

lim supnÑ8

dpen, xq “ lim supnÑ8

dpen, xq “ 0 (2.35)


it holds that

lim supnÑ8

δ`

fpenq, fpenq˘

ď lim supnÑ8

wd,δf`

dpen, enq˘

“ 0. (2.36)

This, Lemma 2.1.20, and the assumption that pF, δq is complete imply that thereexist a function f : A Ñ F which satisfies for all x P A and all penqnPN Ď A withlim supnÑ8 dpen, xq “ 0 that

lim supnÑ8

δ`

fpenq, fpxq˘

“ 0. (2.37)

We observe that the continuity of f implies that for all x P A it holds that fpxq “fpxq. In the next step we show (2.34). The first inequality in (2.34) is clear. To provethe second inequality in (2.34) let h P r0,8s and let x0, y0 P A with dpx0, y0q ď h.Then there exist sequences pxnqnPN Ď A and pynqnPN Ď A with the property thatlimnÑ8 xn “ x0 and limnÑ8 yn “ y0. This implies that for all ε P p0,8q it holds that

δ`

fpx0q, fpy0q˘

“ δ´

limnÑ8

fpxnq, limnÑ8

fpynq¯

“ limnÑ8

δpfpxnq, fpynqq

“ lim infnÑ8

δpfpxnq, fpynqq ď lim infnÑ8

wd,δf`

dpxn, ynq˘

ď wd,δf`

dpx0, y0q ` ε˘

.

(2.38)

This proves the second inequality in (2.34). The second inequality in (2.34), inturn, shows that f is uniformly continuous. The proof of Proposition 2.1.23 is thuscompleted.

Exercise 2.1.24. Prove or disprove the following statement: For every semi-metricspace pE, dq, every complete semi-metric space pF, δq, every set A Ď E, and everyd|AÂ/δ-uniformly continuous function f : AÑ F it holds that

wd|AÂ,δf “ w

d|AÂ,δ

f. (2.39)

Remark 2.1.25. Let pE, dq be a semi-metric space, let pF, δq be a complete metricspace, let A Ď E be a subset of E, and let f : A Ñ F be a d|AÂ/δ-uniformlycontinuous function. Proposition 2.1.23 then proves that there exists a unique f PCpA, F q with

f |A “ f. (2.40)

In the following we often write, for simplicity of presentation, f instead of f .


Lemma 2.1.26. Let K P tR,Cu, let pV, ¨V q and pW, ¨W q be semi-normed K-vector spaces, and let A : V Ñ W be a linear function. Then the following threestatement are equivalent:

(i) It holds that A is continuous in 0.

(ii) It holds that A is continuous.

(iii) It holds that A is uniformly continuous.

The proof of Lemma 2.1.26 is clear and therefore omitted.

2.2 Measurable functionsWe first recall the notion of a measurable function.

Definition 2.2.1 (Measurable function). We say that X is an A/A-measurablefunction (we say that X is a measurable function, we say that X is A/A-measurable)if and only if it holds

(i) that X is a function,

(ii) that A is a sigma-algebra on domainpXq,

(iii) that A is a sigma-algebra on codomainpXq, and

(iv) that @A P A : X´1pAq “ tx P domainpXq : Xpxq P Au P A.

Definition 2.2.2 (Set of all measurable functions). Let pΩ1,F1q and pΩ2,F2q bemeasurable spaces. Then we denote by MpF1,F2q the set of all F1/F2-measurablefunctions.

Definition 2.2.3 (Borel sigma-algebra). Let pE, Eq be a topological space. Then wedenote by BpEq the set given by BpEq “ σEpEq and we call BpEq the Borel sigma-algebra on pE, Eq.

2.2. MEASURABLE FUNCTIONS 73

2.2.1 Nonlinear characterization of the Borel sigma-algebra

Proposition 2.2.5 below demonstrates that if pE, dq is a metric space, then BpEqis the smallest sigma-algebra with respect to which every continuous real-valuedfunction is measurable. We refer to the statement of Proposition 2.2.5 as nonlinearcharacterization of the Borel sigma-algebra. In the proof of Proposition 2.2.5 we usethe following notation.

Definition 2.2.4 (Distance of sets). Let pE, dq be a metric space. Then we denoteby distd : PpEqˆPpEq Ñ r0,8s the function which satisfies for all A,B P PpEq that

distdpA,Bq “

#

infaPA infbPB dpa, bq : A ‰ H and B ‰ H8 : else

. (2.41)

We now present the promised nonlinear characterization result for Borel sigma-algebras for metric spaces.

Proposition 2.2.5 (Nonlinear characterization of the Borel sigma-algebra). LetpE, dq be a metric space. Then

BpEq “ σE`

pϕqϕPCpE,Rq˘

“ σEpϕ : ϕ P CpE,Rqq

“ σE`

ϕ´1pAq P PpEq : ϕ P CpE,Rq, A P BpRq

(˘

.(2.42)

Proof of Proposition 2.2.5. First of all, observe that for every ϕ P CpE,Rq it holdsthat ϕ is BpEq/BpRq-measurable. Hence, we obtain that

BpEq Ě σE`


(˘

. (2.43)

It thus remains to prove that

BpEq Ď σE`


(˘

. (2.44)

For this observe that by definition it holds that

BpEq “ σEptA P PpEq : A is an open set in pE, dquq . (2.45)

It thus remains to prove that

tA P PpEq : A is an open set in pE, dquĎ σE

`


(˘

.(2.46)


For this let B Ĺ E be an open set in pE, dq and let ψ : E Ñ R be the function whichsatisfies for all x P E that

ψpxq “ distdptxu, EzBq. (2.47)

Observe that ψ P CpE,Rq. This implies that

B “ ψ´1pp0,8qq P σE

`


(˘

. (2.48)

The proof of Proposition 2.2.5 is thus completed.

Exercise 2.2.6. Let pΩ,Fq be a measurable space, let pE, dq be a metric space, andlet f : Ω Ñ E be a function. Prove that f is F/BpEq-measurable if and only if itholds for all ϕ P CpE,Rq that ϕ ˝ f is F/BpRq-measurable.

2.2.2 Pointwise limits of measurable functions

Lemma 2.2.7. Let pΩ,Fq be a measurable space, let Y : Ω Ñ R be a function, andlet Xn : Ω Ñ R, n P N, be a sequence of F/BpRq-measurable functions which satisfiesfor all ω P Ω that

Y pωq “ supnPN

Xnpωq. (2.49)

Then it holds that Y is F/BpRq-measurable.

Proof of Lemma 2.2.7. Note that for all c P R it holds that

tY ď cu “

"

supnPN

Xn ď c

*

“č

nPN

tXn ď cul jh n

PF

P F . (2.50)


Lemma 2.2.8. Let pΩ,Fq be a measurable space, let Y : Ω Ñ R be a function,and let Xn : Ω Ñ R, n P N, be a sequence of F/BpRq-measurable functions whichsatisfies for all ω P Ω that lim supnÑ8 |Xnpωq ´ Y pωq| “ 0. Then it holds that Y isF/BpRq-measurable.

2.3. STRONGLY MEASURABLE FUNCTIONS 75

Proof of Lemma 2.2.8. Note that Lemma 2.2.7 implies that for all c P R it holdsthat

tY ě cu “!

limnÑ8

Xn ě c)

“

"

lim supnÑ8

Xn ě c

*

“

#

limnÑ8

«

supmPtn,n`1,... u

Xm

ff

ě c

+

“č

nPN

#«

supmPtn,n`1,... u

Xm

ff

ě c

+

l jh n

PF

P F . (2.51)


The next corollary is an immediate consequence of Exercise 2.2.6 and Lemma 2.2.8;see, e.g., Proposition E.1 in [1] and Proposition A.1.3 in Prévôt & Röckner [6].

Corollary 2.2.9. Let pΩ,Fq be a measurable space, let pE, dq be a metric space, andlet f : Ω Ñ E be a function. Then the following two statements are equivalent:

(i) It holds that f is F/BpEq-measurable.

(ii) There exists a sequence gn : Ω Ñ E, n P N, of F/BpEq-measurable functionswhich satisfies for all ω P Ω that

lim supnÑ8

dpfpωq, gnpωqq “ 0. (2.52)

2.3 Strongly measurable functions

2.3.1 Simple functions

The idea of the Lebesgue integral for real valued functions is to approximate thefunction by suitable simpler functions and then to define the Lebesgue integral ofthe “complicated” function as the limit of the integrals of the simpler functions. Toperform this procedure we use the following definition.

Definition 2.3.1 (Simple functions). We say that f is an F1/F2-simple function(we say that f is F1/F2-simple) if and only if there exist Ω1 and Ω2 such that itholds

(i) that pΩ1,F1q and pΩ2,F2q are measurable spaces,

(ii) that f is an F1/F2-measurable function, and

(iii) that fpΩ1q is a finite set.


2.3.2 Separability

(Unfortunately) Measurable functions can, in general, not be approximated pointwise(see (2.52) in Corollary 2.2.9) by simple functions; see Theorem 2.3.10 below fordetails. To overcome this difficulty, we introduce the notion of a strongly measurablefunction; see Definition 2.3.6 below. In this notion the following definition is used.

Definition 2.3.2 (Separability). We say that E is a separable topological space (wesay that E is separable) if and only if there exist E, E, and F such that it holds

(i) that E “ pE, Eq is a topological space,

(ii) that F is at most countable, and

(iii) that E X F E“ E.

A topological space that is not separable is in a certain sense extremely large.This, in turn, can cause several serious difficulties in the analysis of such spaces. Anexample of a non-separable topological space can be found below. The next lemmaprovides an example for a separable topological space.

Lemma 2.3.3. Let a, b P R with a ă b. Then pCpra, bs,Rq, ¨Cpra,bs,Rqq is separable.

Proof of Lemma 2.3.3. Observe that the set"

f P Cpra, bs,Rq :

ˆ

Dn P N0 : Dλ0, . . . , λn P Q : @x P ra, bs : fpxq “nř

k“0

λkxk

˙*

(2.53)is a countable dense subset of Cpra, bs,Rq. The proof of Lemma 2.3.3 is thus com-pleted.

In the next lemma we provide a further simple example for a separable topologicalspace.

Lemma 2.3.4 (Trivial topology). Let X be a set. Then it holds that the pairpX, tX,Huq is a separable topological space.

Proof of Lemma 2.3.4. Throughout this proof assume w.l.o.g. that X ‰ H and letx P X. Next observe that txu is a finite dense subset of X. The proof of Lemma 2.3.4is thus completed.


Subspaces of separable metric spaces are separable too. This is the subject of thenext lemma.

Lemma 2.3.5. Let pE, dq be a separable metric space and let F Ď E. Then it holdsthat

pF, τpd|FˆF qq (2.54)

is separable.

Proof of Lemma 2.3.5. W.l.o.g. we assume that F ‰ H. Let penqnPN Ď E be asequence of elements in E such that the set ten P E : n P Nu is dense in E. In thenext step let pfnqnPN Ď F be a sequence of elements in F such that for all n P N itholds that

dpfn, enq ď

#

0 : en P F

distdpF, tenuq `1

2n: en R F

. (2.55)

Next observe that for all v P F ztem P E : m P Nu, n P N it holds that

distdptvu, tf1, f2, . . . uq ď distdptvu, tfn, fn`1, . . . uq

“ infmPtn,n`1,... u

dpv, fmq

ď infmPtn,n`1,... u

rdpv, emq ` dpem, fmqs


„

dpv, emq ` distdpF, temuq `1

2m


„

2 dpv, emq `1

2m

ď 2

„

infmPtn,n`1,... u

dpv, emq

`1

2n

“ 2 distdptvu, ten, en`1, . . . uq `1

2n“

1

2n.

(2.56)

Combining this with the fact that @ v P FXtem : m P Nu : distdptvu, tf1, f2, . . . uq “ 0ensures that the set tfn P F : n P Nu is dense in F . The proof of Lemma 2.3.5 isthus completed.


2.3.3 Strongly measurable functions

Definition 2.3.6 (Strongly measurable functions). We say that f is a strongly F/E-measurable function (we say that f is strongly F/E-measurable, we say that f is astrongly measurable function, we say that f is strongly measurable) if and only ifthere exist E and d such that it holds

(i) that E “ pE, dq is a metric space,

(ii) that f is an F/BpEq-measurable function, and

(iii) that pimpfq, τpd|impfqˆimpfqqq is a separable topological space.

Lemma 2.3.5 shows that for every measurable space pΩ,Fq, every separable metricspace pE, dq, and every F/BpEq-measurable function f : Ω Ñ E it holds that f isalso strongly F/pE, dq-measurable.

Exercise 2.3.7. Provide examples of a measurable space pΩ,Fq, a metric spacepE, dEq, and an F/BpEq-measurable function f : Ω Ñ E such that f is not stronglyF/pE, dEq-measurable. Prove that f is F/BpEq-measurable but not strongly F/pE, dEq-measurable.

2.3.4 Pointwise approximations of strongly measurable func-tions

As mentioned above, measurable functions can, in general, not be approximatedpointwise by simple functions. However, strongly measurable functions can be ap-proximated pointwise by simple functions. This is the subject of the Theorem 2.3.10below (cf., e.g., Lemma 1.1 in Da Prato & Zabczyk [3] and Lemma A.1.4 in Prévôt& Röckner [6]). In the proof of Theorem 2.3.10 the following two lemmas are used.

Lemma 2.3.8 (Projections in metric spaces). Let pE, dq be a metric space, let n P N,e1, . . . , en P E, and let Ppe1,...,enq : E Ñ E be the function which satisfies for all x P Ethat

Ppe1,...,enqpxq “ emintkPt1,2,...,nu : dpek,xq“distdptxu,te1,...,enuqu. (2.57)Then

(i) it holds that Ppe1,...,enq is BpEq/PpEq-measurable and

(ii) it holds for all x P E that

dpx, Ppe1,...,enqpxqq “ distdptxu, te1, . . . , enuq. (2.58)


Proof of Lemma 2.3.8. Identity (2.58) is an immediate consequence of (2.57). LetD “ pD1, . . . , Dnq : E Ñ Rn be the function with the property that for all x P E itholds that

Dpxq “ pD1pxq, . . . , Dnpxqq “ pdpx, e1q, . . . , dpx, enqq . (2.59)

Observe that D is continuous and hence that D is BpEq/BpRnq-measurable. Thisimplies that for all k P t1, 2, . . . , nu with ek R tel P E : l P N X r0, k ´ 1su it holdsthat

P´1pe1,...,enq

ptekuq “

x P E : Ppe1,...,enqpxq “ ek(

“

!

x P E : k “ min

l P t1, 2, . . . , nu : dpel, xq “ distdptxu, te1, . . . , enuq(

)

“

"

x P E : k “ min

"

l P t1, 2, . . . , nu : Dlpxq “ minuPt1,...,nu

Dupxq

**

“

"

x P E :

ˆ

Dkpxq ď minuPt1,...,nu

Dupxq and Dkpxq ă minuPt1,...,k´1u

Dupxq

˙*

“

"

x P E :

ˆ

@ l P t1, . . . , k ´ 1u : Dkpxq ă Dlpxq and@ l P t1, . . . , nu : Dkpxq ď Dlpxq

˙*

“

»

—

—

—

–

k´1č

l“1

tx P E : Dkpxq ă Dlpxqul jh n

PBpEq

fi

ffi

ffi

ffi

fl

č

»

—

—

—

–

nč

l“1

tx P E : Dkpxq ď Dlpxqul jh n

PBpEq

fi

ffi

ffi

ffi

fl

P BpEq.

(2.60)

Hence, we obtain that for all f P te1, . . . , enu it holds that

P´1pe1,...,enq

ptfuq P BpEq. (2.61)

This, in turn, implies that for all A P PpEq it holds that

P´1pe1,...,enq

pAq “ P´1pe1,...,enq

`

AX te1, . . . , enu˘

“ YfPAXte1,...,enu P´1pe1,...,enq

ptfuql jh n

PBpEq

P BpEq. (2.62)



Lemma 2.3.9. Let pΩ,Fq be a measurable space, let pE, dq be a metric space, letf : Ω Ñ E be a function, and let gn : Ω Ñ E, n P N, be a sequence of stronglyF/pE, dq-measurable functions which satisfy for all ω P Ω that

lim supnÑ8


Then it holds that f is a strongly F/pE, dq-measurable function.

Proof of Lemma 2.3.9. Corollary 2.2.9 ensures that f is F/BpEq-measurable. It thusremains to prove that fpΩq is separable. This follows from Lemma 2.3.5 and from thefact that YnPNgnpΩq is separable. The proof of Lemma 2.3.9 is thus completed.

We now present the promised pointwise approximation result for strongly mea-surable functions.

Theorem 2.3.10 (Approximations of strongly measurable functions). Let pΩ,Fq bea measurable space, let pE, dq be a metric space, and let f : Ω Ñ E be a function.Then the following four statements are equivalent:

(i) It holds that f is strongly F/pE, dq-measurable.

(ii) There exists a sequence gn : Ω Ñ E, n P N, of strongly F/pE, dq-measurablefunctions which satisfy for all ω P Ω that

lim supnÑ8


(iii) There exists a sequence gn : Ω Ñ E, n P N, of F/BpEq-simple functions whichsatisfy for all ω P Ω that

lim supnÑ8


(iv) There exists a sequence gn : Ω Ñ E, n P N, of F/BpEq-simple functions whichsatisfy for every ω P Ω that dpfpωq, gnpωqq P r0,8q, n P N, decreases monoton-ically to zero.

Proof of Theorem 2.3.10. Throughout this proof assume w.l.o.g. that E ‰ H. Clearly,it holds that ((iv) ñ (iii)) and ((iii) ñ (ii)). Lemma 2.3.9 shows that ((ii) ñ (i)).It thus remains to prove that ((i) ñ (iv)). For this let f : Ω Ñ E be a strongly


F/pE, dq-measurable function. The fact that f is strongly F/pE, dq-measurable en-sures that fpΩq is separable. Hence, there exists a sequence penqnPN Ď fpΩq ofelements in fpΩq which satisfies that

ten P fpΩq : n P Nu Ě fpΩq. (2.66)

In the next step let Ppe1,...,enq : E Ñ E, n P N, and gn : Ω Ñ E, n P N, be thefunctions which satisfy for all x P E, n P N that

Ppe1,...,enqpxq “ emintkPt1,2,...,nu : dpek,xq“distdptxu,te1,...,enuqu (2.67)

andgn “ Ppe1,...,enq ˝ f. (2.68)

Lemma 2.3.8 and the fact that f is F/BpEq-measurable implies that for all n P N itholds that gn is F/BpEq-measurable. In addition, by definition it holds for all n P Nthat

gnpΩq Ď te1, . . . , enu (2.69)

is a finite set. We hence get that for all n P N it holds that gn is an F/BpEq-simplefunction. Moreover, note that (2.58) in Lemma 2.3.8 ensures that for all ω P Ω,n P N it holds that

dpfpωq, gnpωqq “ d`

fpωq, Ppe1,...,enqpfpωqq˘

“ distdptfpωqu, te1, . . . , enuq . (2.70)

This and the fact that @ω P Ω: distdpfpωq, te1, e2, . . . uq “ 0 imply that for all ω P Ω,n P N it holds that

dpfpωq, gnpωqq ě dpfpωq, gn`1pωqq and lim supnÑ8


The proof of Theorem 2.3.10 is thus completed.

2.3.5 Sums of strongly measurable functions

The next result, Corollary 2.3.11, shows that the sum of two strongly measurablefunctions is again a strongly measurable function. Corollary 2.3.11 follows immedi-ately from Theorem 2.3.10.

Corollary 2.3.11. Let pΩ,Fq be a measurable space, let K P tR,Cu, let pV, ¨V qbe a normed K-vector space, and let f, g : Ω Ñ V be strongly F/pV, ¨V q-measurablefunctions. Then f ` g : Ω Ñ V is strongly F/pV, ¨V q-measurable.


Exercise 2.3.12. The statement of Lemma 2.3.13 is in general not correct. Specifythe mistake in the proof of Lemma 2.3.13.

Lemma 2.3.13. Let pΩ,Fq be a measurable space, let pV, ¨V q be an R-Banachspace, let X, Y : Ω Ñ V be F/BpV q-measurable functions, and let Z : Ω Ñ V be thefunction which satisfies for all ω P Ω that Zpωq “ Xpωq ` Y pωq. Then it holds thatZ is F/BpV q-measurable.

Proof of Lemma 2.3.13. Throughout this proof let p : V ˆ V Ñ V be the functionwhich satisfies for all v, w P V that

ppv, wq “ v ` w (2.72)

and let X : Ω Ñ V ˆ V be the function which satisfies for all ω P Ω that

Xpωq “ pXpωq, Y pωqq. (2.73)

Next observe that p : V ˆ V Ñ V is a continuous function from V ˆ V to V . This,in particular, ensures that p is a measurable function. Moreover, note that theassumption that X and Y are measurable ensures that

X : Ω Ñ V ˆ V (2.74)

is also measurable. This and the fact that p : V ˆ V Ñ V is measurable prove thatthe composition function

p ˝X : Ω Ñ V (2.75)

is measurable. Combining this with the fact that

Z “ p ˝X (2.76)

completes the proof of Lemma 2.3.13.

Chapter 3

The Bochner integral

3.1 Sets of integrable functions

3.1.1 Lp-sets of measurable functions for p P r0,8q

Definition 3.1.1 (Lp-sets for p P r0,8q). Let K P tR,Cu, let pΩ,A, µq be a measurespace, let q P p0,8q, and let pV, ¨V q be a normed K-vector space. Then we denoteby L0pµ; ¨V q the set given by

L0pµ; ¨V q “MpA,BpV qq, (3.1)

we denote by ¨Lqpµ;¨V q: L0pµ; ¨V q Ñ r0,8s the function which satisfies for all

f P L0pµ; ¨V q that

fLqpµ;¨V q“

„ż

Ω

fpωqqV µpdωq

1q

, (3.2)

and we denote by Lqpµ; ¨V q the set given by

Lqpµ; ¨V q “

f P L0pµ; ¨V q : fLqpµ;¨V q

ă 8(

. (3.3)

Definition 3.1.2 (Equivalence classes). Let pΩ,F , µq be a measure space, let pS,Sqbe a measurable space, let R be a set, and let f : Ω Ñ R be a function. Then wedenote by rf sµ,S the set given by

rf sµ,S “!

g PMpF ,Sq :“

DA P F :`

µpAq “ 0 and tω P Ω: fpωq ‰ gpωqu Ď A˘‰

)

.

(3.4)

83

84 CHAPTER 3. THE BOCHNER INTEGRAL

Definition 3.1.3 (Lp-sets for p P r0,8q). Let K P tR,Cu, p P r0,8q, q P p0,8q, letpΩ,F , µq be a measure space, and let pV, ¨V q be a normed K-vector space. Then wedenote by Lppµ; ¨V q the set given by

Lppµ; ¨V q “

rf sµ,BpV q ĎMpF ,BpV qq : f P Lppµ; ¨V q(

(3.5)

and we denote by ¨Lqpµ;¨V q: L0pµ; ¨V q Ñ r0,8s the function which satisfies for

all f P L0pµ; ¨V q that

rf sµ,BpV qLqpµ;¨V q“ fLqpµ;¨V q

. (3.6)

3.1.2 Lp-spaces of strongly measurable functions for p P r0,8q

Definition 3.1.4 (Lp-spaces for p P r0,8q). Let K P tR,Cu, q P p0,8q, let pΩ,A, µqbe a measure space, and let pV, ¨V q be a normed K-vector space. Then we denoteby L0pµ; ¨V q the set given by

L0pµ; ¨V q “ tf PMpΩ, V q : f is strongly A/pV, ¨V q-measurableu, (3.7)

we denote by ¨Lqpµ;¨V q: L0pµ; ¨V q Ñ r0,8s the function which satisfies for all

f P L0pµ; ¨V q that

fLqpµ;¨V q“

„ż

Ω

fpωqqV µpdωq

1q

P r0,8s, (3.8)

and we denote by Lqpµ; ¨V q the set given by

Lqpµ; ¨V q “

f P L0pµ; ¨V q : fLqpµ;¨V q

ă 8(

. (3.9)

Corollary 2.3.11 proves, in the setting of Definition 3.1.4, that for all p P r0,8qit holds that Lppµ; ¨V q is a K-vector space.

Definition 3.1.5 (Equivalence classes of strongly measurable functions). Let K P

tR,Cu, let pV, ¨V q be a normed K-vector space, let pΩ,F , µq be a measure space,let R Ď V be a set, and let f : Ω Ñ R be a function. Then we denote by tfuµ,¨V theset given by

tfuµ,¨V

“

!

g P L0pµ; ¨V q :

“

DA P F :`

µpAq “ 0 and tω P Ω: fpωq ‰ gpωqu Ď A˘‰

)

.

(3.10)

3.1. SETS OF INTEGRABLE FUNCTIONS 85

Definition 3.1.6 (Lp-spaces for p P r0,8q). Let K P tR,Cu, p P r0,8q, q P p0,8q,let pΩ,A, µq be a measure space, and let pV, ¨V q be a normed K-vector space. Thenwe denote by Lppµ; ¨V q the set given by

Lppµ; ¨V q “!

tfuµ,¨VĎ L0

pµ; ¨V q : f P Lppµ; ¨V q

)

(3.11)

and we denote by ¨Lqpµ;¨V q: L0pµ; ¨V q Ñ r0,8s the function which satisfies for

all f P L0pµ; ¨V q that

tfuµ,¨V Lqpµ;¨V q“ fLqpµ;¨V q

. (3.12)

Lemma 3.1.7 (Theorem of Fischer-Riesz for equivalence classes of strongly measur-able functions). Let K P tR,Cu, p P r1,8q, let pΩ,F , µq be a measure space, and letpV, ¨V q be a K-Banach space. Then it holds that Lppµ; ¨V q is a K-Banach space.

Lemma 3.1.8. Let K P tR,Cu, p P r1,8q, let pΩ,F , µq be a finite measure space,and let pV, ¨V q be a normed K-vector space. Then it holds that the set

tfuµ,¨V : f is an F/BpV q-simple function(

(3.13)

is dense in Lppµ; ¨V q.

Proof of Lemma 3.1.8. Throughout this proof let f P Lppµ; ¨V q. Theorem 2.3.10proves that there exists a sequence gn : Ω Ñ V , n P N, of F/BpV q-simple functionswith the property that for all ω P Ω it holds that fpωq ´ gnpωqV , n P N, decreasesmonotonically to zero. Lebesgue’s theorem of dominated convergence hence showsthat

lim supnÑ8

f ´ gnLppµ;¨V q“ lim sup

nÑ8

„ż

Ω

fpωq ´ gnpωqpV µpdωq

1p

“ 0. (3.14)



3.2 Existence and uniqueness of the Bochner inte-gral

Theorem 3.2.1 (Bochner integral). Let pΩ,F , µq be a finite measure space, let K PtR,Cu, and let pV, ¨V q be a K-Banach space. Then

(i) there exists a unique continuous K-linear function I : L1pµ; ¨V q Ñ V whichsatisfies for all F/BpV q-simple f : Ω Ñ V that

Ipfq “ř

vPfpΩq

µpf´1ptvuqq ¨ v (3.15)

and

(ii) it holds for all f P L1pµ; ¨V q that IpfqV ď fL1pµ;¨V q.

Proof of Theorem 3.2.1. Throughout this proof let S Ď L1pµ; ¨V q be the set of allF/BpV q-simple functions and let J : S Ñ V be the mapping which satisfies for allf P S that

Jpfq “ř

vPfpΩq

µpf´1ptvuqq ¨ v. (3.16)

Next observe that the triangle inequality proves that for all f P S it holds that

JpfqV ďř

vPfpΩq

µpf´1ptvuqq ¨ vV “ fL1pµ;¨V q. (3.17)

This, the fact that J is linear, and Lemma 2.1.26 imply that J is uniformly continu-ous. In addition, we note that Item (iv) in Theorem 2.3.10 and Lebesgue’s theoremof dominated convergence ensure that

SL1pµ;¨V q

“ L1pµ; ¨V q. (3.18)

The assumption that V is complete hence allows us to apply Proposition 2.1.23to obtain that there exists a unique I P CpL1pµ; ¨V q, V q with the property thatI|S “ J . This proves (i). In addition, observe that (i), (3.17), and (3.18) establish(ii). The proof of Theorem 3.2.1 is thus completed.

3.3. DEFINITION OF THE BOCHNER INTEGRAL 87

3.3 Definition of the Bochner integralDefinition 3.3.1. Let pΩ,F , µq be a finite measure space, let K P tR,Cu, and letpV, ¨V q be a K-Banach space. Then we denote by

ş

Ωp¨q dµ : L1pµ; ¨V q Ñ V the

continuous K-linear function with the property that for all F/BpV q-simple f : Ω Ñ Vit holds that

ż

Ω

f dµ “ř

vPfpΩq

µpf´1ptvuqq ¨ v. (3.19)

Corollary 3.3.2 (Triangle inequality for the Bochner integral). Let pΩ,F , µq be afinite measure space, let K P tR,Cu, let pV, ¨V q be a K-Banach space, and letf P L1pµ; ¨V q. Then

›

›

›

›

ż

Ω

f dµ

›

›

›

›

V

ď

ż

Ω

fV dµ. (3.20)

Corollary 3.3.2 is an immediate consequence of Theorem 3.2.1.

Chapter 4

Nonlinear partial differentialequations

4.1 Diagonal linear operators on Hilbert spaces

4.1.1 Closedness of diagonal linear operators

Proposition 4.1.1 (Closedness of diagonal linear operators). Let K P tR,Cu, letpH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, and let A : DpAq Ď H Ñ H be a diagonallinear operator. Then it holds that A is a closed linear operator.

Proof of Proposition 4.1.1. Throughout this proof let x, y P H, let vn P DpAq, n P N,be a sequence which satisfies

lim supnÑ8

x´ vnH “ lim supnÑ8

y ´ AvnH “ 0, (4.1)

let B Ď H be an orthonormal basis of H, and let λ : B Ñ K be a function whichsatisfies

DpAq “

#

v P H :ÿ

bPB

|λb 〈b, v〉H |2ă 8

+

(4.2)

and

@ v P DpAq : Av “ÿ

bPB

λb 〈b, v〉H b. (4.3)

89

90 CHAPTER 4. NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS

Next note that

0 “ lim supnÑ8

y ´ Avn2H “ lim sup

nÑ8

«

ÿ

bPB

|〈b, y〉H ´ λb 〈b, vn〉H |2

ff

“ lim infnÑ8

«

ÿ

bPB

|〈b, y〉H ´ λb 〈b, vn〉H |2

ff

ěÿ

bPB

”

lim infnÑ8

|〈b, y〉H ´ λb 〈b, vn〉H |2ı

“ÿ

bPB

|〈b, y〉H ´ λb 〈b, x〉H |2 .

(4.4)

This and the fact thatř

bPB |〈b, y〉H |2ă 8 ensure that x P DpAq and Ax “ y. The

proof of Proposition 4.1.1 is thus completed.

4.1.2 Laplace operators on bounded domains

4.1.2.1 Laplace operators with Dirichlet boundary conditions

In this section we provide functional analytic descriptions of Laplace operators withsuitable boundary conditions and thereby present a few important examples of di-agonal linear operators.

Definition 4.1.2 (Laplace operator with Dirichlet boundary conditions). We saythat A is the Laplace operator with Dirichlet boundary conditions on L2pBorelp0,1q; |¨|Rqif and only if there exist en P L2pBorelp0,1q; |¨|Rq, n P N, such that it holds

(i) that @n P N : en “ rp0, 1q Q x ÞÑ?

2 sinpnπxq P RsBorelp0,1q,BpRq and

(ii) that A is the diagonal linear operator on L2pBorelp0,1q; |¨|Rq which satisfies

DpAq “

#

v P L2pBorelp0,1q; |¨|Rq :

8ÿ

n“1

n4ˇ

ˇ 〈en, v〉L2pBorelp0,1q;|¨|Rq

ˇ

ˇ

2

Ră 8

+

(4.5)

and

@ v P DpAq : Av “8ÿ

n“1

´π2n2 〈en, v〉L2pBorelp0,1q;|¨|Rqen. (4.6)


Proposition 4.1.3. Let A : DpAq Ď L2pBorelp0,1q; |¨|Rq Ñ L2pBorelp0,1q; |¨|Rq be theLaplace operator with Dirichlet boundary conditions on L2pBorelp0,1q; |¨|Rq. Then

(i) it holds that A is a diagonal linear operator,

(ii) it holds that σP pAq “ t´π2, ´4π2, ´9π2, ´16π2, . . . u,

(iii) it holds that

DpAq “ H2pp0, 1q,Rq

l jh n

Sobolev space

XH10 pp0, 1q,Rq

l jh n

Sobolev space

“

$

’

’

’

’

’

&

’

’

’

’

’

%

v P H2pp0, 1q,Rq : lim

xŒ0vpxq

l jh n

“vp0`q

“ limxÕ1

vpxql jh n

“vp1´q

“ 0

,

/

/

/

/

/

.

/

/

/

/

/

-

,

(4.7)

and

(iv) it holds for all v P DpAq that Av “ v2.

4.1.2.2 Laplace operators with Neumann boundary conditions

Definition 4.1.4 (Laplace operator with Neumann boundary conditions). We saythat A is the Laplace operator with Neumann boundary conditions on L2pBorelp0,1q; |¨|Rqif and only if there exist en P L2pBorelp0,1q; |¨|Rq, n P N0, such that it holds

(i) that e0 “ rp0, 1q Q x ÞÑ 1 P RsBorelp0,1q,BpRq,

(ii) that @n P N : en “ rp0, 1q Q x ÞÑ?

2 cospnπxq P RsBorelp0,1q,BpRq, and

(iii) that A is the diagonal linear operator on L2pBorelp0,1q; |¨|Rq which satisfies

DpAq “

#


8ÿ

n“1

n4ˇ


ˇ

ˇ

2

Ră 8

+

(4.8)

and

@ v P DpAq : Av “8ÿ

n“0



Proposition 4.1.5. Let A : DpAq Ď L2pBorelp0,1q; |¨|Rq Ñ L2pBorelp0,1q; |¨|Rq be theLaplace operator with Neumann boundary conditions on L2pBorelp0,1q; |¨|Rq. Then


(ii) it holds that σP pAq “ t0, ´π2, ´4π2, ´9π2, ´16π2, . . . u,

(iii) it holds that

DpAq “

$

’

’

’

’

’

&

’

’

’

’

’

%


xŒ0v1pxq

l jh n

“v1p0`q

“ limxÕ1

v1pxql jh n

“v1p1´q

“ 0

,

/

/

/

/

/

.

/

/

/

/

/

-

, (4.10)

and


4.1.2.3 Laplace operators with periodic boundary conditions

Definition 4.1.6 (Laplace operator with periodic boundary conditions). We say thatA is the Laplace operator with periodic boundary conditions on L2pBorelp0,1q; |¨|Rq ifand only if there exist en P L2pBorelp0,1q; |¨|Rq, n P Z, such that it holds

(i) that e0 “ rp0, 1q Q x ÞÑ 1 P RsBorelp0,1q,BpRq,

(ii) that @n P N : en “ rp0, 1q Q x ÞÑ?

2 sinp2nπxq P RsBorelp0,1q,BpRq,

(iii) that @n P N : e´n “ rp0, 1q Q x ÞÑ?

2 cosp2nπxq P RsBorelp0,1q,BpRq,

and

(iv) that A is the diagonal linear operator on L2pBorelp0,1q; |¨|Rq which satisfies

DpAq “

#


ÿ

nPZ

n4ˇ


ˇ

ˇ

2

Ră 8

+

(4.11)

and

@ v P DpAq : Av “ÿ

nPZ



Proposition 4.1.7. Let A : DpAq Ď L2pBorelp0,1q; |¨|Rq Ñ L2pBorelp0,1q; |¨|Rq be theLaplace operator with periodic boundary conditions on L2pBorelp0,1q; |¨|Rq. Then


(ii) it holds that σP pAq “ t0, ´4 ¨ π2, ´4 ¨ 22 ¨ π2, ´4 ¨ 32 ¨ π2, ´4 ¨ 42 ¨ π2, . . . u,

(iii) it holds that

DpAq “ H2P pp0, 1q,Rq

l jh n

Sobolev space

“

$

’

’

’

’

’

&

’

’

’

’

’

%


xŒ0vpxq

l jh n

“vp0`q

“ limxÕ1

vpxql jh n

“vp1´q

, limxŒ0

v1pxql jh n

“v1p0`q

“ limxÕ1

v1pxql jh n

“v1p1´q

,

/

/

/

/

/

.

/

/

/

/

/

-

,

(4.13)

and


4.1.3 Spectral decomposition for a diagonal linear operator

Proposition 4.1.8 (The eigenspaces of diagonal linear operators). Let K P tR,Cube a field, let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, let A : DpAq Ď H Ñ H be alinear operator, let B Ď H be an orthonormal basis of H, and let λ : B Ñ K be afunction which satisfies

DpAq “

#

v P H :ÿ

bPB

|λb 〈b, v〉H |2ă 8

+

(4.14)


ÿ

bPB

λb 〈b, v〉H b. (4.15)

Then

(i) it holds for all µ P σP pAq “ Impλq that

Kernpµ´ Aq “ spantb P B : λb “ µu “ spanpλ´1ptµuqq, (4.16)


(ii) it holds for all µ P σP pAq “ Impλq that”

spantb P B : λb “ µuı

k

”

spantb P B : λb ‰ µuı

“ H, (4.17)

and

(iii) it holds for all µ1, µ2 P σP pAq, v1 P Kernpµ1 ´ Aq, v2 P Kernpµ2 ´ Aq withµ1 ‰ µ2 that

〈v1, v2〉H “ 0. (4.18)

Proof of Proposition 4.1.8. Let µ P σP pAq be arbitrary. We first prove that

Kernpµ´ Aq Ď spantb P B : λb “ µu. (4.19)

For this observe that for all

v P Kernpµ´ Aq “ tw P Dpµ´ Aq “ DpAq : pµ´ Aqw “ 0u (4.20)

it holds that

0 “ 02H “ pµ´ Aqv2H “

ÿ

bPB

|pµ´ λbq 〈b, v〉H |2“

ÿ

bPB

|µ´ λb|2|〈b, v〉H |

2

“ÿ

bPB,λb‰µ

|µ´ λb|2|〈b, v〉H |

2 .(4.21)

Hence, we obtain that for all v P Kernpµ ´ Aq and all b P B with λb ‰ µ it holdsthat 〈b, v〉H “ 0. This implies that for all v P Kernpµ´ Aq it holds that

v P”

spantb P B : λb ‰ µuıK

. (4.22)

This and the identity”

spantb P B : λb ‰ µuı

k

”

spantb P B : λb “ µuı

“ H (4.23)

prove that (4.19) is indeed fufilled. Next we establish that

Kernpµ´ Aq Ě spantb P B : λb “ µu. (4.24)

For this observe that for all

v P spantb P B : λb “ µu “”


(4.25)


it holds thatÿ

bPB

|λb 〈b, v〉H |2“

ÿ

bPB,λb“µ

|λb 〈b, v〉H |2“ |µ|2

ÿ

bPB,λb“µ

|〈b, v〉H |2“ |µ|2 v2H ă 8. (4.26)

This shows that

spantb P B : λb “ µu “”


Ď DpAq “ Dpµ´ Aq. (4.27)

Next note that for all

v P spantb P B : λb “ µu “”


(4.28)

it holds that

pµ´ Aq v “ÿ

bPB

pµ´ λbq 〈b, v〉H b “ÿ

bPB,λb“µ

pµ´ λbq 〈b, v〉H b “ 0. (4.29)

The proof of Proposition 4.1.8 is thus completed.

The next result, Theorem 4.1.9, establishes a spectral decomposition for diagonallinear operators. It follows immediately from Proposition 4.1.8 above.

Theorem 4.1.9 (Spectral decomposition for diagonal linear operators). Let K P

tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, and let A : DpAq Ď H Ñ H be adiagonal linear operator. Then

(i) it holds that

DpAq “

$

&

%

v P H :ÿ

λPσP pAq

|λ|2›

›PKernpλ´Aq,Hpvq›

›

2

Hă 8

,

.

-

(4.30)

and

(ii) it holds for all v P DpAq that

Av “ÿ

λPσP pAq

λ ¨ PKernpλ´Aq,Hpvq. (4.31)


Exercise 4.1.10. Let K P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, and letA : DpAq Ď H Ñ H be a diagonal linear operator. Prove that A is symmetric if andonly if σP pAq Ď R.

Proposition 4.1.11 (Orthonormal basis of eigenfunctions). Let K P tR,Cu, letpH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, let A : DpAq Ď H Ñ H be a diagonal linearoperator, let B Ď DpAq be an orthonormal basis of H, and let λ : BÑ K be a functionwhich satisfies for all b P B that Ab “ λbb. Then

(i) it holds that DpAq “

v P H :ř

bPB |λb 〈b, v〉H |2ă 8

(

and

(ii) it holds for all v P DpAq that Av “ř

bPB λb 〈b, v〉H b.

Proof of Proposition 4.1.11. Throughout this proof let B : DpBq Ď H Ñ H be thelinear operator which satisfies

DpBq “

#

v P H :ÿ

bPB

|λb 〈b, v〉H |2ă 8

+

(4.32)

and@ v P DpBq : Bv “

ÿ

bPB

λb 〈b, v〉H b. (4.33)

Next observe that the assumptions that A is a linear operator, that B Ď DpAq, andthat @ b P B : Ab “ λbb imply that for all v P spanpBq it holds that

Av “ Bv. (4.34)

Moreover, note that Proposition 4.1.1 implies that A is a closed linear operator. Thisand (4.34) ensure that for all v P DpBq it holds that v P DpAq and Av “ Bv. Hence,we obtain that

GraphpBq Ď GraphpAq. (4.35)

Furthermore, observe that the assumptions that B Ď DpAq and that @ b P B : Ab “λbb and Exercise 1.9.5 assure that

Impλq “ σP pBq Ď σP pAq. (4.36)

Item (iii) in Proposition 4.1.8 proves that for all µ P σP pAqz Impλq, v P Kernpµ´Aq,b P B it holds that

〈v, b〉H “ 0. (4.37)


This and the assumption that B is an orthonormal basis ensure that for all µ PσP pAqz Impλq, v P Kernpµ ´ Aq it holds that v “ 0. Hence, we obtain for allµ P σP pAqz Impλq that Kernpµ ´ Aq “ t0u. This implies that σP pAqz Impλq “ H.Combining this with (4.36) proves that

σP pAq “ Impλq “ σP pBq. (4.38)

Next observe that the assumption that @ b P B : Ab “ λbb implies that for all µ PσP pAq it holds that λ´1ptµuq “ tb P B : λb “ µu Ď Kernpµ ´ Aq. Proposition 4.1.8hence ensures for all µ P σP pAq that

spantb P B : λb “ µu Ď Kernpµ´ Aq. (4.39)

This, (4.38), and again Proposition 4.1.8 prove for all µ P σP pAq “ σP pBq “ Impλqthat

Kernpµ´Bq “ spantb P B : λb “ µu Ď Kernpµ´ Aq. (4.40)

Item (iii) in Proposition 4.1.8 and the assumption that B is an orthonormal basis ofHtherefore assure for all µ P σP pAq “ σP pBq “ Impλq that Kernpµ´Bq “ KernpµÁq.Combining this and (4.38) with Theorem 4.1.9 proves that A “ B. The proof ofProposition 4.1.11 is thus completed.

Issue 4.1.12. LetK P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be aK-Hilbert space, let A : DpAq ĎH Ñ H be a closed linear operator, let B Ď DpAq be an orthonormal basis of H, andlet λ : B Ñ K be a function which satisfies for all b P B that Ab “ λbb. Is A then adiagonal linear operator?

4.1.4 Fractional powers of a diagonal linear operator

Definition 4.1.13 (Nonnegative fractional powers of a diagonal linear operator).Let K P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, let r P r0,8q, and letA : DpAq Ď H Ñ H be a diagonal linear operator with σP pAq Ď r0,8q. Then wedenote by Ar : DpArq Ď H Ñ H the linear operator which satisfies

DpArq “

$

&

%

v P H :ÿ

λPσP pAq

›

›λr ¨ PKernpλÁq,Hpvq›

›

2

Hă 8

,

.

-

(4.41)

and@ v P DpArq : Arv “

ÿ

λPσP pAq

λr ¨ PKernpλÁq,Hpvq. (4.42)


Definition 4.1.14 (Negative fractional powers of a diagonal linear operator). LetK P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, let r P p´8, 0q, and letA : DpAq Ď H Ñ H be a diagonal linear operator with σP pAq Ď p0,8q. Then wedenote by Ar : DpArq Ď H Ñ H the linear operator which satisfies

DpArq “

$

&

%

v P H :ÿ

λPσP pAq

›

›λr ¨ PKernpλ´Aq,Hpvq›

›

2

Hă 8

,

.

-

(4.43)

and@ v P DpArq : Arv “

ÿ

λPσP pAq

λr ¨ PKernpλ´Aq,Hpvq. (4.44)

The next lemma collects a simple property of fractional powers of a diagonallinear operator. It follows immediately from Definition 4.1.13 and Definition 4.1.14.

Lemma 4.1.15 (Diagonality of fractional powers of a diagonal linear operators).Let K P tR,Cu, r P R, let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, let A : DpAq ĎH Ñ H be a diagonal linear operator with σP pAq Ď r0,8q, and assume that

`

r Pr0,8q or σP pAq Ď p0,8q

˘

. Then Ar is a diagonal linear operator.

4.1.5 Domain Hilbert space associated to a diagonal linearoperator

Lemma 4.1.16. Let K P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, letA : DpAq Ď H Ñ H be a diagonal linear operator with σP pAq Ď p0,8q. Thenthe triple

`

DpAq, 〈Ap¨q, Ap¨q〉H , Ap¨qH˘

is a K-inner product space.

The proof of Lemma 4.1.16 is clear and therefore omitted. If the point spectrum ofthe diagonal linear operator A in Lemma 4.1.16 in addition satisfies infpσP pAqq ą 0,then the triple

`


is even a K-Hilbert space. This is thesubject of the next lemma.

Lemma 4.1.17. Let K P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, andlet A : DpAq Ď H Ñ H be a diagonal linear operator with σP pAq Ď p0,8q andinfpσP pAqq ą 0. Then

(i) it holds that the triple`


is a K-Hilbert space and

(ii) it holds for all v P DpAq that

vH ďAvH

infpσP pAqq. (4.45)


Proof of Lemma 4.1.17. First of all, note that for all v P DpAq it holds that

Av2H “ÿ

µPσP pAq

›

›µ ¨ PKernpµÁq,Hpvq›

›

2

H“

ÿ

µPσP pAq

|µ|2 ¨›

›PKernpµÁq,Hpvq›

›

2

H

ě

„

infµPσP pAq

|µ|2

»

–

ÿ

µPσP pAq

›

›PKernpµÁq,Hpvq›

›

2

H

fi

fl “

„

infµPσP pAq

µ

2

v2H .

(4.46)

This proves (4.45). Moreover, note that Lemma 4.1.16 ensures that the triple`


is a K-inner product space. It thus remains to provethat the normed K-vector space

`

DpAq, Ap¨qH˘

is complete. For this let pvnqnPN ĎDpAq be a Cauchy sequence in

`

DpAq, Ap¨qH˘

. Inequality (4.45) hence impliesthat pvnqnPN is a Cauchy sequence in pH, ¨Hq too. This and the fact that pH, ¨Hqis complete shows that there exists a vector v P H such that

lim supnÑ8

vn ´ vH “ 0. (4.47)

Next note that for all n P N it holds that

8 ą lim infmÑ8

Apvn ´ vmq2H

“ lim infmÑ8

»

–

ÿ

µPσP pAq

|µ|2›

›PKernpµÁq,Hpvn ´ vmq›

›

2

H

fi

fl

ěÿ

µPσP pAq

|µ|2”

lim infmÑ8

›

›PKernpµÁq,Hpvn ´ vmq›

›

2

H

ı

“ÿ

µPσP pAq

|µ|2„

›

›

›PKernpµÁq,H

´

vn ´ limmÑ8

vm

¯›

›

›

2

H

“ÿ

µPσP pAq

|µ|2”

›

›PKernpµÁq,Hpvn ´ vq›

›

2

H

ı

.

(4.48)

Combining this and the fact that

lim supnÑ8

lim infmÑ8

Apvn ´ vmq2H “ 0 ă 8 (4.49)

with the fact that @n P N : vn P DpAq shows that v P DpAq and

lim supnÑ8

Apvn ´ vqH “ 0. (4.50)



Exercise 4.1.18. Give an example of an R-Hilbert space pH, 〈¨, ¨〉H , ¨Hq and adiagonal linear operator A : DpAq Ď H Ñ H such that σP pAq Ď p0,8q and such thatthe triple

`


(4.51)

is not an R-Hilbert space.

4.1.6 Completion of metric spaces

4.1.6.1 An extension of a metric space

Lemma 4.1.19. Let pE, dq be a metric space, let φ : MpN, Eq Ñ PpMpN, Eqq be thefunction which satisfies for all a PMpN, Eq that

φpaq “

x PMpN, Eq : lim supnÑ8 dpxn, anq “ 0(

, (4.52)

and let E be the set given by

E “

A P PpMpN, Eqq :`

D a PMpN, Eq : A “ φpaq˘(

. (4.53)

Then

(i) there exists a unique function δ : E ˆ E Ñ r0,8q which satisfies for all a, b PMpN, Eq that

δpφpaq, φpbqq “ lim supnÑ8 dpan, bnq, (4.54)

(ii) it holds that the function

E Q e ÞÑ φpN Q n ÞÑ e P Eq P E (4.55)

is injective, and

(iii) it holds that pE , δq is a metric space.

Proof of Lemma 4.1.19. Observe that for all a, b, A,B P MpN, Eq with φpaq “ φpAqand φpbq “ φpBq it holds that

lim supnÑ8 dpan, bnq ď lim supnÑ8 rdpan, Anq ` dpAn, Bnq ` dpBn, bnqs

ď lim supnÑ8 dpan, Anq ` lim supnÑ8 dpAn, Bnq ` lim supnÑ8 dpBn, bnq

“ lim supnÑ8 dpAn, Bnq.

(4.56)


Hence, we obtain that for all a, b, A,B PMpN, Eq with φpaq “ φpAq and φpbq “ φpBqit holds that

lim supnÑ8 dpan, bnq “ lim supnÑ8 dpAn, Bnq. (4.57)

This establishes item (i). Item (ii) is an immediate consequence from the fact thatfor all a, b P E with a ‰ b it holds that dpa, bq ą 0. It thus remains to prove item (iii).For this observe that for all a, b PMpN, Eq with δpφpaq, φpbqq “ 0 it holds that

lim supnÑ8 dpan, bnq “ δpφpaq, φpbqq “ 0. (4.58)

This, in turn, ensures that for all a, b, x P MpN, Eq with δpφpaq, φpbqq “ 0 it holdsthat

`“

lim supnÑ8 dpan, xnq “ 0‰

ô“

lim supnÑ8 dpbn, xnq “ 0‰˘

. (4.59)

Hence, we obtain that for all a, b PMpN, Eq with δpφpaq, φpbqq “ 0 it holds that

φpaq “ φpbq. (4.60)

Next observe that for all a, b, c PMpN, Eq it holds that

δpφpaq, φpcqq “ lim supnÑ8 dpan, cnq

ď lim supnÑ8 dpan, bnq ` lim supnÑ8 dpbn, cnq

“ δpφpaq, φpbqq ` δpφpbq, φpcqq.

(4.61)

Combining this with (4.60) establishes item (iii). The proof of Lemma 4.1.19 is thuscompleted.

4.1.6.2 An artificial completion

Lemma 4.1.20. Let pE, dq be a metric space, let φ : MpN, Eq Ñ PpMpN, Eqq be thefunction which satisfies for all a PMpN, Eq that

φpaq “ tx PMpN, Eq : lim supnÑ8 dpxn, anq “ 0u , (4.62)

and let E be the set given by

E “!

A P PpMpN, Eqq :´

D a P

x PMpN, Eq : lim supn,mÑ8 dpxn, xmq “ 0(

: A “ φpaq¯)

. (4.63)

Then


(i) there exists a unique function δ : E ˆ E Ñ r0,8q which satisfies for all a, b Ptx PMpN, Eq : lim supn,mÑ8 dpxn, xmq “ 0u that

δpφpaq, φpbqq “ lim supnÑ8 dpan, bnq, (4.64)

(ii) it holds that the function

E Q e ÞÑ φpN Q n ÞÑ e P Eq P E (4.65)

is injective, and

(iii) it holds that pE , δq is a complete metric space.

Proof of Lemma 4.1.20. Observe that Lemma 4.1.19 ensures that item (iv) holds,that item (ii) holds, and that the pair pE , δq is a metric space. It thus remains toprove that the metric space pE , δq is complete. For this let apkq P MpN, Eq, k P N,be a sequence in MpN, Eq which satisfies

lim supk,lÑ8

δpφpapkqq, φpaplqqq “ 0 (4.66)

and let e “ pekqkPN PMpN, Eq be a sequence in E which satisfies for all k P N that

lim supnÑ8

dpek, apkqn q ď

12k. (4.67)

Note that for all k, l P N it holds that

dpek, elq ď lim supnÑ8

“

dpek, apkqn q ` dpa

pkqn , aplqn q ` dpa

plqn , elq

‰

ď lim supnÑ8

“

12k` dpapkqn , aplqn q `

12l

‰

“ 12k` 1

2l` δpφpapkqq, φpaplqqq.

(4.68)

Combining this with (4.66) assures that

lim supk,lÑ8

dpek, elq “ 0. (4.69)

The triangle inequality hence ensures that

lim supkÑ8

δpφpeq, φpapkqqq “ lim supkÑ8

lim supnÑ8

dpen, apkqn q

ď lim supkÑ8

lim supnÑ8

“

dpen, ekq ` dpek, apkqn q

‰

ď lim supkÑ8

lim supnÑ8

dpen, ekq ` lim supkÑ8

lim supnÑ8

dpek, apkqn q

“ lim supkÑ8

lim supnÑ8

dpek, apkqn q ď lim sup

kÑ8

12k“ 0.

(4.70)



4.1.6.3 The natural completion

Definition 4.1.21 (The completion of a metric space). Let pE, dq be a metric space.Then we denote by EE,d the set given by

EE,d “ E Y

#

A P PpMpN, Eqq :ˆ

D a P!

x PMpN, Eq :”

lim supn,mÑ8 dpxn, xmq “ 0ı

^

”

E e P E : lim supnÑ8 dpxn, eq “ 0ı)

:

A “!

x PMpN, Eq : lim supnÑ8 dpxn, anq “ 0)

˙

+

, (4.71)

we denote by DE,d : EE,d ˆ EE,d Ñ r0,8q the function which satisfies

(i) that for all e, f P E it holds that

DE,dpe, fq “ dpe, fq (4.72)

and

(ii) that for all a, b P tx P MpN, Eq : rlim supn,mÑ8 dpxn, xmq “ 0s ^ rE e PE : lim supnÑ8 dpxn, eq “ 0su, e P E it holds that

DE,d

`

tx PMpN, Eq : lim supnÑ8 dpxn, anq “ 0u, e˘

“ DE,d

`

e, tx PMpN, Eq : lim supnÑ8 dpxn, anq “ 0u˘

“ lim supnÑ8 dpan, eq,

(4.73)

and

DE,d

`

tx PMpN, Eq : lim supnÑ8 dpxn, anq “ 0u,

tx PMpN, Eq : lim supnÑ8 dpxn, bnq “ 0u˘

“ lim supnÑ8 dpan, bnq, (4.74)

and we call pEE,d,DE,dq the natural completion of pE, dq.

Proposition 4.1.22 (Properties of the natural completion). Let pE, dq be a metricspace. Then

(i) it holds that pEE,d,DE,dq is a complete metric space,

(ii) it holds that E Ď EE,d,

(iii) it holds that DE,d|EˆE “ d, and

(iv) it holds that EEE,d“ EE,d.


Proof of Proposition 4.1.22. The statement of Proposition 4.1.22 is an immediateconsequence of Lemma 4.1.20. The proof of Proposition 4.1.22 is thus completed.

4.1.6.4 Further completions

Theorem 4.1.23 (Existence and “uniqueness” of completions of a metric space).Let pE, dq be a metric space. Then

(i) there exists a complete metric space pE , δq such that E Ď E, EE “ E, andδ|EˆE “ d and

(ii) for all metric spaces pEk, δkq, k P t1, 2u, with @ k P t1, 2u : E Ď Ek, EEk “ Ek,and δk|EˆE “ dk there exists a unique continuous function φ : E1 Ñ E2 whichsatisfies for all e P E that φpeq “ e.

Proof of Theorem 4.1.23. Item (i) is an immediate consequence of Proposition 4.1.22.Item (ii) follows from Proposition 2.1.23. The proof of Theorem 4.1.23 is thus com-pleted.

4.1.7 Interpolation spaces associated to a diagonal linear op-erator

We now introduce the concept of a family of interpolation spaces associated to adiagonal linear operator.

Theorem 4.1.24 (Interpolation spaces associated to a diagonal linear operator). LetK P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, and let A : DpAq Ď H Ñ Hbe a symmetric diagonal linear operator with infpσP pAqq ą 0. Then there exists anup to isometric isomorphisms unique family pHr, 〈¨, ¨〉Hr , ¨Hrq, r P R, of K-Hilbertspaces which satisfy

(i) @ r, s P R with r ě s : Hr Ď Hs “ HrHs,

(ii) @ r P r0,8q : pHr, 〈¨, ¨〉Hr , ¨Hrq “ pDpArq, 〈Arp¨q, Arp¨q〉H , Arp¨qHq, and

(iii) @ r P p´8, 0s, v P H : vHr “ ArvH .

4.2. THE SOBOLEV SPACE W 1,20 pp0, 1q,Rq “ H1

0 pp0, 1q,Rq 105

Definition 4.1.25 (Interpolation spaces associated to a diagonal linear operator).LetK P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be aK-Hilbert space, and let A : DpAq Ď H Ñ Hbe a symmetric diagonal linear operator with infpσP pAqq ą 0. Then we say that H isa family of K-Hilbert spaces spaces associated to A if and only if there exist a familyHr “ pHr, 〈¨, ¨〉Hr , ¨Hrq, r P R, of K-Hilbert spaces such that it holds

(i) that H “ pHrqrPR,

(ii) that @ r, s P R with r ě s : Hr Ď Hs “ HrHs,

(iii) that @ r P r0,8q : pHr, 〈¨, ¨〉Hr , ¨Hrq “ pDpArq, 〈Arp¨q, Arp¨q〉H , Arp¨qHq, and

(iv) that @ r P p´8, 0s, v P H : vHr “ ArvH .

4.2 The Sobolev space W 1,20 pp0, 1q,Rq “ H1

0pp0, 1q,Rq

4.2.1 Weak derivatives in W 1,20 pp0, 1q,Rq “ H1

0pp0, 1q,Rq

Lemma 4.2.1 (On weak derivatives and integration by parts). Let pH, 〈¨, ¨〉H , ¨Hq “pL2pBorelp0,1q; |¨|Rq, 〈¨, ¨〉L2pBorelp0,1q;|¨|Rq

, ¨L2pBorelp0,1q;|¨|Rqq, let A : DpAq Ď H Ñ H be

the Laplace operator with Dirichlet boundary conditions on H, let v0 P DppÁq12q,

and let en P H, n P N, satisfy for all n P N that en “ rt?

2 sinpnπxquxPp0,1qsBorelp0,1q,BpRq.Then there exists a unique v1 P H such that

(i) it holds for all ϕ P C8cptpp0, 1q,Rq that⟨v1, rϕsBorelp0,1q,BpRq

⟩H“ ´

⟨v0, rϕ

1sBorelp0,1q,BpRq

⟩H, (4.75)

(ii) it holds that

lim supnÑ8

›

›

›

›

v1 ńř

k“1

kπ 〈ek, v0〉H rt?

2 cospkπxquxPp0,1qsBorelp0,1q,BpRq

›

›

›

›

H

“ 0, (4.76)

and

(iii) it holds that

›

›pÁq12v0H “ v1H “

˜

8ÿ

k“1

`

k2π2|〈ek, v0〉H |

2˘

¸12

. (4.77)


Proof of Lemma 4.2.1. Throughout this proof let pfnqnPN Ď H, pvpnq0 qnPN Ď H,pvpnq1 qnPN Ď H, v1 P H satisfy for all n P N that

fn “ rt?

2 cospnπxquxPp0,1qsBorelp0,1q,BpRq, (4.78)

vpnq0 “

nÿ

k“1

〈ek, v0〉H ek, (4.79)

vpnq1 “

nÿ

k“1

kπ 〈ek, v0〉H fk, (4.80)

and

v1 “

8ÿ

k“1

kπ 〈ek, v0〉H fk. (4.81)

Note that Item (ii) is an immediate consequence of (4.81). Next observe that inte-gration by parts ensures for all ϕ P C8cptpp0, 1q,Rq, n P N that⟨

vpnq1 , rϕsBorelp0,1q,BpRq

⟩H“

⟨řnk“1 kπ 〈ek, v0〉H fk, rϕsBorelp0,1q,BpRq

⟩H

“

nÿ

k“1

kπ 〈ek, v0〉H⟨fk, rϕsBorelp0,1q,BpRq

⟩H

“

nÿ

k“1

kπ 〈ek, v0〉Hż 1

0

?2 cospkπxqϕpxq dx

“ ´

nÿ

k“1

〈ek, v0〉Hż 1

0

?2 sinpkπxqϕ1pxq dx

“ ´

nÿ

k“1

〈ek, v0〉H⟨ek, rϕ

1sBorelp0,1q,BpRq

⟩H

“ ´⟨vpnq0 , rϕ1sBorelp0,1q,BpRq

⟩H.

(4.82)

This proves for all ϕ P C8cptpp0, 1q,Rq that⟨v1, rϕsBorelp0,1q,BpRq

⟩H“ ´

⟨v0, rϕ

1sBorelp0,1q,BpRq

⟩H. (4.83)


0 pp0, 1q,Rq 107

This establishes Item (i). It thus remains to prove Item (iii). For this we observethat the fact that tfn : n P Nu is an orthonormal set in H yields that

›

›pÁq12v0

2H “

8ÿ

k“1

|⟨ek, pÁq

12v0

⟩H|2“

8ÿ

k“1

`

k2π2|〈ek, v0〉H |

2˘

“ v12H . (4.84)

This establishes Item (iii). The proof of Lemma 4.2.1 is thus completed.

The following result, Lemma 4.2.2 below, is an immediate consequence of the factthat

C8cptpp0, 1q,RqL2pBorelp0,1q;|¨|Rq

“ L2pBorelp0,1q; |¨|Rq (4.85)

and of Lemma 4.2.1 above.

Lemma 4.2.2 (Weak derivatives in the Hilbert space H12). Let pH, 〈¨, ¨〉H , ¨Hq “pL2pBorelp0,1q; |¨|Rq, 〈¨, ¨〉L2pBorelp0,1q;|¨|Rq


the Laplace operator with Dirichlet boundary conditions on H, and let v P DppÁq12q.Then there exists a unique v1 P H which satisfies for all ϕ P C8cptpp0, 1q,Rq that⟨

v1, rϕsBorelp0,1q,BpRq⟩H“ ´

⟨v, rϕ1sBorelp0,1q,BpRq

⟩H. (4.86)

Definition 4.2.3 (Weak derivatives in the Hilbert spaceH12). Let pH, 〈¨, ¨〉H , ¨Hq “pL2pBorelp0,1q; |¨|Rq, 〈¨, ¨〉L2pBorelp0,1q;|¨|Rq


the Laplace operator with Dirichlet boundary conditions on H, and let v P DppÁq12q.Then we denote by v1 P H the set which satisfies for all ϕ P C8cptpp0, 1q,Rq that⟨

v1, rϕsBorelp0,1q,BpRq⟩H“ ´

⟨v, rϕ1sBorelp0,1q,BpRq

⟩H. (4.87)


4.2.2 A special case of the Sobolev embedding theorem

Lemma 4.2.4 (An embedding into continuous functions). Let pH, x¨, ÿH , ¨Hq “pL2pBorelp0,1q; |¨|Rq, x¨, ÿL2pBorelp0,1q;|¨|Rq


the Laplace operator with Dirichlet boundary conditions on H, let en P H, n P N,satisfy for all n P N that en “ rt

?2 sinpnπxquxPp0,1qsBorelp0,1q,BpRq, and let γ P p14,8q,

v P DppÁqγq. Then

(i) there exists a unique continuous function ϕ : p0, 1q Ñ R which satisfies ϕ P v,

(ii) it holds that ϕ is uniformly continuous,

(iii) it holds that

lim supNÑ8

supxPp0,1q

ˇ

ˇ

ˇ

ˇ

ˇ

ϕpxq Ńÿ

n“1

xen, vyH?

2 sinpnπxq

ˇ

ˇ

ˇ

ˇ

ˇ

“ 0, (4.88)

and

(iv) it holds that

supxPp0,1q

|ϕpxq| ď?

28ÿ

n“1

|xen, vyH |

ď

?2

π2γ

«

8ÿ

n“1

1n4γ

ff12

pÁqγvH ď

«

8ÿ

n“1

1n4γ

ff12

pÁqγvH .

(4.89)

Proof of Lemma 4.2.4. First, observe that

8ÿ

n“1

„

supxPr0,1s

|xen, vyH?

2 sinpnπxq|

“

8ÿ

n“1

ˆ

|xen, vyH |

„

supxPr0,1s

|?

2 sinpnπxq|

˙

“?

2

˜

8ÿ

n“1

|xen, vyH |

¸

.

(4.90)


0 pp0, 1q,Rq 109

Therefore, we obtain that8ÿ

n“1

„

supxPr0,1s

|xen, vyH?

2 sinpnπxq|

“?

2

˜

8ÿ

n“1

|xpÁqγpÁq´γen, vyH |

¸

“?

2

˜

8ÿ

n“1

|xpÁq´γen, pÁqγvyH |

¸

“?

2

˜

8ÿ

n“1

pn2π2q´γ|xen, pÁq

γvyH |

¸

“

?2

π2γ

«

8ÿ

n“1

1

n2γ|xen, pÁq

γvyH |

ff

.

(4.91)

The Cauchy-Schwarz inequality hence shows that8ÿ

n“1

„

supxPr0,1s

|xen, vyH?

2 sinpnπxq|

ď

?2

π2γ

«

8ÿ

n“1

1

n4γ

ff12«8ÿ

n“1

|xen, pÁqγvyH |

2

ff12

“

?2

π2γ

«

8ÿ

n“1

1

n4γ

ff12

pÁqγvH ď

«

8ÿ

n“1

1

n4γ

ff12

pÁqγvH ă 8.

(4.92)

This and the fact that pCpr0, 1s,Rq, ¨Cpr0,1s,Rqq is an R-Banach space ensure thatthere exists a continuous function φ : r0, 1s Ñ R which satisfies

lim supNÑ8

supxPr0,1s

ˇ

ˇ

ˇ

ˇ

ˇ

φpxq Ńÿ

n“1

xen, vyH?

2 sinpnπxq

ˇ

ˇ

ˇ

ˇ

ˇ

“ 0. (4.93)

Equation (4.93), in particular, assures that

lim supNÑ8

˜

ż 1

0

ˇ

ˇ

ˇφpxq ´

Nř

n“1

xen, vyH?

2 sinpnπxqˇ

ˇ

ˇ

2

dx

¸

“ 0. (4.94)

This implies that

lim supNÑ8

›

›

›

›

›

rφ|p0,1qsBorelp0,1q,BpRq ´

Nÿ

n“1

xen, vyHen

›

›

›

›

›

H

“ 0. (4.95)

Therefore, we obtain that

rφ|p0,1qsBorelp0,1q,BpRq “ v. (4.96)

Combining this with (4.92) and (4.93) completes the proof of Lemma 4.2.4.


4.3 Nonlinear evolution equations

4.3.1 Complete function spaces

Lemma 4.3.1. Let X be a non-empty set, let pE, dq be a complete metric space, letE Ď MpX,Eq be the set given by E “ tf P MpX,Eq : infePE supxPX dpfpxq, eq ă 8u,and let δ : EˆE Ñ r0,8q be the function which satisfies for all f, g P E that δpf, gq “supxPX dpfpxq, gpxqq. Then the pair pE , δq is a complete metric space.

Proof of Lemma 4.3.1. First, note that for all f, g, h P E it holds that ppδpf, gq “0q ô pf “ gqq, δpf, gq “ δpg, fq, and

δpf, gq ` δpg, hq “ supxPX

dpfpxq, gpxqq ` supxPX

dpgpxq, hpxqq

ě supxPX

“

dpfpxq, gpxqq ` dpgpxq, hpxqq‰

ě supxPX

dpfpxq, hpxqq “ δpf, hq.

(4.97)

This proves that the pair pE , δq is a metric space. It remains to prove that pE , δqis complete. For this let pfnqnPN Ď E be a Cauchy sequence in pE , δq. Then for allx P X it holds that pfnpxqqnPN Ď E is a Cauchy sequence in pE, dq. The assumptionthat pE, dq is a complete metric space hence ensures that there exists a functiong : X Ñ E such that for all x P X it holds that

lim supnÑ8

dpfnpxq, gpxqq “ 0. (4.98)

This and the assumption that pfnqnPN Ď E is a Cauchy sequence yield that

0 “ infNPN

supn,mPNXrN,8q

supxPX

dpfnpxq, fmpxqq

“ infNPN

supnPNXrN,8q

supxPX

supmPNXrN,8q

dpfnpxq, fmpxqq

ě infNPN

supnPNXrN,8q

supxPX

limmÑ8

dpfnpxq, fmpxqq

“ infNPN

supnPNXrN,8q

supxPX

dpfnpxq, gpxqq

“ lim supnÑ8

supxPX

dpfnpxq, gpxqq.

(4.99)

This proves that there exists a natural number n P N such that

supxPX

dpfnpxq, gpxqq ď 1. (4.100)

4.3. NONLINEAR EVOLUTION EQUATIONS 111

The fact that fn P E therefore proves that

infePE

supxPX

dpgpxq, eq ď infePE

supxPX

“

dpgpxq, fnpxqq ` dpfnpxq, eq‰

ď supxPX

dpgpxq, fnpxqq ` infePE

supxPX

dpfnpxq, eq

ď 1` infePE

supxPX

dpfnpxq, eq ă 8.

(4.101)

This ensures that g P E . Next note that (4.99) assures that lim supnÑ8 δpfn, gq “ 0.The proof of Lemma 4.3.1 is thus completed.

Lemma 4.3.2. Let pX,X q be a topological space, let pE, dq be a metric space,let g : X Ñ E be a function, and let fn P CpX,Eq, n P N, satisfy lim supnÑ8supptdpfnpxq, gpxqq : x P Xu Y t0uq “ 0. Then g P CpX,Eq.

Proof of Lemma 4.3.2. Throughout this proof assume w.l.o.g. that X ‰ H, let x0 P

X, and let ε P p0,8q. Observe that the assumption that

lim supnÑ8

supxPX

dpfnpxq, gpxqq “ 0 (4.102)

ensures that there exists a natural number N P N such that for all x P X, n PpNX rN,8qq it holds that

dpfnpxq, gpxqq ăε3. (4.103)

Moreover, the fact that fN P CpX,Eq implies that there exists a set A P X withx0 P A such that for all x P A it holds that

dpfNpx0q, fNpxqq ăε3. (4.104)

This and (4.103) prove that for all x P A it holds that

dpgpx0q, gpxqq ď dpgpx0q, fNpx0qq ` dpfNpx0q, fNpxqq ` dpfNpxq, gpxqq

ă ε3` ε

3` ε

3“ ε.

(4.105)

This proves that the function g is continuous at the point x0. As x0 P X was anarbitrary point in X, we obtain that g P CpX,Eq. The proof of Lemma 4.3.2 is thuscompleted.

The next result, Corollary 4.3.3, follows directly from Lemma 4.3.1 and Lemma 4.3.2.


Corollary 4.3.3. Let pX,X q be a topological space, let pE, dq be a complete metricspace, let E ĎMpX,Eq be the set given by

E “ tf P CpX,Eq : @ e P E : supxPX dpfpxq, eq ă 8u , (4.106)

and let δ : EˆE Ñ r0,8q be the function which satisfies for all f, g P E that δpf, gq “supptdpfpxq, gpxqq : x P Xu Y t0uq. Then the pair pE , δq is a complete metric space.

The next result, Corollary 4.3.4, is an immediate consequence of Corollary 4.3.3.

Corollary 4.3.4. Let pX,X q be a topological space, let pE, dq be a complete metricspace, let e P E, R P r0,8q, let E ĎMpX,Eq be the set given by

E “ tf P CpX,Eq : supxPX dpfpxq, eq ď Ru , (4.107)

and let δ : EˆE Ñ r0,8q be the function which satisfies for all f, g P E that δpf, gq “supptdpfpxq, gpxqq : x P Xu Y t0uq. Then the pair pE , δq is a complete metric space.

4.3.2 Measurability properties

The next result, Lemma 4.3.5, is proved as, e.g., Hytönen et al. [4, Lemma 1.1.12].

Lemma 4.3.5. Let pE, dq be a separable metric space, let pF, δq be a metric space,and let f PMpBpEq,BpF qq. Then f is strongly BpEqpF, δq-measurable.

Lemma 4.3.6. Let pV, ¨V q and pW, ¨W q be R-Banach spaces and let T P p0,8q,S PMpBpp0, T qq,BpLpW,V qqq, w P W . Then it holds that the function p0, T q Q s ÞÑST´sw P V is strongly Bpp0, tqqpV, ¨V q-measurable.

Proof of Lemma 4.3.6. Throughout this proof let ε : LpW,V q Ñ V and g : p0, T q ÑV be the functions which satisfy for all A P LpW,V q that

εpAq “ Aw and g “ ε ˝ S. (4.108)

Note that for all a, b P R, A,B P LpW,V q it holds that

εpaA` bBq “ aAw ` bBw “ a εpAq ` b εpBq (4.109)

and

εpAqV “ AwV ď ALpW,V qwW . (4.110)


Therefore, we obtain thatε P LpLpW,V q, V q. (4.111)

This, in particular, proves that ε is BpLpW,V qqBpV q-measurable. This and theassumption S PMpBpp0, T qq,BpLpW,V qqq yield that

g “ ε ˝ S PMpBpp0, T qq,BpV qq. (4.112)

The fact that pp0, T q Q s ÞÑ pT ´sq P p0, T qq PMpBpp0, T qq,Bpp0, T qqq hence ensuresthat

pp0, T q Q s ÞÑ ST´sw P V q PMpBpp0, T qq,BpV qq. (4.113)


Lemma 4.3.7. Let pV, ¨V q and pW, ¨W q be R-Banach spaces and let T P p0,8q,S PMpBpp0, T qq,BpLpW,V qqq, y P Cpr0, T s,W q satisfy that

infαPp0,1q

suptPp0,T q

tαStLpW,V q ă 8. (4.114)

Then

(i) it holds that the function p0, T q Q s ÞÑ ST´s ys P V is strongly Bpp0, tqqpV, ¨V q-measurable and

(ii) it holds thatşT

0ST´s ysV ds ă 8.

Proof of Lemma 4.3.7. Throughout this proof let α P p0, 1q and x P Mpp0, T q, V qsatisfy for all s P p0, T q that xpsq “ ST´s ys and

suptPp0,T q

tαStLpW,V q ă 8, (4.115)

let t¨uh : R Ñ R, h P p0,8q, be the functions which satisfy for all h P p0,8q, s P Rthat

tsuh “ max pp´8, ss X t0, h,´h, 2h,´2h, . . .uq , (4.116)

and let xn : p0, T q Ñ V , n P N, be the functions which satisfy for all s P p0, T q, n P Nthat

xnpsq “ ST´s ytsuTn

. (4.117)


Next observe that the assumption that y P Cpr0, T s,W q proves for all s P p0, T q that

lim supnÑ8

xnpsq ´ xpsqV ď ST´sLpW,V qytsuTn

´ ysV “ 0. (4.118)

Moreover, note that for all h P p0,8q, s P p0, T q it holds that

ST´s ytsuh “

tTh

u1ÿ

k“0

1rkh,pk`1qhqpsqST´s ykh. (4.119)

Lemma 4.3.6 and Corollary 2.3.11 therefore prove that the functions xn, n P N,are strongly Bpp0, T qqpV, ¨V q-measurable. Combining this together with (4.118)and Lemma 2.3.9 yields that the function x is also strongly Bpp0, T qqpV, ¨V q-measurable. This establishes Item (i). Next observe that (4.115) proves that

ż T

0

ST´sLpW,V q ds ď“

supsPp0,T q sαSsLpW,V q

‰

ż T

0

pT ´ sq´α ds

“T 1´α

p1´ αq

“


‰

ă 8.

(4.120)

This together with the assumption that y P Cpr0, T s,W q ensures thatż T

0

ST´s ysV ds ď

ż T

0

ST´sLpW,V qysW ds

ď“

supsPr0,T s ysW‰

ż T

0

ST´sLpW,V q ds ă 8.

(4.121)

This proves Item (ii). The proof of Lemma 4.3.7 is thus completed.


4.3.3 Local existence of mild solutions

Theorem 4.3.8. Let pV, ¨V q and pW, ¨W q be R-Banach spaces, and let T P p0,8q,F P CpV,W q, S P MpBpp0, T qq,BpLpW,V qqq, o P Cpr0, T s, V q satisfy for all r Pr0,8q that sup

`

F pvq´F pwqWv´wV

: v, w P V, v ‰ w, vV ` wV ď r(

Y t0u˘

` infαPp0,1qsuptPp0,T q t

αStLpW,V q ă 8. Then there exists a real number τ P p0, T s such that thereexists a unique continuous function x : r0, τ s Ñ V such that

(i) it holds for all t P p0, τ s that the function p0, tq Q s ÞÑ St´sF pxsq P V is stronglyBpp0, tqq/pV, ¨V q-measurable,

(ii) it holds for all t P r0, τ s thatşt

0St´s F pxsqV ds ă 8, and

(iii) it holds for all t P r0, τ s that

xt “

ż t

0

St´s F pxsq ds` ot. (4.122)

Proof of Theorem 4.3.8. Throughout this proof let Ψ PMpr0,8q, r0,8qq be the func-tion which satisfies for all r P r0,8q that

Ψprq “ sup´!

F pvq´F pwqWv´wV

: v, w P V, v ‰ w, vV ` wV ď r)

Y t0u¯

, (4.123)

let α P p0, 1q be a real number which satisfies that suptPp0,T q tαStLpW,V q ă 8, let

R P r0,8q be the real number given by R “ supsPr0,T s osV , let Ξτ Ď Cpr0, τ s, V q, τ Pp0, T s, be the sets which satisfy for all τ P p0, T s that

Ξτ “

x P Cpr0, τ s, V q : suptPr0,τ s xtV ď R ` 1(

, (4.124)

and let Φτ : Cpr0, τ s, V q Ñ Cpr0, τ s, V q, τ P p0, T s, be the functions which satisfy forall τ P p0, T s, t P r0, τ s, x P Cpr0, τ s, V q that

pΦτ pxqqptq “

ż t

0

St´s F pxsq ds` ot. (4.125)

Observe that Lemma 4.3.7 and the assumption that o P Cpr0, T s, V q ensure that thefunctions Φτ , τ P p0, T s, are well-defined. Moreover, observe that for all τ P p0, T s,


t P r0, τ s it holds that

pΦτ p0qqptqV

ď otV `

ż t

0

St´s F p0qV ds ď otV `

ż t

0

St´sLpW,V qF p0qW ds

ď otV `“


‰

F p0qW

ż t

0

pt´ sq´α ds

ď R `t1´α

1´ α

“


‰

F p0qW ă 8.

(4.126)

Hence, we obtain for all τ P p0, T s that

Φτ p0qCpr0,τ s,V q ď R `τ 1´α

1´ α

“


‰

F p0qW ă 8. (4.127)

Next note that for all τ P p0, T s, t P r0, τ s, x, y P Ξτ it holds that

pΦτ pxqqptq ´ pΦτ pyqqptqV “

›

›

›

›

ż t

0

St´srF pxsq ´ F pysqs ds

›

›

›

›

V

ď

ż t

0

St´sLpW,V qF pxsq ´ F pysqV ds

ď ΨpsupsPr0,τ srxsV ` ysV sq“


‰

¨

ż t

0

pt´ sq´αxs ´ ysV ds

ďt1´α

1´ αΨp2R ` 2q

“


‰

x´ yCpr0,τ s,V q ă 8.

(4.128)

Therefore, we obtain that for all τ P p0, T s, x, y P Ξτ it holds that

Φτ pxq ´ Φτ pyqCpr0,τ s,V q

ďτ 1´α

1´ αΨp2R ` 2q

“


‰

x´ yCpr0,τ s,V q ă 8.(4.129)

Combining this and (4.127) ensures that for all τ P p0, T s, x P Ξτ it holds that

Φτ pxqCpr0,τ s,V q ď Φτ pxq ´ Φτ p0qCpr0,τ s,V q ` Φτ p0qCpr0,τ s,V q

ďτ 1´α

1´ αΨp2R ` 2q

“


‰

xCpr0,τ s,V q

`R `τ 1´α

1´ α

“


‰

F p0qW

ď R `τ 1´α

1´ α

“


‰“

Ψp2R ` 2qpR ` 1q ` F p0qW‰

ă 8.

(4.130)


The fact that α ă 1 together with (4.129) therefore implies that there exists a realnumber τ P p0, T s such that for all x, y P Ξτ it holds that

Φτ pxqCpr0,τ s,V q ď R ` 1 (4.131)

and

Φτ pxq ´ Φτ pyqCpr0,τ s,V q ď12x´ yCpr0,τ s,V q. (4.132)

This ensures that Φτ pΞτ q Ď Ξτ . The Banach fixed point theorem, Corollary 4.3.4,and (4.132) hence prove that there exists a unique function x P Ξτ such that

Φτ pxq “ x. (4.133)

This completes the proof of Theorem 4.3.8.

Lemma 4.3.9 (On the nonlinearity in the Burgers equation). Let pH, x¨, ÿH , ¨Hq “pL2pBorelp0,1q; |¨|Rq, x¨, ÿL2pBorelp0,1q;|¨|Rq

, ¨L2pBorelp0,1q;|¨|Rqq, let c P R, let A : DpAq Ď

H Ñ H be the Laplace operator with Dirichlet boundary conditions on H, and let

pV, x¨, ÿV , ¨V q “`

DppÁq12q,⟨pÁq12p¨q, pÁq12p¨q

⟩H , pÁq

12p¨qH

˘

. (4.134)

Then

(i) it holds that V Ď L8pBorelp0,1q;Rq,

(ii) it holds for all v P V thatv1 ¨ v P H, (4.135)

and

(iii) there exists a continuous function F : V Ñ H which satisfies for all v, w P Vthat

F pvq “ ć ¨v1 ¨v and F pvq´F pwqH ď|c|?

3

“

vV `wV‰

v´wV . (4.136)

Proof of Lemma 4.3.9. Observe that Lemma 4.2.4 and the Hölder inequality ensurethat for all v P V , w P H it holds that

v P L8pBorelp0,1q;Rq, (4.137)

v ¨ w P H, (4.138)


and

v ¨ wH ď vL8pBorelp0,1q;RqwH ď

?2

π

«

8ÿ

n“1

1

n2

ff12

vV wH

“

?2

π

„

π2

6

12

vV wH “

?2

?6vV wH “

1?

3vV wH .

(4.139)

Combining this with Lemma 4.2.1 shows that for all v, w P V it holds that

v P L8pBorelp0,1q;Rq, (4.140)

v1 ¨ v P H, (4.141)

and

v1 ¨ v ´ w1 ¨ wH ď v1¨ v ´ v1 ¨ wH ` v

1¨ w ´ w1 ¨ wH

“ v1 ¨ pv ´ wqH ` pv ´ wq1¨ wH

ďv1H v ´ wV?

3`pv ´ wq1H wV?

3

“vV v ´ wV?

3`v ´ wV wV?

3“v ´ wV

“

vV ` wV‰

?3

.

(4.142)


The next result, Lemma 4.3.10, is an immediate consequence of Theorem 1.10.7in Subsection 1.10.3 above.


Lemma 4.3.10 (Linear dynamics in the Burgers equation). Let pH, x¨, ÿH , ¨Hq “pL2pBorelp0,1q; |¨|Rq, x¨, ÿL2pBorelp0,1q;|¨|Rq


the Laplace operator with Dirichlet boundary conditions on H, and let

pV, x¨, ÿV , ¨V q “`

DppÁq12q,⟨pÁq12p¨q, pÁq12p¨q

⟩H , pÁq

12p¨qH

˘

. (4.143)

Then

(i) it holds for all t P p0, T q, v P H that

etAv P V (4.144)

and

(ii) there exists a continuous function S : p0, T q Ñ LpH,V q which satisfies for allt P p0, T q, v P H that

Stv “ etAv (4.145)

and

infαPp0,1q

supsPp0,T q

`

sα SsLpH,V q˘

ď supsPp0,T q

`?s SsLpH,V q

˘

ă 8. (4.146)

The next result, Corollary 4.3.11, follows directly from Theorem 4.3.8, Lemma 4.3.9,and Lemma 4.3.10.

Corollary 4.3.11 (Burgers equations). Let pH, x¨, ÿH , ¨Hq “ pL2pBorelp0,1q; |¨|Rq,x¨, ÿL2pBorelp0,1q;|¨|Rq

, ¨L2pBorelp0,1q;|¨|Rqq, let A : DpAq Ď H Ñ H be the Laplace operator

with Dirichlet boundary conditions on H, let pHr, 〈¨, ¨〉Hr , ¨Hrq, r P R, be a familyof interpolation spaces associated to Á, let ξ P H12, c P R, and let F : H12 Ñ H bethe function which satisfies for all v P H12 that F pvq “ ć ¨ v1 ¨ v (cf. Lemma 4.3.9).Then there exists a real number T P p0,8q such that there exists a unique continuousfunction x : r0, T s Ñ H12 such that

(i) it holds for all t P p0, T s that the function p0, tq Q s ÞÑ ept´sqAF pxsq P H12 iscontinuous,

(ii) it holds for all t P r0, T s thatşt

0ept´sqA F pxsqH12

ds ă 8, and

(iii) it holds for all t P r0, T s that

xt “ etAξ `

ż t

0

ept´sqA F pxsq ds. (4.147)


Remark 4.3.12 (Burgers equations). Let pH, x¨, ÿH , ¨Hq “ pL2pBorelp0,1q; |¨|Rq,x¨, ÿL2pBorelp0,1q;|¨|Rq

, ¨L2pBorelp0,1q;|¨|Rqq, let A : DpAq Ď H Ñ H be the Laplace operator

with Dirichlet boundary conditions on H, let pHr, 〈¨, ¨〉Hr , ¨Hrq, r P R, be a familyof interpolation spaces associated to Á, let ξ P H12, T P p0,8q, let F : H12 Ñ Hbe the function which satisfies for all v P H12 that F pvq “ ´v1 ¨ v (cf. Lemma 4.3.9),and let x : r0, T s Ñ H12 be a continuous function such that

(i) it holds for all t P p0, T s that the function p0, tq Q s ÞÑ ept´sqAF pxsq P H12 iscontinuous,

(ii) it holds for all t P r0, T s thatşt

0ept´sqA F pxsqH12

ds ă 8, and

(iii) it holds for all t P r0, T s that

xt “ etAξ `

ż t

0

ept´sqA F pxsq ds. (4.148)

Then x is referred to as a mild solution of the Burgers equation

B

Btupt, xq “ B2

Bx2upt, xq ´ upt, xq ¨B

Bxupt, xq (4.149)

with up0, xq “ ξpxq and upt, 0q “ upt, 1q “ 0 for t P r0, T s, x P p0, 1q.

Bibliography

[1] Cohn, D. L. Measure theory. Birkhäuser Boston Inc., Boston, MA, 1993. Reprintof the 1980 original.

[2] Cox, S., Hutzenthaler, M., Jentzen, A., van Neerven, J., and Welti,T. Convergence in Hölder norms with applications to Monte Carlo methods ininfinite dimensions. arXiv:1605.00856 (2016), 38 pages.

[3] Da Prato, G., and Zabczyk, J. Stochastic equations in infinite dimensions,vol. 44 of Encyclopedia of Mathematics and its Applications. Cambridge Univer-sity Press, Cambridge, 1992.

[4] Hytönen, T., van Neerven, J., Veraar, M., and Weis, L. Analysis inBanach Spaces: Volume I: Martingales and Littlewood-Paley Theory. Springer,2016.

[5] Pazy, A. Semigroups of linear operators and applications to partial differentialequations, vol. 44 of Applied Mathematical Sciences. Springer-Verlag, New York,1983.

[6] Prévôt, C., and Röckner, M. A concise course on stochastic partial dif-ferential equations, vol. 1905 of Lecture Notes in Mathematics. Springer, Berlin,2007. 144 pages.

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