Partial Differential Equations and Semigroups of Bounded Linear Operators

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1 Partial Differential Equations and Semigroups of Bounded Linear Operators (Currently under construction!) Arnulf Jentzen December 23, 216

2 2 Preface These lecture notes are based on the lecture notes Stochastic Partial Differential Equations: Analysis and Numerical Approximations. These lecture notes are far away from being complete and remain under construction. In particular, these lecture notes do not yet contain a suitable comparison of the presented material with existing results, arguments, and notions in the literature. Furthermore, these lecture notes do not contain a number of proofs, arguments, and intuitions. For most of this additional material, the reader is referred to the lectures of the course L Partial Differential Equations and Semigroups of Bounded Linear Operators in the spring semester 216. Special thanks are due to Sonja Cox, Ryan Kurniawan, Primoz Pusnik, Diyora Salimova, and Timo Welti for their very helpful advice and their very useful assistance. The students of the courses L Numerical Analysis of Stochastic Partial Differential Equations are gratefully acknowledged for pointing out a number of misprints to me. Zürich, September 216 Arnulf Jentzen

3 3 Exercises Solutions to the exercises can be turned in the designated mailbox in the anteroom HG G 53.x. Exercise Exercises Deadline sheet 1 Exercises 1.3.5, 1.9.3, 1.9.4, and , 15:15 AM 2 Exercises , , , and , 15:15 AM 3 Exercises 2.3.7, 4.1.1, and , 15:15 AM

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5 Contents 1 Semigroups of bounded linear operators Preliminaries Definition of a semigroup of bounded linear operators Types of semigroups The generator of a semigroup Global a priori bounds for semigroups Strongly continuous semigroups A priori bounds for strongly continuous semigroups The Baire category theorem on complete metric spaces The uniform boundedness principle Local a priori bounds Global a priori bounds Existence of solutions of linear ordinary differential equations in Banach spaces Pointwise convergence in the space of bounded linear operators Domains of generators of strongly continuous semigroups Generators of strongly continuous semigroups A generalization of matrix exponentials to infinite dimensions A characterization of strongly continuous semigroups Uniformly continuous semigroups Matrix exponential in Banach spaces Continuous invertibility of bounded linear operators in Banach spaces Generators of uniformly continuous semigroup A characterization result for uniformly continuous semigroups An a priori bound for uniformly continuous semigroups The Hille-Yosida theorem Yosida approximations

6 6 CONTENTS Scalar shifts of generators of strongly continuous semigroups Convergence of linear-implicit Euler approximations Properties of Yosida approximations Convergence of Yosida approximations Semigroups generated by Yosida approximations A characterization for generators of strongly continuous contraction semigroups Diagonal linear operators on Hilbert spaces Semigroups generated by diagonal linear operators A characterization for strongly continuous semigroups generated by diagonal linear operators Contraction semigroups generated by diagonal linear operators Smoothing effect of the semigroup Semigroup generated by the Laplace operator Nonlinear functions and nonlinear spaces Continuous functions Topological spaces Topological spaces induced by distance-type functions Semi-metric spaces Continuity properties of functions Uniform continuity Hölder continuity Modulus of continuity Properties of the modulus of continuity Convergence of the modulus of continuity Extensions of uniformly continuous functions Measurable functions Nonlinear characterization of the Borel sigma-algebra Pointwise limits of measurable functions Strongly measurable functions Simple functions Separability Strongly measurable functions Pointwise approximations of strongly measurable functions Sums of strongly measurable functions

7 CONTENTS 7 3 The Bochner integral Sets of integrable functions ˆLp -sets of measurable functions for p P r, 8q L p -spaces of strongly measurable functions for p P r, 8q Existence and uniqueness of the Bochner integral Definition of the Bochner integral Nonlinear partial differential equations Diagonal linear operators on Hilbert spaces Closedness of diagonal linear operators Laplace operators on bounded domains Laplace operators with Dirichlet boundary conditions Laplace operators with Neumann boundary conditions Laplace operators with periodic boundary conditions Spectral decomposition for a diagonal linear operator Fractional powers of a diagonal linear operator Domain Hilbert space associated to a diagonal linear operator Completion of metric spaces An extension of a metric space An artificial completion The natural completion Further completions Interpolation spaces associated to a diagonal linear operator The Sobolev space W 1,2 pp, 1q, Rq Hpp, 1 1q, Rq Weak derivatives in W 1,2 pp, 1q, Rq Hpp, 1 1q, Rq A special case of the Sobolev embedding theorem Nonlinear evolution equations Complete function spaces Measurability properties Local existence of mild solutions

8 8 CONTENTS

9 Chapter 1 Semigroups of bounded linear operators In this chapter we mostly follow the presentations in Pazy [5]. 1.1 Preliminaries Definition (Power set). Let A be a set. Then we denote by PpAq the power set of A (the set of all subsets of A). Definition (Set of functions). Let A and B be sets. MpA, Bq the set of all functions from A to B. Then we denote by 1.2 Definition of a semigroup of bounded linear operators Definition (Semigroups of bounded linear operators). Let K P tr, Cu and let 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 is a semigroup) if and only if S P Mpr, 8q, LpV qq is a function from r, 8q to LpV q which satisfies for all t 1, t 2 P r, 8q that S Id V and S t1 S t2 S l jh t1`t n 2. (1.1) semigroup property 9

10 1 CHAPTER 1. SEMIGROUPS OF BOUNDED LINEAR OPERATORS 1.3 Types of semigroups Definition (Contraction semigroups). Let K P tr, Cu, let pv, } } V q be a normed K-vector space, and let S : r, 8q Ñ LpV q be a semigroup. Then we say that S is contractive if and only if it holds that sup }S t } LpV q ď 1. (1.2) tpr,8q Definition (Strongly continuous semigroups). Let K P tr, Cu, let pv, } } V q be a normed K-vector space, and let S : r, 8q Ñ LpV q be a semigroup. Then we say that S is strongly continuous if and only if it holds for every v P V that the function is continuous. r, 8q Q t ÞÑ S t v P V (1.3) Definition (Uniformly continuous semigroups). Let K P tr, Cu, let pv, } } V q be a normed K-vector space, and let S : r, 8q Ñ LpV q be a semigroup. Then we say that S is uniformly continuous if and only if the function is continuous. r, 8q Q t ÞÑ S t P LpV q (1.4) Example (Matrix exponential). Let d P N and let A P R dˆd be an arbitrary d ˆ d-matrix. Then the function is a uniformly continuous semigroup. r, 8q Q t ÞÑ e At P R dˆd (1.5) Clearly, it holds that every uniformly continuous semigroup is also strongly continuous. However, not every strongly continuous semigroup is uniformly continuous too. This is the subject of the next exercise. Exercise Give an example of an R-Banach space pv, } } V q and a strongly continuous semigroup S : r, 8q Ñ LpV q so that S is not a uniformly continuous semigroup. Prove that your function S does indeed fulfill the desired properties.

11 1.4. THE GENERATOR OF A SEMIGROUP The generator of a semigroup Definition (Generator). Let K P tr, Cu, let pv, } } V q be a normed K-vector space, and let S : r, 8q Ñ LpV q be a semigroup. Then we denote by G S : DpG S q Ď V Ñ V the function with the property that DpG S q " v P V : ˆ j St v v and with the property that for all v P DpG S q it holds that j St v v G S v lim tœ t t j* converges as p, 8q Q t Œ and we call G S the generator of S (we call G S the infinitesmal generator of S). (1.6) (1.7) In the next notion we label all linear operators that are generators of strongly continuous semigroups. Definition (Generator of a strongly continuous semigroup). Let K P tr, Cu and let pv, } } V q be a normed K-vector space. Then we say that A is the generator of a strongly continuous semigroup on V (we say that A is the generator of a strongly continuous semigroup) if and only if there exists a strongly continuous semigroup S : r, 8q Ñ LpV q of bounded linear operators on V such that G S A. (1.8) We complete this section with a simple exercise which aims to illustrate and relate the different concepts introduced above. Exercise Let K P tr, Cu, let pv, } } V q be a normed K-vector space with # V ą 1, and let S : r, 8q Ñ LpV q be the function which satisfies for all t P r, 8q that # S t Id V : t : t ą. (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 DpG S q and G S.

12 12 CHAPTER 1. SEMIGROUPS OF BOUNDED LINEAR OPERATORS 1.5 Global a priori bounds for semigroups In the next result, Proposition 1.5.1, we present a global a priori bound for semigroups of bounded linear operators. Proposition (Global a priori bound). Let K P tr, Cu, let pv, } } V q be a normed K-vector space, and let S : r, 8q Ñ LpV q be a semigroup. Then it holds for all t P r, 8q, ε P p, 8q that sup spr,ts }S s } LpV q ď sup spr,εs }S s } LpV q e t lnp}s ε} 1{ε LpV qq `. (1.1) Proof of Proposition Note that for all t P r, 8q, ε P p, 8q, n P N Xp t {ε 1, t {εs it holds that }S t } LpV q LpV Snε`pt nεq q Snε S LpV pt nεq q ď }S nε } LpV LpV q Spt nεq q }rs ε s n } LpV LpV q Spt nεq q ď }S ε } n LpV LpV q Spt nεq q ď max ( 1, }S ε } n LpV q supspr,εs }S s } LpV q ( n sup spr,εs }S s } LpV q max 1, }Sε } LpV q ď ( t{ε sup spr,εs }S s } LpV q max 1, }Sε } LpV q ) sup spr,εs }S s } LpV q max!e, exp t ln }S ε } 1{ε LpV q ď sup spr,εs }S s } LpV q exp t max (, ln`}s ε } 1{ε LpV q. (1.11) This completes the proof of Proposition Strongly continuous semigroups A priori bounds for strongly continuous semigroups In Corollary below we present a global priori bound for strongly continuous semigroups. The proof of Corollary uses the local a priori bound in Lemma below. The proof of Lemma 1.6.7, in turn, exploits the uniform boundedness principle. This is the subject of the next result.

13 1.6. STRONGLY CONTINUOUS SEMIGROUPS The Baire category theorem on complete metric spaces Lemma (A set contains an open ball). Let pe, d E q be a metric space and let A P PpEq. Then EzA E if and only if there exist ε P p, 8q, x P E such that ty P E : d E px, yq ă εu Ď A. (1.12) Proof of Lemma Observe that EzA E ô EzA is dense in E x P E ε P p, 8q: D y P pezaq: d E px, yq ă ε x P E ε P p, 8q: pezaq X ty P E : d E px, yq ă εu H. (1.13) This implies that EzA E ô D x P E : D ε P p, 8q: pezaq X ty P E : d E px, yq ă εu H ô D x P E : D ε P p, 8q: ty P E : d E px, yq ă εu Ď A. (1.14) The proof of Lemma is thus completed. Corollary (A characterization for density). Let pe, d E q be a metric space and let A P PpEq. Then the following three statements are equivalent: (i) It holds that A E. (ii) It holds for all ε P p, 8q, x P E that A X ty P E : d E px, yq ă εu H. (iii) It holds for all non-empty open sets O P PpEq that A X O H. Proof of Corollary Note that Item (ii) and Item (iii) are equivalent. Moreover, observe that Lemma shows that Item (i) and Item (ii) are equivalent. The proof of Corollary is thus completed. Lemma Let pe, d E q be a metric space, let U P PpEq, let V P PpEq be an open set, and assume that U V E. Then U X V E.

14 14 CHAPTER 1. SEMIGROUPS OF BOUNDED LINEAR OPERATORS Proof of Lemma First of all, observe that the assumption that V is an open set proves that for every open set W Ď E it holds that V X W (1.15) is an open set. Next observe that Corollary proves that for every non-empty open set W Ď E and every set E P PpEq with E E it holds that pe X W q H. (1.16) This and the assumption that V E assure that for every non-empty open set W Ď E it holds that pv X W q H. (1.17) Combining this with (1.15) ensures that for every non-empty open set W Ď E it holds that V X W (1.18) is a non-empty open set. The assumption that U E together with (1.16) hence implies that for all non-empty open sets W Ď E it holds that pu X V q X W U X pv X W q H. (1.19) Combining this with Corollary completes the proof of Lemma Theorem (Baire category theorem for complete metric spaces). Let pe, d E q be a complete metric space and let A n Ď E, n P N, be closed subsets of E which satisfy Ez ry npn A n s E. Then there exists a natural number n P N such that EzA n E. Proof of Theorem Throughout this proof let N n P PpNq, n P N Y t8u, be the sets which satisfy for all n P N Y t8u that N n k P pn X r, nsq: EzA k E ( (1.2) and let U Ď E be a non-empty open set (why does such a set exist?). We intend to prove that N 8 N. We may thus assume w.l.o.g. that N 8 is not a finite set. The fact that for every n P N 8 it holds that EzA n is an open set, the fact n P N 8 : EzA n E, and Lemma assure that for all n P N it holds that X kpnn reza k s E. (1.21) Hence, we obtain that for all n P N and all non-empty open sets B P PpEq it holds that ` XkPNn reza k s X B H. (1.22)

15 1.6. STRONGLY CONTINUOUS SEMIGROUPS 15 Combining this with the fact that every n P N it holds that X kpnn reza k s (1.23) is an open set ensures that for every n P N and every non-empty open set B P PpRq it holds that ` XkPNn reza k s X B (1.24) is a non-empty open set. This assures that for every n P N and every non-empty open set B P PpRq there exist w P E and ε P p, 8q which satisfy ` tv P E : d E pw, vq ă εu Ď XkPNn reza k s X B. (1.25) This implies the existence of functions e pe n q npn : N Ñ E and r pr n q npn : N Ñ p, 8q which satisfy for all n P N that r n ă 1 2 n and tv P E : d E pe n, vq ă 2r n u Ď px kpnn reza k sq X Hence, we obtain that for all n P N it holds that U X ` X kpnn 1 v P E : de pe k, vq ă r k ( ı. (1.26) tv P E : d E pe n, vq ă r n u Ď tv P E : d E pe n, vq ă 2r n u Ď ` X kpnn 1 v P E : de pe k, vq ă r k (. (1.27) Therefore, we get that for all n P N, k P N n 1 it holds that tv P E : d E pe n, vq ă r n u Ď ( v P E : d E pe k, vq ă r k. (1.28) This assures that for all m, n P N 8 with m ď n it holds that tv P E : d E pe n, vq ă r n u Ď tv P E : d E pe m, vq ă r m u. (1.29) In particular, we obtain that for all m, n P N 8 with m ď n it holds that e n P tv P E : d E pe m, vq ă r m u. (1.3) Therefore, we get that for all m, n P N 8 with m ď n it holds that d E pe m, e n q ă r m. (1.31)

16 16 CHAPTER 1. SEMIGROUPS OF BOUNDED LINEAR OPERATORS This together with the fact that lim sup mñ8 r m shows that e n P E, n P N 8, is a Cauchy sequence in pe, d E q. The completeness of pe, d E q hence establishes the existence of an e P E which satisfies lim sup d E pe, e n q. (1.32) N 8Q Combining this with (1.31) implies that for all m P N 8 it holds that d E pe m, eq lim sup d E pe m, e n q ď lim sup r m r m. (1.33) N 8Q N 8Q This and (1.26) assure that for all n P N 8 it holds that e P tv P E : d E pe n, vq ď r n u Ď tv P E : d E pe n, vq ă 2r n u Ď Therefore, we get that ı e P X npn8 px kpnn reza k sq X U px kpnn reza k sq X U. (1.34) px kpn8 reza k sq X U. (1.35) This establishes, in particular, that px kpn8 reza k sq X U H. (1.36) As U was an arbitrary non-empty open set, we obtain from Corollary that X kpn8 reza k s E. (1.37) This together with the fact that X kpn reza k s EzrY kpn A k s E ensures that N 8 N. Hence, we obtain that rnzn 8 s H. (1.38) The proof of Theorem is completed. Corollary (Baire category theorem for complete metric spaces on the density of the countable intersection of dense open sets). Let pe, d E q be a complete metric space and let O n Ď E, n P N, be open and dense subsets of E. Then it holds that X npn O n is a dense subset of E.

17 1.6. STRONGLY CONTINUOUS SEMIGROUPS The uniform boundedness principle Theorem (Uniform boundedness principle). Let K P tr, Cu, let pu, } } U q be a K-Banach space, let pv, } } V q be a normed K-vector space, and let A Ď LpU, V q be a non-empty set which satisfies for all u P Uztu that sup }Au} V ă 8. (1.39) APA Then ˆ" sup }A} LpU,V q sup APA sup APA }Au}V }u} U ı * : u P Uztu Y tu ă 8. (1.4) Proof of Theorem Throughout this proof let U n Ď U, n P N, be the sets which satisfy for all n P N that " * U n u P U : sup }Au} V ď n APA. (1.41) Note that the triangle inequality shows that for all A P A, n, K P N, v P U, pu k q kpn Ď U n with lim sup kñ8 }v u k } U it holds that }Av} V ď }Apv u K q} V ` }Au K } V ď }A} LpU,V q }v u K } U ` n. (1.42) This implies that for all A P A, n P N, v P U, pu k q kpn Ď U n with lim sup kñ8 }v u k } U it holds that }Av} V lim sup }Av} V ď lim sup }A}LpU,V q }v u K } U ` n n. (1.43) KÑ8 KÑ8 This establishes that for all n P N, v P U, pu k q kpn Ď U n with lim sup kñ8 }v u k } U it holds that sup }Av} V ď n. (1.44) APA Hence, we obtain that for all n P N, v P U, pu k q kpn Ď U n with lim sup kñ8 }v u k } U it holds that v P U n. This proves that for every n P N it holds that U n is a closed set. In addition, note that assumption (1.39) ensures that Y npn U n U. (1.45) This shows that UzrY npn U n s H H U. (1.46)

18 18 CHAPTER 1. SEMIGROUPS OF BOUNDED LINEAR OPERATORS Theorem and the fact that for every n P N it holds that U n is a closed set therefore show that there exists a natural number n P N such that UzU n U. (1.47) Combining this with Lemma implies that there exist v P U n and ε P p, 8q such that tu P U : }u v} U ă 2εu Ď U n. (1.48) This shows that for all A P A, u P U with }u} U ď 1 it holds that Hence, we obtain that }Au} V 1 ε }Apεuq} V 1 ε }Apv ` εuq Av} V ď 1 ε r}apv ` εuq} V ` }Av} V s ď 1 ε pn ` nq 2n ε. (1.49) sup APA }A} LpU,V q sup sup APA upu,}u} U ď1 }Au} V ď 2n ε ă 8. (1.5) The proof of Theorem is thus completed Local a priori bounds Lemma (Local a priori bound). Let K P tr, Cu, let pv, } } V q be a K-Banach space, and let S : r, 8q Ñ LpV q be a semigroup which satisfies for all v P V that lim sup tœ }S t v v} V. Then lim sup tœ }S t } LpV q lim sup }S s } LpV q ă 8. (1.51) tœ spr,ts Proof of Lemma We prove Lemma by a contradiction. More specifically, we assume in the following that lim sup }S s } LpV q 8. (1.52) tœ spr,ts This and the fact that }S } LpV q 1 ă 8 imply that for all t P p, 8q it holds that sup }S s } LpV q 8. (1.53) spp,ts Hence, there exists a strictly decreasing sequence t n P p, 8q, n P N, with lim t n and with the property that for all n P N it holds that }S tn } LpV q ě n. (1.54)

19 1.6. STRONGLY CONTINUOUS SEMIGROUPS 19 This ensures that sup }S tn } LpV q 8. (1.55) npn Theorem hence implies that there exists a vector v P V such that sup }S tn v} V 8. (1.56) npn Combining this and the fact n P N: }S tn v} V ă 8 implies that lim sup }S tn v} V 8. (1.57) This and the assumption w P V : lim sup tœ }S t w w} V show that 8 ą }v} V lim rs tn vs lim }S tn v} V lim sup }S tn v} V 8. (1.58) V This contradiction completes the proof of Lemma Global a priori bounds The next result, Corollary 1.6.8, proves a stronger version of Lemma Observe that Lemma and Corollary apply to strongly continuous semigroups on Banach spaces. Corollary (Global a priori bound). Let K P tr, Cu, let pv, } } V q be a K- Banach space, and let S : r, 8q Ñ LpV q be a semigroup which satisfies for all v P V that lim sup tœ }S t v v} V. Then it holds for all t P r, 8q, ε P p, 8q that sup spr,ts }S s } LpV q ď sup spr,εs }S s } LpV q e t lnp}s ε} 1{ε LpV qq ` ă 8. (1.59) Corollary is an immediate consequence of Proposition and Lemma above Existence of solutions of linear ordinary differential equations in Banach spaces Lemma (Invariance of the domain of the generator). Let K P tr, Cu, let pv, } } V q be a normed K-vector space, and let S : r, 8q Ñ LpV q be a semigroup. Then it holds for all t P r, 8q, v P DpG S q that S t`dpgs q Ď DpG S q and G S S t v S t G S v. (1.6)

20 2 CHAPTER 1. SEMIGROUPS OF BOUNDED LINEAR OPERATORS Proof of Lemma Observe that for all t P r, 8q, v P DpG S q it holds that j jj jj Ss rs t vs rs t vs Ss v v Ss v v lim lim S t S t lim S t G S v. sœ s sœ s sœ s (1.61) This completes the proof of Lemma Lemma Let K P tr, Cu, let pv, } } V q be a K-Banach space, let S : r, 8q Ñ LpV q be a strongly continuous semigroup, and let v P DpG S q. Then (i) it holds that the function r, 8q Q t ÞÑ S t v P V is continuously differentiable and (ii) it holds for all t P r, 8q that d rs dt tvs G S S t v S t G S v. (1.62) Proof of Lemma Observe that for all s, t P r, 8q with s t it holds that S s v S t v S t G S v s t V j S Ss mints,tu v S t mints,tu v mints,tu S t G S v s t V ď S Smaxts,tu mints,tu v v V minps,tq Svj maxts, tu mints, tu G ` S mints,tu S t GS v (1.63) V ď Sminps,tq LpV S maxts,tu mints,tu v v q maxts, tu mints, tu G Sv ` S mints,tu S t GS v V. V Corollary and the fact that S is strongly continuous hence imply that for all t P r, 8q it holds that lim sup S s v S t v S t G S v r,8qzttuqsñt s t V «ff «ff ď sup }S s } LpV q lim sup S maxts,tu mints,tu v v spr,t`1s r,8qzttuqsñt maxts, tu mints, tu G Sv (1.64) V ` lim sup S mints,tu S t GS v V. r,8qzttuqsñt This and Lemma complete the proof of Lemma

21 1.6. STRONGLY CONTINUOUS SEMIGROUPS Pointwise convergence in the space of bounded linear operators Lemma (A characterization of pointwise convergence in the space of bounded linear operators). Let K P tr, Cu, let pv, } } V q be a K-Banach space, and let ps n q npn Ď LpV q. v P V : lim sup }S n v S v} V if and only if it holds for all non-empty compact sets K Ď V that lim sup sup vpk }S n v S v} V. Proof of Lemma The proof of the ð direction in the statement of Lemma is clear. It thus remains to prove the ñ direction in the statement of Lemma To this end we assume that for all v P V it holds that lim sup }S n v S v} V (1.65) and we assume that there exists a non-empty compact set K Ď V such that lim sup sup }S n v S v} V ą. (1.66) vpk In the next step we note that there exists a sequence pv n q npn Ď K such that for all n P N it holds that }S n v n S v n } V sup }S n v S v} V. (1.67) vpk The compactness of K ensures that there exist a w P K and a strictly increasing sequence pn k q kpn Ď N such that lim sup }v nk w} V. (1.68) kñ8 Next note that (1.65) ensures that lim sup }S nk w S w} V. (1.69) kñ8

22 22 CHAPTER 1. SEMIGROUPS OF BOUNDED LINEAR OPERATORS This, (1.67), (1.68), and Theorem imply that lim sup }S nk w S w} V kñ8 lim sup }S nk pw v nk q ` ps nk S q v nk ` S pv nk wq} V kñ8 ě lim sup }psnk S q v nk } V }S nk pw v nk q} V }S pv nk wq} V kñ8 ě lim sup kñ8 ě lim sup kñ8 lim sup kñ8 }ps nk S q v nk } V lim sup kñ8 sup }ps nk S q v} V vpk }ps nk S q v} V ą. sup vpk }S nk pw v nk q} V lim sup }S pv nk wq} V kñ8 j sup }S nk } LpV q kpn This condradiction completes the proof of Lemma lim sup }w v nk } V kñ8 (1.7) Domains of generators of strongly continuous semigroups In this subsection we prove that the generator of a strongly continuous semigroup is densely defined; see Corollary below. In the proof of Corollary we use the following result, Lemma Lemma and its proof can, e.g., be found as Theorem 2.4 (b) in Pazy [5] and Corollary and its proof can, e.g., be found as Corollary 2.5 in Pazy [5]. Lemma (Fundamental theorem of calculus for strongly continuous semigroups). Let K P tr, Cu, t P r, 8q, let pv, } } V q be a K-Banach space, let S : r, 8q Ñ LpV q be a strongly continuous semigroup, and let v P V. Then it holds that ż t S s v ds P DpG S q and G Sˆż t S s v ds S t v v. (1.71) Proof of Lemma Throughout this proof we assume w.l.o.g. that t P p, 8q. Next we observe that for all u P p, tq it holds that rs u Id V s u ż t j S s v ds 1 u ż t rs u`s v S s vs ds 1 u ż t`u t S s v ds 1 u ż u S s v ds. (1.72)

23 1.6. STRONGLY CONTINUOUS SEMIGROUPS 23 Continuity of the function r, 8q Q s ÞÑ S s v P V hence proves that ş t S sv ds P DpG S q and that t ż rsu Id V s t jj G Sˆż S s v ds lim S s v ds S t v S v S t v v. (1.73) uœ u The proof of Lemma is thus completed. We are now ready to prove that the generator of a strongly continuous semigroup is densely defined. Corollary Let K P tr, Cu, let pv, } } V q be a K-Banach space, and let S : r, 8q Ñ LpV q be a strongly continuous semigroup. Then it holds that DpG S q is dense in V. Proof of Corollary Let v P V be arbitrary. The assumption that S is a strongly continuous semigroup together with the fundamental theorem of calculus ensures that ż ˆ1 t lim S s v ds S v v. (1.74) tœ t In addition, Lemma proves that for all t P p, 8q it holds that 1 t ż t S s v ds P DpG S q. (1.75) This and (1.74) imply that v P DpG S q. The proof of Corollary is thus completed Generators of strongly continuous semigroups In this section we show that a strongly continuous semigroup is uniquely determined by its generator; see Proposition below. In Proposition we use the assumption 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 following remark. Remark Let A 1, A 2, B be sets and let f 1 : A 1 Ñ B and f 2 : A 2 Ñ B be functions. Then it holds that Graphpf 1 q Ď Graphpf 2 q if and only if (A 1 Ď A 2 and f 2 A1 f 1 ).

24 24 CHAPTER 1. SEMIGROUPS OF BOUNDED LINEAR OPERATORS We are now ready to show that a strongly continuous semigroup is uniquely determined by its generator. Proposition (The generator determines the semigroup). Let K P tr, Cu, let pv, } } V q be a K-Banach space, and let S, S : r, 8q Ñ LpV q be strongly continuous semigroups with GraphpG S q Ď GraphpG Sq. Then it holds that S S and G S G S. Proof of Proposition Let v P DpG S q Ď DpG Sq, t P p, 8q and let η : r, ts Ñ V be the function which satisfies for all s P r, ts that ηpsq S t s S s v. (1.76) Next note that for all s, u P r, ts with s u it holds that ηpuq ηpsq S t u S u v u s S t s S s v V u s V j S Su v S s v St u t u ` S t s Ss v u s u s V ˆ St u S ı j j S u v S s v t s ` u s S Su v S s v St u t s ` S t s Ss v. u s u s V (1.77) The triangle inequality hence implies that for all s, u P r, ts with s u it holds that ηpuq ηpsq u s V St u S ı j j S u v S s v t s ` u s S Su v S s v t s u s ď St u S ı j S u v S s v t s u s V j ` S Su v S s v St u t s S t s Ss v. u s pt uq pt sq V St u S t s Ss v pt uq pt sq V (1.78) This implies that for all s P r, ts, n P N and all pu m q mpn P MpN, r, tsztsuq with

25 1.6. STRONGLY CONTINUOUS SEMIGROUPS 25 lim sup mñ8 u m s it holds that ηpu n q ηpsq u n s V j ď S Sun v S s v t s u n s St un S t s Ss v pt u n q pt sq ` sup! St un w S t s w V : w P tg S S s vu Y! V Sum v S sv u m s )) : m P N. (1.79) Next observe that Lemma and Lemma prove that for all s P r, ts it holds that «St u lim S ff t s Ss v G pr,tsztsuqquñs pt uq pt sq S St s S s v S t s G SS s v S t s G S S s v (1.8) and lim pr,tsztsuqquñs ` lim sup sup! St un w S t s w V : w P tg S S s vu Y j Su v S s v G S S s v. (1.81) u s Putting (1.8) (1.81) into (1.79) proves that for all s P r, ts and all pu n q npn r, tsztsu with lim sup u n s it holds that lim sup ηpu n q ηpsq u n s V j ď lim sup S Sun v S s v St un t s S t s Ss v u n s pt u n q pt sq )) : m P N St s G S S s v S t s G S S s v V ` lim sup sup! St un w S t s w V : w P tg S S s vu Y lim sup sup! St un w S t s w V : w P tg S S s vu Y! V Sum v S sv u m s! Sum v S sv u m s! Sum v S sv u m s )) : m P N )) : m P N. Ď (1.82) Next observe that the assumption that S is a strongly continuous semigroup ensures that for all s P r, ts, w P V and all pu n q npn Ď r, tsztsu with lim sup u n s it holds that lim sup St un w S t s w V. (1.83)

26 26 CHAPTER 1. SEMIGROUPS OF BOUNDED LINEAR OPERATORS Moreover, note that Lemma shows that for all s P r, ts and all pu n q npn Ď r, tsztsu with lim sup u n s it holds that ) tg S S s vu Y : m P N (1.84)! Sum v S sv u m s is a compact set. Combining this and Lemma with (1.83) and (1.82) proves that η is differentiable and that for all s P r, ts it holds that η 1 psq. The fundamental theorem of calculus hence implies that S t v ηpq ηptq S t v. (1.85) As v P DpG S q was arbitrary, we obtain that S t DpGS q S t DpGS q. Corollary hence proves that S t S t. This completes the proof of Proposition Lemma (Closedness of generators of strongly continuous semigroups). Let K P tr, Cu, let pv, } } V q be a K-Banach space, and let S : r, 8q Ñ LpV q be a strongly continuous semigroup. Then it holds that G S is a closed linear operator. Proof of Lemma Throughout this proof let x, y P V and let pv n q npn Ď DpG S q be a sequence which satisfies that lim sup }x v n } V lim sup }y G S v n } V. (1.86) Next note that Lemma and the fundamental theorem of calculus show that for all t P r, 8q, n P N it holds that S t v n v n ż t S s G S v n ds. (1.87) Furthermore, observe that (1.86) and Corollary imply that for all t P r, 8q it holds that ż t ż t ż t lim sup S s y ds S s G S v n ds ď lim sup }S s y S s G S v n } V ds V ď t (1.88) sup spr,ts }S s } LpV q lim sup }y G S v n } V. This, (1.86), and (1.87) prove that for all t P r, 8q it holds that S t x x ż t S s y ds. (1.89)

27 1.6. STRONGLY CONTINUOUS SEMIGROUPS 27 Again the fundamental theorem of calculus hence shows that lim sup S t x x t tœ y lim sup ˆ1 V t tœ ż t S s y ds y. (1.9) V This ensures that x P DpG S q and G S x y. completed. The proof of Lemma is thus A generalization of matrix exponentials to infinite dimensions Definition and Proposition ensure that the next concept, Definition , makes sense. Definition (Generalized matrix exponential). Let K P tr, Cu, t P r, 8q, let pv, } } V q be a K-Banach space, and let A: DpAq Ď V Ñ V be a generator of a strongly continuous semigroup. Then we denote by e ta P LpV q the linear operator which satisfies for every strongly continuous semigroup S : r, 8q Ñ LpV q with G S A that e ta S t. (1.91) A characterization of strongly continuous semigroups Lemma (Characterization of strongly continuous semigroups). Let K P tr, Cu, let pv, } } V q be a K-Banach space, and let S : r, 8q Ñ LpV q be a semigroup on V. Then it holds that S is strongly continuous if and only if it holds for all v P V that lim sup }S t v v} V. (1.92) tœ Proof of Lemma It is clear that a strongly continuous semigroup satisfies condition (1.92). In the following we thus assume that S : r, 8q Ñ LpV q is a semigroup which fulfills for all v P V that lim sup }S t v v} V. (1.93) tœ

28 28 CHAPTER 1. SEMIGROUPS OF BOUNDED LINEAR OPERATORS Corollary hence implies that for all t P r, 8q it holds that lim sup }S s v S t v} V lim sup Smints,tu `Ss mints,tu v S v t mints,tu V r,8qqsñt r,8qqsñt lim sup Smints,tu `S t s v v V r,8qqsñt Sminps,tq ď lim sup LpV q S t s v v ı V ď r,8qqsñt «sup upr,t`1s }S u } LpV q ff «lim sup S t s v v ff V. r,8qqsñt (1.94) The proof of Lemma is thus completed. 1.7 Uniformly continuous semigroups Lemma Let K P tr, Cu, let pv, } } V q be a normed K-vector space, and let S : r, 8q Ñ LpV q be a semigroup which satisfies that lim sup }S t S } LpV q. (1.95) tœ Then it holds for all t P r, 8q that sup spr,ts }S s } LpV q ă 8. Proof of Lemma The assumption that lim sup tœ }S t S } LpV q that there exists a real number ε P p, 8q such that ensures sup }S s } LpV q ă 8. (1.96) spr,εs Combining this with Proposition completes the proof of Lemma Lemma Let K P tr, Cu, let pv, } } V q be a normed K-vector space, and let S : r, 8q Ñ LpV q be a semigroup. Then S is uniformly continuous if and only if lim sup tœ }S t S } LpV q. Proof of Lemma Clearly, it holds that if S is uniformly continuous, then it holds that lim tœ }S t S } LpV q. It thus remains to prove that the condition lim tœ }S t S } LpV q ensures that S is uniformly continuous. We thus assume in the following that lim sup }S t S } LpV q. (1.97) tœ

29 1.7. UNIFORMLY CONTINUOUS SEMIGROUPS 29 Lemma hence implies that for all t P r, 8q it holds that sup }S s } LpV q ă 8. (1.98) spr,ts This and (1.97) show that for all t P r, 8q it holds that lim sup r,8qqsñt ď «lim sup r,8qqsñt }S s S t } LpV q lim sup r,8qqsñt ff «Srmaxps,tq minps,tqs S LpV q Smints,tu Srmaxts,tu mints,tus S LpV q ff sup }S s } LpV q. spr,t`1s (1.99) The proof of Lemma is thus completed Matrix exponential in Banach spaces The next result, Lemma 1.7.3, demonstrates one way how uniformly continuous semigroups can be constructed. Lemma (Matrix exponential in Banach spaces). Let K P tr, Cu, let pv, } } V q be 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 pe ta q tpr,8q Ď LpV q is a uniformly continuous semigroup, (iii) it holds for all t P r, 8q that e ta LpV q ď ř 8 ptaqn LpV q ď e t }A} LpV q ă 8, and n n! (iv) it holds for all t P r, 8q that 8ÿ e ta ptaq n. (1.1) n! n Proof of Lemma First of all, observe that 8ÿ ptaq n 8ÿ t n }A} n LpV q n! ď e t}a} LpV q ă 8. (1.11) LpV q n! n n Next let S : r, 8q Ñ LpV q be the function which satisfies for all t P r, 8q that S t 8ÿ n patq n. (1.12) n!

30 3 CHAPTER 1. SEMIGROUPS OF BOUNDED LINEAR OPERATORS In addition, note that for all t 1, t 2 P r, 8q it holds that «ff «ff ÿ 8 pat 1 q n ÿ 8 pat 2 q n 8ÿ A n`m pt 1 q n pt 2 q m S t1 S t2 n! n! n! m! n n n,m «8ÿ ÿ A k pt 1 q n pt 2 q m 8ÿ A k kÿ n! m! k! k 8ÿ k n,mpn, n`m k A k k! rt 1 ` t 2 s k S t1`t 2. k n k! n! pk nq! pt 1q n pt 2 q pk nq ff (1.13) This shows that S is a semigroup. Moreover, observe that for all t P r, 8q it holds that 8ÿ patq n 8ÿ patq pn 1q }S t S } LpV q ď t }A} n! LpV q n! n 1 LpV q n 1 LpV q «ff ÿ 8 patq n ÿ 8 }At} n LpV q (1.14) t }A} LpV q ď t }A} pn ` 1q! LpV q pn ` 1q! LpV q n ď t }A} LpV q e t}a} LpV q. This together with Lemma proves that S is uniformly continuous. Furthermore, note that for all t P p, 8q it holds that ř8 ı ptaq S t S n «ff n 1 n! 8 A t A LpV q t A ÿ ptaq pn 1q A n! n 1 LpV q LpV q «ff «ff 8 A ÿ ptaq n A pn ` 1q! A ÿ 8 ptaq n pn ` 1q! n LpV q n 1 LpV q «ff «ff ÿ 8 ptaq pn 1q ÿ 8 ptaq n t A2 t pn ` 1q! A2 ď t }A 2 } pn ` 2q! LpV q e t}a} LpV q. LpV q LpV q n 1 Therefore, we obtain that n n (1.15) G S A. (1.16) This, in turn, establishes Item (i). Proposition , (1.16), and the fact that S is a uniformly continuous semigroup hence prove Item (ii) and Item (iv). Next note

31 1.7. UNIFORMLY CONTINUOUS SEMIGROUPS 31 that Item (iii) follows from Item (iv) and (1.11). The proof of Lemma is thus completed. Lemma (Groups of bounded linear operators generated by bounded operators). Let K P tr, Cu, let pv, } } V q be a K-Banach space, let A P LpV q, and let S : R Ñ LpV q be the function which satisfies for all t P R that 8ÿ t n A n S t. (1.17) n! n Then (i) it holds for all t 1, t 2 P R that S t1 S t2 S t1`t 2, (ii) it holds that R Q t ÞÑ S t P LpV q is infinitely often differentiable, and (iii) it holds for all n P N, t P R that S pnq t A n S t S t A n. Proof of Lemma First of all, note that for all t 1, t 2 P R it holds that «ff «ff 8ÿ pat 1 q n 8ÿ pat 2 q n 8ÿ S t1 S t2 n! n! n n «8ÿ ÿ A k pt 1 q n pt 2 q m 8ÿ A k ÿ k n! m! k! k 8ÿ k n,mpn, n`m k A k k! rt 1 ` t 2 s k S t1`t 2. k n,m n A n`m pt 1 q n pt 2 q m n! m! k! n! pk nq! pt 1q n pt 2 q pk nq ff (1.18) This proves Item (i). It thus remains to prove Item (ii) and Item (iii). For this note

32 32 CHAPTER 1. SEMIGROUPS OF BOUNDED LINEAR OPERATORS that Item (i) ensures that for all t P R it holds that lim sup RzttuQhÑ ««lim sup RzttuQhÑ lim sup «}St`h S t has t } LpV q ř 8 n 2 8ÿ RzttuQhÑ n 1 h h n A n n! h LpV q h n }A} n LpV q pn ` 1q! ff ff ff ď lim sup }S t } LpV q ď RzttuQhÑ «}A} LpV q }S t } LpV q «ff ÿ 8 h n }A} n LpV q lim sup h RzttuQhÑ pn ` 2q! n ď lim sup ` h e }ha} LpV q }A} 2 LpV q }S t} LpV q. RzttuQhÑ «}Sh Id V ha} LpV q }S t } LpV q lim sup 8ÿ RzttuQhÑ n 2 }A} 2 LpV q }S t} LpV q h h n 1 }A} n LpV q n! ff ff }S t } LpV q (1.19) Hence, we obtain that for all t P R it holds that S 1 t AS t S t A. Induction hence establishes Item (ii) and Item (iii). The proof of Lemma is thus completed Continuous invertibility of bounded linear operators in Banach spaces Lemma (Geometric series in Banach spaces and inversion of bounded linear operators). Let K P tr, Cu, let pv, } } V q be a K-Banach space, and let A P LpV q be a bounded linear operator with }Id V A} LpV q ă 1. Then it holds (i) that A is bijective, (ii) that A 1 P LpV q, (iii) that ř 8 n }rid V As n } LpV q ă 8, and (iv) that A 1 8ÿ rid V As n. (1.11) n

33 1.7. UNIFORMLY CONTINUOUS SEMIGROUPS 33 Proof of Lemma Throughout this proof let Q P LpV q and S n P LpV q, n P N, be the bounded linear operators which satisfy for all n P N that Q Id V A and S n Note that the assumption that }Q} LpV q ă 1 ensures that 8ÿ k Q k LpV q ď 8ÿ k nÿ Q k. (1.111) k }Q} k LpV q 1 1 }Q}LpV q ă 8. (1.112) This implies that S n, n P N, is a Cauchy-sequence in LpV q. Completeness of LpV q hence shows that S n, n P N, converges in LpV q. Next we claim that for all n P N it holds that AS n Id V Q n`1. (1.113) We show (1.113) by induction on n P N. For this observe that AS A Id V A Id V Q. (1.114) This proves the base case n in (1.113). Next note that for all n P N with AS n 1 Id V Q n it holds that «ff «ff ««ffff nÿ nÿ n 1 ÿ AS n A Q k A Id V ` Q k A Id V `Q Q k k k 1 k A rid V `QS n 1 s A ` QAS n 1 A ` Q rid V Q n s A ` Q Q n`1 Id V Q n`1. (1.115) This proves (1.113) by induction on n P N. Next note that (1.113) and the fact that AQ QA imply that for all n P N it holds that AS n S n A Id V Q n`1. (1.116) This and the fact that ps n q npn Ď LpV q converges shows that ı ı A lim S n lim S n A Id V lim Q n Id V. (1.117) l jh n l jh n PLpV q PLpV q This implies that A is bijective and that A 1 lim S n P LpV q. The proof of Lemma is thus completed.

34 34 CHAPTER 1. SEMIGROUPS OF BOUNDED LINEAR OPERATORS Generators of uniformly continuous semigroup Lemma (The generator of a uniformly continuous semigroup). Let K P tr, Cu, let pv, } } V q be a K-Banach space, and let S : r, 8q Ñ LpV q be a uniformly continuous semigroup. Then G S P LpV q. Proof of Lemma The assumption that S is uniformly continuous assures that for all t P r, 8q it holds that ż lim sup 1 t`s ż S u du S t lim sup 1 t`s p,8qqsñ s rs u S t s du t LpV q p,8qqsñ s t LpV q ż 1 t`s j «ff ď lim sup }S u S t } p,8qqsñ s LpV q du ď lim sup sup }S u S t } LpV q t p,8qqsñ uprt,t`ss ««(1.118) ď }S t } LpV q }S t } LpV q «lim sup p,8qqsñ ffff sup }S u S } LpV q upr,ss ff lim sup }S s S } LpV q. p,8qqsñ This implies that there exists a real number ε P p, 8q such that ż 1 ε S s ds S ă 1. (1.119) ε LpV q Lemma hence shows that ş ε S s ds P LpV q is bijective and that ż ε S s dsj 1 P LpV q. (1.12) Therefore, we obtain that for all t P p, εq it holds that j ż S t Id V St S ε j ż ε j 1 S s ds S s ds t t ş ε rs j ż t`s S s s ds ε «ş t`ε S t s ds S s dsj 1 şε S ff ż s ds ε S s ds t t «ş ε`t S ε s ds şt S ff ż s ds ε j 1 S s ds t ż 1 ε`t S s ds 1 ż t j ż ε S s ds S s dsj 1. t t ε j 1 (1.121)

35 1.7. UNIFORMLY CONTINUOUS SEMIGROUPS 35 This together with (1.118) shows that ż S t Id ε V lim sup rs ε S s S s dsj 1. (1.122) t LpV q p,8qqtñ Hence, we obtain that G S P LpV q and ż ε G S rs ε S s S s dsj 1. (1.123) The proof of Lemma is thus completed A characterization result for uniformly continuous semigroups Theorem (Characterization of uniformly continuous semigroups). Let K P tr, Cu, let pv, } } V q be a K-Banach space, and let S : r, 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 sup tœ }S t S } LpV q. (iii) It holds that G S P LpV q. Proof of Theorem Lemma implies that Item (i) and Item (ii) are equivalent. Moreover, Lemma ensures that Item (i) implies Item (iii). It thus remains to prove that (iii) implies (i). We thus assume in the following that G S P LpV q. Next let S : r, 8q Ñ LpV q be the function which satisfies for all t P r, 8q that S t 8ÿ n ptg S q n. (1.124) n! Observe that Lemma shows that S is a uniformly continuous semigroup and that G S G S. (1.125) In particular, we obtain that S is a strongly continuous semigroup. Next note that the assumption that G S P LpV q ensures that for all v P V it holds that j }St v v tg S v} lim sup V lim sup S t v v p,8qqtñ t G S v p,8qqtñ t. (1.126) V

36 36 CHAPTER 1. SEMIGROUPS OF BOUNDED LINEAR OPERATORS This implies that for all v P V it holds that lim sup }S t v v tg S v} V lim sup t }S j tv v tg S v} V. (1.127) p,8qqtñ p,8qqtñ t Hence, we obtain that for all v P V it holds that lim sup }S t v v} V lim sup }S t v v tg S v ` tg S v} V p,8qqtñ p,8qqtñ ď lim sup r}s t v v tg S v} V ` t }G S v} V s p,8qqtñ ď lim sup }S t v v tg S v} V ` lim sup rt }G S v} V s p,8qqtñ p,8qqtñ lim sup }S t v v tg S v} V. p,8qqtñ (1.128) This allows us to apply Lemma to obtain that S is a strongly continuous semigroup. This, the fact that S is also a stongly continuous semigroup, and (1.125) enables us to apply Proposition to obtain that S S. (1.129) This and the fact that S is uniformly continuous complete the proof of Theorem is thus completed An a priori bound for uniformly continuous semigroups Combining Lemma and Theorem immediately results in the following estimate. Proposition (A priori bounds for uniformly continuous semigroups). Let K P tr, Cu, let pv, } } V q be a K-Banach space, and let S : r, 8q Ñ LpV q be a uniformly continuous semigroup. Then it holds for all t P r, 8q that G S P LpV q and sup }S s } LpV q ď e t }G S} LpV q ă 8. (1.13) spr,ts

37 1.8. THE HILLE-YOSIDA THEOREM The Hille-Yosida theorem Yosida approximations Definition (Resolvent set 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 linear operator. Then we 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 1 v P V s P LpV q ( (1.131) and we call ρpaq the resolvent set of A. Definition (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 linear operator. Then we say that A is the family of Yosida approximations of A if and only if A P MpρpAq, LpV qq is a function from ρpaq to LpV q which satisfies for all λ P ρpaq, v P V that A λ v λ 2 pλ Aq 1 v λv. (1.132) Remark (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 pa λ q λpρpaq Ď LpV q be the family of Yosida approximations of A. Then it holds for all λ P ρpaqztu, v P V that A λ v Ap1 1{λ Aq 1 v λapλ Aq 1 v. (1.133) Remark (An intuition for the Yosida approximations). 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 linear operator which satisfies that p, 8q Ď ρpaq, and let pa λ q λpρpaq Ď LpV q be the family of Yosida approximations of A. Then it holds for all v P DpAq and all sufficiently large n P N that A n v ApId V 1 n Aq 1 v «Av (1.134)

38 38 CHAPTER 1. SEMIGROUPS OF BOUNDED LINEAR OPERATORS Remark (Boundedness of implicit Euler approximations for linear differential equations). 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 such that p, 8q Ď ρpaq and sup hpp,8q }p1 haq 1 } LpV q ď 1, and let py h n q npn Ď V, h P p, 8q, satisfy for all h P p, 8q, n P N that Then it holds for all h P p, 8q, n P N that }Yn`1} h V ď sup rpp,8q Y h n`1 p1 haq 1 Y h n. (1.135) p1 raq 1 LpV q j}y h n } V ď }Yn h } V. (1.136) Scalar shifts of generators of strongly continuous semigroups Lemma (Scalar shifts of generators of strongly continuous semigroups). Let K P tr, Cu, λ P K, let pv, } } V q be a K-Banach space, and let A: DpAq Ď V Ñ V be 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 continuous semigroup and (ii) it holds for all t P r, 8q that e tpa λq e λt e ta. Proof of Lemma Throughout this proof let S : r, 8q Ñ LpV q be the function which satisfies for all t P r, 8q that S t e λt e ta. (1.137) Next note that for all t 1, t 2 P r, 8q it holds that S Id V and S t1 S t2 e λt 1 e λt 2 e t 1A e t 2A e λpt 1`t 2 q e pt 1`t 2 qa S t1`t 2. (1.138) Moreover, observe that the fact that r, 8q Q t ÞÑ e ta P LpV q is a strongly continuous semigroup ensures that for every v P V it holds that the function r, 8q Q t ÞÑ S t v P V (1.139)

39 1.8. THE HILLE-YOSIDA THEOREM 39 is continuous. This and (1.138) show that S is a strongly continuous semigroup. In addition, note that for all v P DpAq it holds that lim sup S t v v pa λqv tœ t lim sup 1 pe λt e ta v vq pa λqv t V V tœ lim sup 1 pe λt e ta v vq ` λe ta v Av ` λpv e ta vq t V tœ lim sup 1 e λt e ta v 1 t t eta v ` λe ta v ` 1 t eta v 1v Av ` λpv t eta vq V tœ lim sup 1 pe λt 1q ` λ e ta v ` 1 t t peta v vq Av ` λpv e ta vq (1.14) V tœ j j ď lim supˇ 1 pe λt 1q ` λˇˇ lim sup }e ta v} t V tœ tœ ` lim sup 1 t peta v vq Av j j V ` λ lim sup }v e ta v} V. tœ tœ This implies that DpAq Ď DpG S q. Furthermore, observe that for all v P DpG S q it holds that lim sup e ta v v pg S ` λqv tœ t V lim sup 1 t peta v vq λe ta v G S v ` λpe ta v vq V tœ lim sup 1 t eta v 1e λt e ta v λe ta v ` 1e λt e ta v 1v G t t t Sv ` λpe ta v vq V tœ lim sup 1 p1 e λt q λ e ta v ` 1pe λt e ta v vq G t t S v ` λpe ta v vq V tœ j j ď lim supˇ 1 p1 e λt q λˇˇ lim sup }e ta v} t V tœ tœ ` lim sup 1 ps t tv vq G S v j j V ` λ lim sup }e ta v v} V. tœ tœ (1.141) This proves that DpAq Ě DpG S q. Combining (1.14) and (1.141) hence establishes that for all v P DpAq it holds that DpAq DpG S q and G S v pa λqv. (1.142) The proof of Lemma is thus completed.

40 4 CHAPTER 1. SEMIGROUPS OF BOUNDED LINEAR OPERATORS Convergence of linear-implicit Euler approximations Lemma 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 which satisfies DpAq V, p, 8q Ď ρpaq, and lim sup p1 haq 1 LpV q lim sup p1 1{λ Aq 1 LpV q ď 1. (1.143) hœ λñ8 Then it holds for all v P V that lim sup hœ }p1 haq 1 v v} V lim sup }p1 1{λ Aq 1 v v} V. (1.144) λñ8 Proof of Lemma Note that it holds for all v P DpAq that lim sup }p1 1{λ Aq 1 v v} V lim sup 1 }Ap1 λ 1{λ Aq 1 v} V λñ8 λñ8 lim sup ď λñ8 1 λ }p1 1{λ Aq 1 Av} V ď lim sup λñ8 j lim sup }p1 1{λ Aq 1 1 } LpV q lim sup }Av} λ V λñ8 λñ8 }p1 1{λ Aq 1 1 } LpV q }Av} λ V j ď lim sup 1 }Av} λ V. λñ8 (1.145) This implies that for all x P V and all pv n q npn Ď DpAq with lim sup }x v n } V it holds that lim sup }p1 1{λ Aq 1 x x} V λñ8 lim sup ď 2 lim sup lim sup }p1 1{λ Aq 1 px v n q ` p1 1{λ Aq 1 v n v n ` v n x} V λñ8 }x v n } ` lim sup The proof of Lemma is thus completed Properties of Yosida approximations Convergence of Yosida approximations lim sup }p1 1{λ Aq 1 v n v n } V. (1.146) λñ8 Corollary (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 operator which satisfies DpAq V, p, 8q Ď ρpaq, and lim sup hœ }p1 haq 1 } LpV q ď 1, and let pa λ q λpρpaq Ď LpV q be the family of Yosida approximations of A. Then it holds for all v P DpAq that lim sup }A λ v Av} V. (1.147) λñ8