Lecture Notes Spectral Theory

Home , Compact embedding, Fredholm alternative, Fredholm operator, Operator (mathematics), Resolvent set, Spectral radius

Roland Schnaubelt

These lecture notes are based on my course from the summer semester 2015. I kept the numbering and the contents of the results presented in the lectures (except for minor corrections and improvements). Typically, the proofs and calculations in the notes are a bit shorter than those given in the lecture. Moreover, the drawings and many additional, mostly oral remarks from the lectures are omitted here. On the other hand, I have added several lengthy proofs not shown in the lectures (mainly about Sobolev spaces). These are indicated in the text. I like to thank Bernhard Konrad for his support in preparing the version of these lecture notes from 2010 and Heiko Hoﬀmann and Martin Meyries for several corrections and comments.

Karlsruhe, 25. Mai 2020 Roland Schnaubelt Contents

Chapter 1. General spectral theory1 1.1. Closed operators1 1.2. The spectrum4 Chapter 2. Spectral theory of compact operators 14 2.1. Compact operators 14 2.2. The Fredholm alternative 17 2.3. The Dirichlet problem and boundary integrals 22 2.4. Closed operators with compact resolvent 26 2.5. Fredholm operators 28 Chapter 3. Sobolev spaces and weak derivatives 34 3.1. Basic properties 34 3.2. Density and embedding theorems 44 3.3. Traces 58 3.4. Diﬀerential operators in Lp spaces 63 Chapter 4. Selfadjoint operators 67 4.1. Basic properties 67 4.2. The spectral theorems 72 Chapter 5. Holomorphic functional calculi 81 5.1. The bounded case 81 5.2. Sectorial operators 86 Bibliography 94

ii CHAPTER 1

General spectral theory

General notation. X ‰ t0u, Y ‰ t0u and Z ‰ t0u are Banach spaces over C with norms k¨k (or k¨kX etc.). The space BpX,Y q “ tT : X Ñ Y T is linear and continuousu is endowed with the operator norm kˇT k “ supkxkď1kT xk, and we abbreviate BpXq :“ BpX,Xq. ˇ Let DpAq be a linear subspace of X and A :DpAq Ñ Y be linear. Then A, or pA, DpAqq, is called linear operator from X to Y (and on X if X “ Y ) with domain DpAq. We denote by NpAq “ tx P DpAq Ax “ 0u and RpAq “ ty P Y D xˇ P DpAq with y “ Axu ˇ the kernel and range of A. ˇ ˇ 1.1. Closed operators We recall one of the basic examples of an unbounded operator: Let X “ Cpr0, 1sq be endowed with the supremum norm and let Af “ f 1 with DpAq “ 1 C pr0, 1sq. Then A is linear, but not bounded. Indeed, the functions un P 1 ? DpAq given by unpxq “ p { nq sinpnxq for n P N satisfy kunk8 Ñ 0 and 1 ? kAunk8 ě |unp0q| “ n Ñ 8 as n Ñ 8. 1 1 However, if fn P DpAq “ C pr0, 1sq fulﬁll fn Ñ f and Afn “ fn Ñ g in Cpr0, 1sq as n Ñ 8, then f P DpAq and Af “ g (see Analysis 1 & 2). This observation leads us to the following basic deﬁnition. Definition 1.1. Let A be a linear operator from X to Y . The operator A is called closed if for all xn P DpAq, n P N, such that there exists x “ limnÑ8 xn in X and y “ limnÑ8 Axn in Y , we have x P DpAq and Ax “ y.

Hence, limnÑ8pAxnq “ AplimnÑ8 xnq if both pxnq and pAxnq converge and A is closed. Example 1.2. a) It is clear that every operator A P BpX,Y q is closed (with DpAq “ X). On X “ Cpr0, 1sq the operator Af “ f 1 with DpAq “ C1pr0, 1sq is closed, as seen above. b) Let X “ Cpr0, 1sq. The operator Af “ f 1 with DpAq “ tf P C1pr0, 1sq fp0q “ 0u. is closed in X. Indeed, let fn P DpAq andˇf, g P X be such that fn Ñ f 1 ˇ and Afn “ fn Ñ g in X as n Ñ 8. Again by Analysis 1 & 2, the function 1 1 f belongs to C pr0, 1sq and f “ g. Since 0 “ fnp0q Ñ fp0q as n Ñ 8, we

1 1.1. Closed operators 2 obtain f P DpAq. This means that A is closed on X. In the same way we 1 see that A1f “ f with 1 1 DpA1q “ tf P C pr0, 1sq fp0q “ f p0q “ 0u is closed in X. There are many more variants.ˇ ˇ c) Let X “ Cpr0, 1sq and Af “ f 1 with 1 1 DpAq “ Cc pp0, 1sq “ tf P C pr0, 1sq supp f Ď p0, 1su, where the support supp f of f is the closure of ˇtt P r0, 1s fptq ‰ 0u in r0, 1s. ˇ This operator is not closed. In fact, consider the maps fn P DpAq given by ˇ ˇ 0, 0 ď t ď 1{n, fnptq “ 2 #pt ´ 1{nq , 1{n ď t ď 1, 1 1 2 for every n P N. Then, fn Ñ f and fn Ñ f in X as n Ñ 8, where fptq “ t . However, supp f “ r0, 1s and so f R DpAq. p d d d) Let X “ L pR q, 1 ď p ď 8, and m : R Ñ C be measurable. Deﬁne Af “ mf with DpAq “ tf P X mf P Xu. This is the maximal domain. Then A isˇ closed. Indeed, let fn Ñ f and Afn “ mfn Ñ g in X as n Ñ 8. Thenˇ there is a subsequence such that d fnj pxq Ñ fpxq and mpxqfnj pxq Ñ gpxq for a.e. x P R , as j Ñ 8. Hence, p d mf “ g in L pR q and we thus obtain f P DpAq and Af “ g. 1 e) Let X “ L pr0, 1sq, Y “ C, and Af “ fp0q with DpAq “ Cpr0, 1sq. Then A is not closed from X to Y . In fact, look at fn P DpAq given by

1 ´ nt, 0 ď t ď 1{n, fnptq “ #0, 1{n ď t ď 1, for n P N. Then kfnk1 “ 1{p2nq Ñ 0 as n Ñ 8, but Afn “ fnp0q “ 1. ♦ Definition 1.3. Let A be a linear operator from X to Y . The graph of A is given by GpAq “ tpx, Axq P X ˆ Y x P DpAqu. The graph norm of A is deﬁned by kxkA “ kxˇkX `kAxkY . We write rDpAqs if we equip DpAq with k¨kA. ˇ

Of course, k¨kA is equivalent to k¨kX if A P BpX,Y q. We endow X ˆ Y with the norm kpx, yqkXˆY “ kxkX `kykY . Recall that a seqeunce in X ˆY converges if and only if its components in X and in Y converge. Lemma 1.4. Every linear operator A from X to Y satisﬁes the following assertions. (a) GpAq Ď X ˆ Y is a linear subspace. (b) rDpAqs is a normed vector space and A P BprDpAqs,Y q. (c) A is closed if and only if GpAq is closed in X ˆ Y if and only if rDpAqs is a Banach space. (d) Let A be injective and put DpA´1q :“ RpAq. Then, A is closed from X to Y if and only if A´1 is closed from Y to X. 1.1. Closed operators 3

Proof. Parts (a) and (b) follow from the deﬁnitions, and assertion (c) implies (d) since GpA´1q “ tpy, A´1yq y P RpAqu “ tpAx, xq x P DpAqu is closed in Y ˆ X if and only ifˇ GpAq is closed in X ˆ Yˇ . We next show (c). ˇ ˇ The operator A is closed if and only if for all xn P DpAq, n P N, and px, yq P X ˆ Y with pxn, Axnq Ñ px, yq in X ˆ Y as n Ñ 8, we have x P DpAq and Ax “ y; i.e., px, yq P GpAq. This property is equivalent to the closedness of GpAq. Since kpx, AxqkXˆY “ kxkX ` kAxkY , a Cauchy sequence or a converging sequence in GpAq corresponds to a Cauchy or a converging sequence in rDpAqs, respectively. So rDpAqs is complete if and only if pGpAq, k¨kXˆY q is complete if and only if GpAq Ď X ˆY is closed. The next result is a variant of the open mapping theorem. Theorem 1.5 (Closed Graph Theorem). Let X and Y be Banach spaces and A be a closed operator from X to Y . Then A is bounded (i.e., kAxk ď c kxk for some c ě 0 and all x P DpAq) if and only if DpAq is closed in X. In particular, a closed operator with DpAq “ X belongs to BpX,Y q. Proof. “ð”: Let DpAq be closed in X. Then DpAq is a Banach space for k¨kX and k¨kA. Since kxkX ď kxkA for all x P DpAq, a corollary to the open mapping theorem (see e.g. Corollary 4.29 in [Sc2]) shows that there is a constant c ą 0 such that kAxkY ď kxkA ď c kxkX for all x P DpAq. “ñ”: Let A be bounded and let xn P DpAq converge to x P X with respect to k¨kX . Then kAxn ´ AxmkY ď c kxn ´ xmkX , and so the sequence pAxnqn is Cauchy in Y . There thus exists y :“ limnÑ8 Axn in Y . The closedness of A shows that x P DpAq; i.e., DpAq is closed in X. We next show that Theorem 1.5 is wrong without completeness and give an example of a non-closed, everywhere deﬁned operator.

Remark 1.6. a) Let T be given by pT fqptq “ tfptq, t P R, on CcpRq with supremum norm. This linear operator is everywhere deﬁned, unbounded and closed. In fact, take fn, f, g P CcpRq such that fnptq Ñ fptq and pT fnqptq “ tfnptq Ñ gptq uniformly for t P R as n Ñ 8. Then gptq “ tfptq for all t P R; i.e., g “ T f and T is closed. Further, pick ϕn P CcpRq with }ϕn}8 “ 1 and ϕnpnq “ 1. Since }T ϕn}8 ě |T ϕnpnq| “ n, the operator T is unbounded. b) Let X be an inﬁnite dimensional Banach space and let B be an algebraic basis of X, see Theorem III.5.1 in [La]. (Hence, for each x P X there are unique α1, . . . , αn P C, b1, . . . , bn P B and n P N such that x “ α1b1 ` ¨ ¨ ¨ ` αnbn.) We may assume that kbk “ 1 for all b P B. Choose a countable subset B0 “ tbk k P Nu of B and set T bk “ kbˇ k for each bk P B0, and T b “ 0 for each b P BzB0. ˇ Then T can be extended to a linear operator on X which is unbounded, since kT bkk “ k and kbkk “ 1. Thus T is not closed by Theorem 1.5. ♦ Proposition 1.7. Let A be closed from X to Y , T P BpX,Y q, and S P BpZ,Xq. Then the following operators are closed. (a) B “ A ` T with DpBq “ DpAq, (b) C “ AS with DpCq “ tz P Z Sz P DpAqu. ˇ ˇ 1.2. The spectrum 4

Proof. (a) Let xn P DpBq, n P N, and x P X, y P Y such that xn Ñ x in X and Bxn “ Axn ` T xn Ñ y in Y as n Ñ 8. Since T is bounded, there exists T x “ limnÑ8 T xn and so Axn Ñ y ´T x as n Ñ 8. The closedness of A then yields x P DpAq “ DpBq and Ax “ y ´ T x; i.e., Bx “ Ax ` T x “ y. (b) Let zn P DpCq, n P N, and z P Z, y P Y such that zn Ñ z in Z and ASzn Ñ y in Y as n Ñ 8. By the boundedness of S, the vectors xn :“ Szn converge to Sz. Since Axn Ñ y and A is closed, we obtain Sz P DpAq and ASz “ y; i.e., z P DpCq and Cz “ y. Corollary 1.8. Let A be linear on X and λ P C. Then the following assertions hold. (a) A is closed on X if and only if λI ´ A is closed on X. (b) If λI ´ A is bijective with pλI ´ Aq´1 P BpXq, then A is closed. Proof. Assertion (a) is a consequence of Proposition 1.7 since A “ ´ppλI ´ Aq ´ λIq. For the second part, Lemma 1.4 shows that λI ´ A is closed, and then assertion (a) yields (b). The following examples show that closedness can be lost when taking sums or products of closed operators. See the exercises for further related results.

2 Example 1.9. a) Let E “ CbpR q and Ak “Bk with

DpAkq “ tf P E the partial derivative Bkf exists and belongs to Eu, for k P t1, 2u. Set Bˇ “B1 ` B2 on ˇ 1 2 1 2 DpBq :“ DpA1q X DpA2q “ Cb pR q “ tf P C pR q f, B1f, B2f P Eu.

By an exercise, A1 and A2 are closed. However, B is notˇ closed. 1 ˇ 1 Indeed, take φn P Cb pRq converging uniformly to some φ P CbpRqzC pRq. 2 Set fnpx, yq “ φnpx ´ yq and fpx, yq “ φpx ´ yq for px, yq P R and n P N. We then obtain f P E, fn P DpBq, kfn ´ fk8 “ kφn ´ φk8 Ñ 0 and 1 1 Bfn “ φn ´ φn “ 0 Ñ 0 as n Ñ 8, but f R DpBq. b) Let X “ Cpr0, 1sq, Af “ f 1 with DpAq “ C1pr0, 1sq and m P Cpr0, 1sq such that m “ 0 on r0, 1{2s. Deﬁne T P BpXq by T f “ mf for all f P X. Then the operator TA with DpTAq “ DpAq is not closed. To see this, take functions fn P DpAq such that fn “ 1 on r1{2, 1s and 1 1 fn Ñ f in X with f R C pr0, 1sq. Then, T Afn “ mfn “ 0 converges to 0, but f R DpAq. ♦

1.2. The spectrum Definition 1.10. Let A be a closed operator on X. The resolvent set of A is given by

ρpAq “ tλ P C λI ´ A :DpAq Ñ X is bijectiveu, and its spectrum by ˇ ˇ σpAq “ CzρpAq. We further deﬁne the point spectrum of A by

σppAq “ tλ P C D v P DpAqzt0u with λv “ Avu Ď σpAq, ˇ ˇ 1.2. The spectrum 5 where we call λ P σppAq an eigenvalue of A and the corresponding v an eigenvector or eigenfunction of A. For λ P ρpAq the operator Rpλ, Aq :“ pλI ´ Aq´1 : X Ñ X and the set tRpλ, Aq λ P ρpAqu are called the resolvent. Remark 1.11. a)ˇ Let A be closed on X and λ P ρpAq. The resolvent ˇ Rpλ, Aq has the range DpAq. Corollary 1.8 and Lemma 1.4 further show that Rpλ, Aq is closed and thus it belongs to BpXq by Theorem 1.5. b) Let A be a linear operator such that λI Á :DpAq Ñ X has a bounded inverse for some λ P C. Then A is closed by Corollary 1.8. In this case, the closedness assumption in Definition 1.10 is redundant. ♦ λt We set eλptq “ e for λ P C, t P J, and any interval J Ď R. d Example 1.12. a) Let X “ C and T P BpXq. Then σpT q only consists of the eigenvalues λ1, . . . , λm of T , where m ď d. b) Let X “ Cpr0, 1sq and Au “ u1 with DpAq “ C1pr0, 1sq. Then σpAq “ σppAq “ C. Indeed, eλ belongs to DpAq and Aeλ “ λeλ for each λ P C. c) Let X “ Cpr0, 1sq and Au “ u1 with DpAq “ tu P C1pr0, 1sq up0q “ 0u. Then A is closed by Example 1.2. Moreover, σpAq is empty. In fact, let ˇ λ P C and f P X. We then have u P DpAq and pλI ´ Aqu “ f ˇif and only if u P C1pr0, 1sq, u1ptq “ λuptq ´ fptq, t P r0, 1s, and up0q “ 0, which is equivalent to t λpt´sq uptq “ ´ e fpsq ds “: pRλqfptq, ż0 for all 0 ď t ď 1. Hence, σpAq “ H and Rpλ, Aq “ Rλ. ♦ We see in Examples 1.21 and 1.25 that both for unbounded and bounded operators the point spectrum can be empty though the spectrum is void, provided that dim X “ 8. Let U Ď C be open. The derivative of f : U Ñ Y at λ P U is given by 1 f 1pλq “ lim pfpµq ´ fpλqq P Y, µÑλ µ ´ λ if the limit exists in Y . In the next theorem we collect the most basic properties of the spectrum and the resolvent of closed operators. Theorem 1.13. Let A be a closed operator on X and let λ P ρpAq. Then the following assertions hold. (a) ARpλ, Aq “ λRpλ, AqÍ, ARpλ, Aqx “ Rpλ, AqAx for all x P DpAq, and 1 pRpλ, Aq ´ Rpµ, Aqq “ Rpλ, AqRpµ, Aq “ Rpµ, AqRpλ, Aq µ ´ λ if µ P ρpAqztλu. The formula in display is called the resolvent equation. (b) The spectrum σpAq is closed, where Bpλ, 1{kRpλ,Aqkq Ď ρpAq and 8 n n`1 Rpµ, Aq “ pλ ´ µq Rpλ, Aq “: Rµ n“0 ÿ 1.2. The spectrum 6

if |λ ´ µ| ă 1{kRpλ,Aqk “: rλ. This series converges in BpX, rDpAqsq, absolutely and uniformly on Bpλ, δrλq for each δ P p0, 1q. Moreover, cpλq kRpµ, Aqk ď BpX,rDpAqsq 1 ´ δ for all µ P Bpλ, δ{kRpλ,Aqkq and a constant cpλq given by (1.1). (c) The function ρpAq Ñ BpX, rDpAqsq; λ ÞÑ Rpλ, Aq, is infinitely often differentiable with d n n n`1 dλ Rpλ, Aq “ p´1q n! Rpλ, Aq for every n P N. 1 (d) k`Rp˘λ, Aqk ě dpλ,σpAqq . Proof. (a) The first assertions follow from x “ pλI ´ AqRpλ, Aqx “ Rpλ, AqpλI ´ Aqx, where x P X in the first equality and x P DpAq in the second one. For µ P ρpAq, we further have pλRpλ, Aq ´ ARpλ, AqqRpµ, Aq “ Rpµ, Aq, Rpλ, AqpµRpµ, Aq ´ ARpµ, Aqq “ Rpλ, Aq. Subtracting these two equations and also interchanging λ and µ, we deduce the resolvent equation. (b) Let |µ ´ λ| ď δ{kRpλ,Aqk for some δ P p0, 1q and x P X with kxk ď 1. It follows n n`1 kpλ ´ µq Rpλ, Aq xkA δn ď kARpλ, AqRpλ, Aqnxk ` kRpλ, Aqn`1xk kRpλ, Aqkn ď δn pkλRpλ, A`qk ` 1 ` kRpλ, Aqkq “: δncpλq. ˘(1.1)

Due to this inequality and e.g. Lemma 4.23 in [Sc2], the series Rµ converges and can be estimated as asserted. Part (a) yields pµI ´ AqRpλ, Aq “ pµ ´ λqRpλ, Aq ` I. Using this fact and that A belongs to BprDpAqs,Xq, we infer 8 8 n`1 n`1 n n pµI ´ AqRµ “ ´ pλ ´ µq Rpλ, Aq ` pλ ´ µq Rpλ, Aq “ I, n“0 n“0 ÿ ÿ and similarly RµpµI ´ Aqx “ x for all x P DpAq. Hence, µ P ρpAq and Rµ “ Rpµ, Aq. Assertion (c) is a consequence of the power series expansion, as in the scalar case. Part (b) also implies (d). d Proposition 1.14. Let Ω Ď R , m P CpΩq, X “ CbpΩq, and Af “ mf with DpAq “ tf P X mf P Xu. Then A is closed, ˇ σpAq “ mpΩq, ˇ 1 and Rpλ, Aqf “ λ´m f for all λ P ρpAq and f P X. For every closed (resp., non-empty and compact) subset S Ď C there is a closed (resp., bounded) operator B on a Banach space with σpBq “ S. 1.2. The spectrum 7

Proof. The closedness of A can be shown as in Remark 1.6. Let λ R 1 mpΩq and g P CbpΩq. The function f :“ λ´m g then belongs to CbpΩq and λf ´ mf “ g so that mf “ λf ´ g P CbpΩq. As a result, f P DpAq and f is the unique solution in DpAq of the equation λf ´ Af “ g. This means that 1 λ P ρpAq and Rpλ, Aqg “ λ´m g. In the case that λ “ mpzq for some z P Ω, we obtain ppλI ´ Aqfqpzq “ λfpzq ´ mpzqfpzq “ 0 for every f P DpAq. Consequently, λI ´ A is not surjective and so λ P σpAq. We now conclude that σpAq “ mpΩq since the spectrum is closed. The ﬁnal assertion follows from Example 1.12c) if S “H. Otherwise, consider Ω “ S. Deﬁne A and X as above. Then, σpAq “ S and A is bounded if S is compact (where CbpSq “ CpSq). A similar result is valid in Lp-spaces, see e.g. Example IX.2.6 in [Co2].

Example 1.15. Let X “ C0pR`q “ tf P CpR`q limtÑ8 fptq “ 0u with the supremum norm and Af “ f 1 with ˇ 1 1 ˇ 1 DpAq “ C0 pR`q “ tf P C pR`q f, f P Xu. As in Example 1.2 one sees that A is closed.ˇ Moreover, σpAq “ tλ P ˇ C Re λ ď 0u “: C´ and σppAq “ tλ P C Re λ ă 0u. Proof. First note that for λ P C with Re λ ă 0, the function eλ belongs ˇ ˇ toˇ DpAq and Aeλ “ λeλ. Hence, ˇ tλ P C Re λ ă 0u Ď σppAq Ď σpAq, so that C´ Ď σpAq by the closednessˇ of the spectrum. Next, let Re λ ą 0 and fˇ P X. We then have u P DpAq and λu ´ Au “ f 1 1 if and only if u P X X C pR`q and u ptq “ λuptq ´ fptq for all t ě 0. This equation is uniquely solved by 8 λpt´sq uptq “ e fpsq ds “: pRλfqptq, t ě 0. żt We still have to show that Rλf P X. Let ε ą 0. Then there is a number tε ě 0 such that |fpsq| ď ε for all s ě tε. We can now estimate 8 8 ε |R fptq| ď epRe λqpt´sq|fpsq| ds ď ε e´ Re λr dr “ , λ Re λ żt ż0 for all t ě tε, where we substituted r “ s ´ t. As a result, Rλf P X and so λ P ρpAq with Rλ “ Rpλ, Aq. We have thus shown that σpAq “ C´. 1 Finally, assume there is a number λ P iR and a function v P C0 pR`q with v1 “ λv. It follows vptq “ eλpt´sqvpsq and so |vptq| “ |vpsq| for all t ě s ě 0. Letting t Ñ 8, we infer that |vpsq| “ 0 for all s ě 0; i.e., v “ 0. Hence, A has no eigenvalues on iR. l Complementing Theorem 1.13, we state additional properties of the spectrum if the operator is bounded. Theorem 1.16. Let T P BpXq. Then σpT q is a non-empty compact set. The spectral radius rpT q :“ maxt|λ| λ P σpT qu is given by rpT q “ lim kT nk1{n ˇ“ inf kT nk1{n ď kT k, nÑ8 ˇ nPN 1.2. The spectrum 8 and for λ P C with |λ| ą rpT q we have 8 ń´1 n Rpλ, T q “ λ T “: Rλ. n“0 ÿ n m n m Proof. 1) Since }T ` } ď }T }}T } for all n, m P N, by an elementary lemma (see Lemma VI.1.4 in [We]) there exists the limit lim kT nk1{n “ inf kT nk1{n “: r ď kT k. nÑ8 nPN If |λ| ą r, then 1 r lim supkλńT nk1{n “ lim kT nk1{n “ ă 1. nÑ8 |λ| nÑ8 |λ| Using Lemma 4.23 in [Sc2], as in Analysis 1 one now checks that the series Rλ converges in BpXq for |λ| ą r. Moreover, 8 8 ń n ń´1 n`1 pλI ´ T qRλ “ λ T ´ λ T “ I, n“0 n“0 ÿ ÿ and similarly RλpλI ´ T q “ I. Hence, λ P ρpT q and Rλ “ Rpλ, T q. Due to its closedness, the spectrum σpT q Ď Bp0, rq is compact. Therefore, rpT q exists as the maximum of a compact subset of R, and rpT q ď r. ˚ 2) Take Φ P BpXq and define fΦpλq :“ ΦpRpλ, T qq for λ P D “ CzBp0, rpT qq. Note that fΦ : D Ñ C is complex differentiable and 8 ń´1 n fΦpλq “ λ ΦpT q “: Sλ if |λ| ą r. n“0 ÿ By e.g. Theorem V.1.11 in [Co1], there are unique coeffcients am P C with 8 m fΦpλq “ amλ for λ P D. m“´8 ÿ Hence, the series Sλ converges for all λ P D and so @ λ P D, Φ P BpXq˚ : sup |λń´1ΦpT nq| ă 8. nPN A corollary to the uniform boundedness principle (see e.g. Corollary 5.12 in [Sc2]) thus yields that cpλq :“ sup }λń´1T n} ă 8 nPN for each λ P D. As a result, lim kT nk1{n “ lim |λ| p|λ| kλń´1T nkq1{n ď |λ| lim p|λ| cpλqq1{n “ |λ| nÑ8 nÑ8 nÑ8 for all |λ| ą rpT q. This means that r “ rpT q. 3) Suppose that σpT q “ H. The functions fΦ from part 2) are then holomorphic on C for every Φ P BpXq˚. Step 1) implies that 8 kT kn 2kΦk |f pλq| ď kΦk|λ|´1 ď , Φ |λ|n |λ| n“0 ÿ for all λ P C with |λ| ě 2kT k. Hence, fΦ is bounded and thus constant by Liouville’s theorem from complex analysis. The above estimate then shows 1.2. The spectrum 9 that ΦpRpλ, T qq “ 0 for all λ P C and Φ P BpX˚q. Employing the Hahn- Banach theorem (see e.g. Corollary 5.10 in [Sc2]), we obtain Rpλ, T q “ 0, which is impossible since Rpλ, T q is injective and X ‰ t0u. Example 1.17. a) We define the Volterra operator V on X “Cpr0, 1sq by t V fptq “ fpsq ds ż0 for t P r0, 1s and f P X. Then V belongs to BpXq and t s1 sn´1 1 |V nfptq| ď ... kfk ds ... ds ď kfk , 8 n 1 n! 8 ż0 ż0 ż0 n for all n P N, t P r0, 1s, and f P X. Hence, kV k ď 1{n!. For f “ 1 we obtain n n 1 n 1 kV k ě kV 1k8 “ {n! and so kV k “ {n!. Theorem 1.16 thus yields 1 1{n rpV q “ lim “ 0 and σpV q “ t0u. nÑ8 n! ˆ ˙ 1 Observe that σppV q “ H since V f “ 0 implies that f “ pV fq “ 0. More- over, kV k “ 1 ą rpV q “ 0. p b) Let left shift L given by Lx “ pxn`1q on X P tc0, ` 1 ď p ď 8u has the spectrum σpLq “ Bp0, 1q. Moreover, σ pLq “ Bp0, 1q if X ‰ `8 and p ˇ σpLq “ Bp0, 1q if X “ `8. ˇ Proof. The operator L P BpXq has norm 1 (see e.g. Example 2.9 in [Sc2]), and so σpLq Ď Bp0, 1q. Clearly, Lp1, 0, ¨ ¨ ¨ q “ 0. For |λ| ă 1 the n n`1 sequence v “ pλ qnPN belongs to X and satisfies Lv “ pλ qn “ λv so that Bp0, 1q Ď σppLq Ď σpLq. If |λ| “ 1 and Lx “ λx, then xn`1 “ λxn for all n´1 n P N. Hence, x “ pλ x1qn which belongs to X (for x1 ‰ 0) if and only if X “ `8. The closedness of the spectrum now implies the assertions. l For further investigations we need certain subdivisions of the spectrum. We note that in this context also other definitions are used in the literature. Definition 1.18. Let A be a linear operator on X. Then we call

σappAq “ tλ P C D xn P DpAq with kxnk “ 1 for all n P N and ˇ λxn ´ Axn Ñ 0 as n Ñ 8u ˇ the approximate point spectrum of A and

σrpAq “ tλ P C pλI ´ Aq DpAq is not dense in Xu the residual spectrum of A.ˇ ˇ One calls λ P σappAq an approximate eigenvalue and the corresponding xn approximate eigenvectors. If one has λ P C and xn P DpAq with kxnk ě δ ą 0 for all n P N and λxn ´ Axn Ñ 0 as n Ñ 8, then λ belongs to σappAq ´1 ´1 with approximate eigenvectors }xn} xn, since }xn} ď 1{δ. Proposition 1.19. Let A be closed on X. The following assertions are true (with possibly non-disjoint unions). (a) σappAq “ σppAq Y tλ P C pλI ´ Aq DpAq is not closed in Xu. (b) σpAq “ σappAq Y σrpAq. ˇ (c) BσpAq Ď σappAq. ˇ 1.2. The spectrum 10

Proof. 1) Let λ R σappAq. Then there is a constant c ą 0 with kpλI ´ Aqxk ě ckxk for all x P DpAq. This lower estimate implies that λ R σppAq. Moreover, if yn “ λxn ´ Axn Ñ y in X as n Ñ 8 for some xn P DpAq, then the lower estimate shows that pxnq is Cauchy in X, and so xn tends to some x in X. Hence, Axn “ λxn ´ yn Ñ λx ´ y and the closedness of A yields x P DpAq and λx ´ Ax “ y. Consequently, pλI ´ Aq DpAq is closed. Conversely, if pλI ´ Aq DpAq is closed and λ R σppAq, then the inverse pλI ´ Aq´1 exists and is closed on its closed domain pλI ´ Aq DpAq. The closed graph theorem 1.5 then yields the boundedness of pλI ´ Aq´1. Thus, kxk “ kpλI ´ Aq´1pλI ´ Aqxk ď CkpλI ´ Aqxk for all x P DpAq and a constant C ą 0. This means that λ R σappAq. We thus have shown assertion (a), which implies (b). 2) Let λ PBσpAq. Then there are points λn in ρpAq with λn Ñ λ as n Ñ 8. By Theorem 1.13(d), the norms kRpλn,Aqk tend to 8 as n Ñ 8, and thus there exist yn P X such that kynk “ 1 for all n P N and 0 ‰ an :“ kRpλ ,Aqy k Ñ 8 as n Ñ 8. Set x “ 1 Rpλ ,Aqy P DpAq. We then n n n an n n have kx k “ 1 for all n P and λx ´ Ax “ pλ ´ λ qx ` 1 y converges n N n n n n an n to 0 as n Ñ 8. As a result, λ P σappAq. In the next result we determine the spectra of certain operators which (formally) arise as functions fpAq of A, namely the resolvent of A where fpµq “ pλ ´ µq´1 for λ P ρpAq and a aﬃne transformation of A where fpµq “ αµ ` β for α, β P C. Proposition 1.20. Let A be closed on X, λ P ρpAq, α P Czt0u and β P C. Then the following assertions hold. ´1 1 (a) σpRpλ, Aqqzt0u “ pλ ´ σpAqq “ t λ´ν ν P σpAqu. (b) σ pRpλ, Aqqzt0u “ pλ ´ σ pAqq´1 for j P tp, ap, ru. j j ˇ (c) If x is an eigenvector for the eigenvalueˇ µ ‰ 0 of Rpλ, Aq, then y “ µRpλ, Aqx is an eigenvector for the eigenvalue ν “ λ ´ 1{µ of A. If y P DpAq is an eigenvector for the eigenvalue ν “ λ ´ 1{µ of 1 A with µ P Czt0u, then x “ µ´ pλy ´ Ayq is an eigenvector for the eigenvalue µ of Rpλ, Aq. (d) rpRpλ, Aqq “ 1{dpλ,σpAqq. (e) If A is unbounded, then 0 P σpRpλ, Aqq. (f) σpαA ` βIq “ ασpAq ` β and σjpαA ` βIq “ ασjpAq ` β for j P tp, ap, ru. Proof. For µ P Czt0u we have 1 µI ´ Rpλ, Aq “ λ ´ µ I ´ A µRpλ, Aq. (1.2) Observe that the operator µRpλ,´ A` q : X˘Ñ Dp¯Aq is bijective. Hence, the 1 bijectivity of µI ´ Rpλ, Aq is equivalent to that of pλ ´ µ qI ´ A. As a result, 1 µ belongs to ρpRpλ, Aqq if and only if λ ´ µ belongs to ρpAq if and only if µ “ pλ ´ νq´1 for some ν P ρpAq. We have thus shown (a). In the same way, one derives assertion (b) for j “ p, assertion (c) and 1 that µI ´ Rpλ, Aq and pλ ´ µ qI ´ A have the same range. Using also Propo- sition 1.19, we then deduce (b) for j “ ap and j “ r. 1.2. The spectrum 11

Part (d) is a consequence of (a). In part (e), the inverse Rpλ, Aq´1 “ λI ´ A is unbounded so that 0 P σpRpλ, Aqq. Similar as (a) and (b), the last assertion follows from the equality λ´β λI ´ pαA ` βIq “ α α I ´ A . p Example 1.21. a) Let X “ L pRq, 1 ď` p ď 8, and˘ the translation T ptq be given by pT ptqfqpsq “ fps ` tq for s P R, f P X, and t P R. Then σpT ptqq “ BBp0, 1q for t ‰ 0. Proof. Recall from Example 4.12 in [Sc2] that T ptq is an isometry on 1 X with inverse pT ptqq´ “ T p´tq for every t P R. By Theorem 1.16 we have σpT ptqq Ď Bp0, 1q. Proposition 1.20 further yields σpT ptqq´1 “ σpT ptq´1q “ σpT p´tqq Ď Bp0, 1q so that σpT ptqq Ď BBp0, 1q for all t P R. Fix t ‰ 0. For 8 every λ P iR, the function eλ belongs to CbpRq Ď L pRq and λps`tq λt pT ptqeλqpsq “ e “ e eλpsq for all s P R. Hence, σpT ptqq “ σppT ptqq “ BBp0, 1q for p “ 8. If p P r1, 8q, we use eλ to construct approximate eigenfunctions. For ´1{p1 ´1{p 1 n P N set fn “ n r0,nseλ. We then have kfnkp “ n k r0,nskp “ 1 and (see above) λt ´1{p λt 1 1 ´1{p ´1{p kT ptqfn ´ e fnkp “ n ke p r´t,n´ts ´ r0,nsqkp “ n |2t| Ñ 0, as n Ñ 8. As a result, σpT ptqq “ BBp0, 1q if t ‰ 0. l 1 1 1 1 b) Let X “ C0pRq and Au “ u with DpAq “ C0 pRq :“ tf P C pRq f, f P C0pRqu. Then σpAq “ iR and σppAq “ H. ˇ Proof. As in Example 1.15 one sees that λ P ρpAq if Re λ ‰ 0 andˇ 8 Rpλ, Aqfptq “ eλpt´sqfpsq ds if Re λ ą 0, t ż t Rpλ, Aqfptq “ ´ eλpt´sqfpsq ds if Re λ ă 0, ż´8 for all t P R and f P X. Let Re λ “ 0. Then λ is not an eigenvalue, cf. 1 1 1 Example 1.15. Choose ϕn P Cc pRq with kϕnk8 ď {n and kϕnk8 “ 1, and set un “ ϕneλ for all n P N. Then, kunk8 “ 1, un P DpAq and Aun “ 1 1 1 1 1 ϕneλ ` ϕneλ “ ϕneλ ` λun. Since kϕneλk8 ď {n, we obtain λ P σappRq. l We now introduce the adjoint of a densely defined linear operator in order to obtain a convenient description of the residual spectrum, for instance. Definition 1.22. Let A be a linear operator from X to Y with dense domain. We define its adjoint A˚ from Y ˚ to X˚ by setting DpA˚q “ ty˚ P Y ˚ D z˚ P X˚ @ x P DpAq : xAx, y˚y “ xx, z˚yu, ˚ ˚ ˚ A y “ z . ˇ ˇ Observe that for all x P DpAq and y˚ P DpA˚q we obtain xAx, y˚y “ xx, A˚y˚y. We note that the operator Af “ f 1 with DpAq “ tf P C1pr0, 1sq fp0q “ 0u is not densely defined on X C 0, 1 since D A f X f 0 0 . “ pr sq p q “ t P pˇq “ u ˇ ˇ ˇ 1.2. The spectrum 12

Remark 1.23. Let A be linear from X to Y with DpAq “ X. a) Since DpAq is dense, there is at most one vector z˚ “ A˚y˚ as in Definition 1.22, so that A˚ :DpA˚q Ñ X˚ is a map. It is clear that A˚ is linear. If A P BpX,Y q, then Definition 1.22 coincides with the definition of A˚ in §5.4 of [Sc2], where DpA˚q “ Y ˚. b) The operator A˚ is closed from Y ˚ to X˚. ˚ ˚ ˚ ˚ ˚ ˚ ˚ ˚ ˚ Proof. Let yn P DpA q, y P Y , and z P X such that yn Ñ y in Y ˚ ˚ ˚ ˚ ˚ and zn :“ A yn Ñ z in X as n Ñ 8. Let x P DpAq. We derive ˚ ˚ ˚ ˚ xx, z y “ lim xx, zny “ lim xAx, yny “ xAx, y y. nÑ8 nÑ8 As a result, y˚ P DpA˚q and A˚y˚ “ z˚. l c) If T P BpX,Y q, then the sum A ` T with DpA ` T q “ DpAq has the adjoint pA ` T q˚ “ A˚ ` T ˚ with DppA ` T q˚q “ DpA˚q. Proof. Let x P DpAq and y˚ P Y ˚. We obtain xpA ` T qx, y˚y “ xAx, y˚y ` xx, T ˚y˚y. Hence, y˚ P DppA ` T q˚q if and only if y˚ P DpA˚q, and then pA ` T q˚y˚ “ A˚y˚ ` T ˚y˚. l Theorem 1.24. Let A be a closed operator on X with dense domain. Then the following assertions hold. ˚ (a) σrpAq “ σppA q. (b) σpAq “ σpA˚q and Rpλ, Aq˚ “ Rpλ, A˚q for every λ P ρpAq. Proof. (a) Due to a corollary of the Hahn-Banach theorem (see e.g. Corollary 5.13 in [Sc2]), the set pλI ´ Aq DpAq is not dense in X if and only if there is a vector y˚ P X˚zt0u such that xλx ´ Ax, y˚y “ 0 for every x P DpAq. This equation is equivalent to xAx, y˚y “ xx, λy˚y, which in turn ˚ ˚ ˚ ˚ ˚ ˚ means that y P DpA qzt0u and A y “ λy ; i.e., λ P σppA q. (b) Let λ P ρpAq. Take y˚ P DpA˚q and x P X. We compute xx, Rpλ, Aq˚pλI ´ A˚qy˚y “ xRpλ, Aqx, pλI ´ A˚qy˚y “ xpλI ´ AqRpλ, Aqx, y˚y “ xx, y˚y, using Definition 1.22 and that Rpλ, Aqx belongs to DpAq. It follows Rpλ, Aq˚pλI ´ A˚qy˚ “ y˚ so that λI ´ A˚ is injective. Next, take x˚ P X˚. Set y˚ “ Rpλ, Aq˚x˚. For x P DpAq, we obtain xpλI ´ Aqx, y˚y “ xRpλ, AqpλI ´ Aqx, x˚y “ xx, x˚y. Hence, y˚ P DpA˚q and x˚ “ pλI ´ Aq˚y˚ “ pλI ´ A˚qy˚, where we use Remark 1.23(c). Consequently, the operator λI ´ A˚ is surjective, hence bijective with inverse Rpλ, A˚q “ Rpλ, Aq˚. ˚ ˚ Conversely, let λ P ρpA q. Then λ does not belong to σppA q “ σrpAq, see (a). Take x P DpAq. Due to a corollary of the Hahn-Banach theorem (see e.g. Corollary 5.10 in [Sc2]), there is a functional y˚ P X˚ such that ky˚k “ 1 and xx, y˚y “ kxk. As above, we calculate }x} “ xx, y˚y “ xx, pλI ´ A˚qRpλ, A˚qy˚y “ xpλI ´ Aqx, Rpλ, A˚qy˚y ď kRpλ, A˚qk kλx ´ Axk; i.e., λ does not belong to σappAq. Proposition 1.19 thus yields λ R σpAq. 1.2. The spectrum 13

p Example 1.25. Let X P tc0, ` 1 ď p ď 8u. Let Rx “ p0, x1, x2,... q be the right shift on X. Then σpRq “ Bp0, 1q, σppRq “ H, σrpRq “ Bp0, 1q for 1 ˇ p X “ ` , and σrpRq “ Bp0, 1q for Xˇ P tc0, ` 1 ă p ă 8u. Proof. First, let X ‰ `8. From e.g. Example 5.44 of [Sc2] we know that ˚ 1 ˇ p1 R “ L, where the left shift L acts on ` ifˇ X “ c0 and on ` otherwise. Since σpLq “ Bp0, 1q by Example 1.17, Theorem 1.24 yields σpRq “ σpR˚q “ p σpLq “ Bp0, 1q. Similarly, σrpRq “ σppLq “ Bp0, 1q if X “ c0 or X “ ` 1 with 1 ă p ă 8, and σrpRq “ σppLq “ Bp0, 1q if X “ ` . If X “ `8, then R “ L˚ for L on `1 so that again σpRq “ Bp0, 1q. Clearly, Rx “ 0 yields x “ 0. If λx “ Rx “ p0, x1, x2, ¨ ¨ ¨ q and λ ‰ 0, then 0 “ λx1 and so x1 “ 0. Iteratively one sees that x “ 0. Hence, R has no eigenvalues. l Let A be a linear operator from X to Y . Then a linear operator B from X to Y is called A–bounded (or if relatively bounded with respect to A) if DpAq Ď DpBq and B P BprDpAqs,Y q. Remark 1.26. Let A and B be linear from X to Y with DpAq Ď DpBq. a) The operator B is A–bounded if and only if there are constants a, b ě 0 such that }Bx} ď a }Ax} ` b }x} (1.3) for all x P DpAq. If X “ Y and there exists a point λ in ρpAq, then the A–boundedness of B is also equivalent to the boundedness of BRpλ, Aq. For instance, we then have (1.3) with a :“}BRpλ, Aq} and b :“ |λ| a. b) Let A be closed and let (1.3) be satisﬁed with a ă 1. Then A ` B with DpA ` Bq “ DpAq is also closed, by an exercise. In view of a), the next result also requires that b in (1.3) is suﬃciently small. ♦ Theorem 1.27. Let A be a closed operator on X and λ P ρpAq. Further, let B be linear on X with DpAq Ď DpBq. Assume that kBRpλ, Aqk ă 1. Then A ` B with DpA ` Bq “ DpAq is closed, λ P ρpA ` Bq, 8 Rpλ, A ` Bq “ Rpλ, Aq pBRpλ, Aqqn “ Rpλ, AqpI ´ BRpλ, Aqq´1, n“0 ÿ kRpλ, Aqk kRpλ, A ` Bqk ď . 1 ´ kBRpλ, Aqk Proof. By e.g. Proposition 4.24 of [Sc2], the operator I ´ BRpλ, Aq has the inverse 8 n Sλ “ pBRpλ, Aqq n“0 ÿ in BpXq. Hence, λI ´ A ´ B “ pI ´ BRpλ, AqqpλI ´ Aq :DpAq Ñ X is bijective with the bounded inverse Rpλ, AqSλ. Remark 1.11 thus yields the closedness of A ` B on DpAq, and so λ P ρpA ` Bq. The asserted estimate also follows from Proposition 4.24 of [Sc2]. The smallness condition in the above theorem is sharp in general: Let X “ C, a P C – BpCq, a ‰ 0, and b “ a. Then a is invertible, but a ´ a “ 0 a is not. Here we have λ “ 0 and |bRp0, aq| “ | a | “ 1. CHAPTER 2

Spectral theory of compact operators

2.1. Compact operators We first state a few facts from, e.g., Section 1.3 of [Sc2]. A non-empty subset S Ď X is compact if each sequence in S has a subsequence that converges in S. Equivalently, S is compact if every open covering of S has a finite subcovering. We call S Ď X relatively compact if S is compact which means that each sequence in S has a converging subsequence (with limit in S). Finally, S Ď X is relatively compact if and only if it is totally bounded; i.e., for each ε ą 0 there are finitely many balls in X covering it, where one may chose the centers in S. Recall that compact sets are bounded and closed. The converse is true if and only if X has finite dimension. Definition 2.1. A linear map T : X Ñ Y is called compact if T Bp0, 1q is relatively compact in Y . The set of all compact operators is denoted by B0pX,Y q. Remark 2.2. a) If T is compact, then T Bp0, 1q is bounded and thus T is bounded; i.e., B0pX,Y q Ď BpX,Y q. b) Let T : X Ñ Y be linear. Then the following assertions are equivalent. piq T is compact. piiq T maps bounded sets of X into relatively compact sets of Y . piiiq For every bounded sequence pxnqn in X there exists a convergent sub-

sequence pT xnj qj in Y . Proof. piq ñ piiq: If T is compact and B Ď X is bounded, then B Ď Bp0, rq for some r ě 0 and TB Ď T Bp0, rq “ rT Bp0, 1q, which is compact. Hence, TB is compact. The implications piiq ñ piiiq ñ piq are clear. l c) The space of operators of ﬁnite rank is deﬁned by

B00pX,Y q “ tT P BpX,Y q dim TX ă 8u, see Example 5.16 of [Sc2]. For T P B pX,Yˇ q, the set T Bp0, 1q is relatively 00 ˇ compact, and hence B00pX,Y q Ď B0pX,Y q. d) The identity I : X Ñ X is compact if and only if Bp0, 1q is compact if and only if dim X ă 8. ♦

The next result says that B0pX,Y q is a closed two-sided ideal in BpX,Y q.

Proposition 2.3. The set B0pX,Y q is a closed linear subspace of BpX,Y q. Let T P BpX,Y q and S P BpY,Zq. If one of the operators T or S is compact, then ST is compact.

Proof. Let xk P X, k P N, satisfy supkPNkxkk “: c ă 8. 14 2.1. Compact operators 15

1) Let T,R P B0pX,Y q be compact. If α P C, then αT is also compact.

There further exists a converging subsequence pT xkj qj. Since pxkj qj is still bounded, there is another converging subsequence pRx q . Hence, ppT ` kjl l Rqx q converges and so T ` R P B0pX,Y q. kjl l 2) Let Tn P B0pX,Y q converge in BpX,Y q to some T P BpX,Y q as n Ñ 8.

Since T1 is compact, there is a converging subsequence pT1xν1pjqqj. Because kxν1pjqk ď c for all j, there exists a subsubsequence ν2 Ď ν1 such that pT2xν2pjqqj converges. Note that pT1xν2pjqqj still converges. Iteratively, we obtain subsequences νl Ď νl´1 such that pTnxνlpjqqj converges for all l ě n.

Set um “ xνmpmq. Then pTnumqm converges as m Ñ 8 for each n P N. Let ε ą 0. Fix an index N “ Nε P N such that kTN ´ T k ď ε. Let m ě k ě N. We then obtain

kT um ´ T ukk ď kpT ´ TN qumk ` kTN pum ´ ukqk ` kpTN ´ T qukk

ď 2εc ` kTN pum ´ ukqk.

Therefore pT umq is a Cauchy sequence. So we have shown that T is compact. Hence, B0pX,Y q is closed. 3) Let S P B0pX,Y q. Since pT xkqk is bounded, there is a converging subsequence pST xkj qj, so that ST is compact.

If T P B0pX,Y q, then there is a converging subsequence pT xkj qj and thus

ST xkj converges. Again, ST is compact. Remark 2.4. Strong limits of compact operators may fail to be compact. 2 Consider, e.g., X “ ` and Tnx “ px1, . . . , xn, 0, 0,... q for all x P X and n P N. Then Tn P B00pXq Ď B0pXq but Tnx Ñ x “ Ix as n Ñ 8 for every x P X and I R B0pXq. ♦ Example 2.5. a) Let X P tCpr0, 1sq,Lppr0, 1sq 1 ď p ď 8u, Y “ Cpr0, 1sq, and k P Cpr0, 1s2q. Setting ˇ 1 ˇ T fptq “ kpt, τqfpτq dτ, ż0 for f P X and t P r0, 1s, we deﬁne the integral operator T : X Ñ Y for the kernel k. We claim that T P B0pX,Y q. Proof. By Analysis 2 or 3, the function T f is continuous for all f P X and T : X Ñ Y is linear. Since kT fk8 ď kkk8kfk1 ď kkk8kfkp (using that λpr0, 1sq “ 1), we obtain that T P BpX,Y q and thus TB is bounded in Y . where B :“ BX p0, 1q. Moreover, for t, s P r0, 1s and f P B we have 1 |T fptq ´ T fpsq| ď |kpt, τq ´ kps, τq| |fpτq| dτ ż0 ď sup |kpt, τq ´ kps, τq| kfk1 τPr0,1s

ď sup |kpt, τq ´ kps, τq| kfkp. τPr0,1s The right hand side tends to 0 as |t´s| Ñ 0 uniformly in f P B, because k is uniformly continuous. Therefore TB is equicontinuous. The Arzela-Ascoli theorem (see e.g. Theorem 1.47 in [Sc2]) then implies that TB is relatively compact. Hence, T P B0pX,Y q. l 2.1. Compact operators 16

t b) Let X “ Cpr0, 1sq and V fptq “ 0 fpsq ds for t P p0, 1q and f P X. This deﬁnes a bounded operator V on X with norm 1, see Example 1.17. Let ş 1 f P Bp0, 1q. Then }V f}8 ď 1 and }pV fq }8 “}f}8 ď 1. The Arzela-Ascoli theorem (see e.g. Corollary 1.48 in [Sc2]) then yields the compactness of V . 2 c) Let X “ L pRq. For f P X, we deﬁne

´|t´s| T fptq “ e fpsq ds, t P R. żR Theorem 2.14 of [Sc2] yields that T P BpXq. We claim that T is not compact. 1 Proof. Take fn “ rn,n`1s. For n ą m in N, we compute }fn}2 “ 1 and

n`2 n`1 m`1 2 2 s´t s´t }T fn ´ T fm}2 ě e ds ´ e ds dt żn`1 ˇżn żm ˇ n`2 ˇ ˇ ˇ ´2t n`1 n m`1 m ˇ2 “ eˇ e ´ e ´ e ` e ˇ dt żn`1 1 ´2n´2 ` ´2n´4 n`1 n 2 ˘ ě 2 pe ´ e qpe ´ 2e q 1 ´2 ´4 2 “ 2 pe ´ e qpe ´ 2q ą 0.

Hence, pT fnq has no converging subsequence. l 2 d 2 2d d) Let E “ L pR q and k P L pR q. For f P E, we set

d T fpxq “ kpx, yqfpyq dy, x P R . d żR By Example 5.44 in [Sc2], this defines an operator T P BpEq. We claim that T is compact. 2d Proof. Proposition 4.13 of [Sc2] yields kn P CcpR q that converge to k in E. Let Tn be the corresponding integral operators in BpEq. There is a d closed ball Bn Ď R such that supp kn Ď Bn ˆ Bn. We then have d 0, x P R zBn, Tnfpxq “ k px, yqfpyq dy, x P B . # Bn n n for f P E and n P N. Letş Rnf “ f|Bn . Fix n P N and take a bounded sequence pfkq in E. Arguing as in (a), one finds a subsequence such that pRnTnfkj qj has a limit g0 in CpBnq. Since Bn has finite measure, this se- 2 quence converges also in L pBnq. Hence, Tnfkj tend in E to the 0–extension g of g0 as j P 8, and so Tn is compact. Let f P BEp0, 1q. Hölder’sinequality in the inner integral yields 2 2 }T f ´ Tnf}2 “ pkpx, yq ´ knpx, yqqfpyq dy dx d d żR ˇżR ˇ ˇ ˇ ˇ 2 ˇ 2 ď ˇ |kpx, yq ´ knpx, yq| dy |ˇfpyq| dy dx Rd Rd Rd ż ż 2 ż ď }k ´ kn}2 for all n P N. The operators Tn thus converge to T in BpEq so that T is compact by Proposition 2.3. l 2.2. The Fredholm alternative 17

Summarizing, integral operators are usually compact if the base space is compact or has ﬁnite measure, or if the kernel decays fast enough at inﬁnity. σ Proposition 2.6. Let T P B0pX,Y q and xn Ñ x in X as n Ñ 8 (i.e., ˚ ˚ ˚ ˚ xxn, x y Ñ xx, x y as n Ñ 8 for every x P X ). Then T xn converges to T x in Y as n Ñ 8. ˚ ˚ ˚ ˚ ˚ Proof. Let y P Y . We have xT xn ´ T x, y y “ xxn ´ x, T y y Ñ 0 as ˚ ˚ σ n Ñ 8 for each y P Y , hence T xn Ñ T x as n Ñ 8. Suppose that T xn does not converge to T x. Then there were δ ą 0 and a subsequence such that

kT xnj ´ T xk ě δ ą 0 for all j P N. By compactness, there is a subsubsequence and a y P Y such σ σ that T x Ñ y as l Ñ 8. On the other hand, T x Ñ T x and T x Ñ y. njl njl njl Since weak limits are unique, it follows that y “ T x, which is impossible. Theorem 2.7 (Schauder). An operator T P BpX,Y q is compact if and only if its adjoint T ˚ P BpY ˚,X˚q is compact. ˚ ˚ ˚ Proof. 1) Let T be compact and take yn P Y with supnPNkynk “: c ă 8. The set K :“ T BX p0, 1q is a compact metric space for the restriction ˚ of the norm of Y . Set fn :“ yn|K P CpKq for each n P N. Putting c1 :“ maxyPK kyk ă 8, we obtain ˚ kfnk8 “ max|xy, yny| ď cc1 yPK for every n P N. Moreover, pfnqnPN is equicontinuous since ˚ |fnpyq ´ fnpzq| ď kynk ky ´ zk ď c ky ´ zk for all n P N and y, z P K. The Arzela-Ascoli theorem then yields a subsequence pfnj qj converging in CpKq. We thus deduce that ˚ ˚ ˚ ˚ ˚ ˚ ˚ ˚ ˚ kT y T y k ˚ sup | x, T y y | sup | T x, y y | nj ´ nl X “ x p nj ´ nl y “ x nj ´ nl y kxkď1 kxkď1

ď kfnj ´ fnl kCpKq ˚ ˚ ˚ tends to 0 as j, l Ñ 8. This means that pT ynj qj converges and so T is compact. 2) Let T ˚ be compact. By step 1), the bi-adjoint T ˚˚ is compact. Let ˚˚ JX : X Ñ X be the canonical isometric embedding. Proposition 5.54 in ˚˚ [Sc2] says that T JX “ JY T , and hence JY T is compact by Proposition 2.3.

If pxnq is bounded in X, we thus obtain a converging subsequence pJY T xnj qj which is Cauchy. Since JY is isometric, also pT xnj qj is Cauchy and thus converges; i.e., T is compact.

2.2. The Fredholm alternative We need some fact from functional analysis. For non-empty sets M Ď X ˚ and N˚ Ď X we deﬁne the annihilators M K “ tx˚ P X˚ @y P M : xy, x˚y “ 0u, K ˚ ˚ N˚ “ tx P X @ˇy P N˚ : xx, y y “ 0u. ˇ ˇ ˇ 2.2. The Fredholm alternative 18

˚ These sets are equal to X or X if and only if M “ t0u or N˚ “ t0u, respectively, see e.g. Remark 5.21 in [Sc2]. For T P BpXq we have RpT qK “ NpT ˚q, RpT q “ KNpT ˚q, (2.1) NpT q “ KRpT ˚q, RpT ˚q Ď NpT qK. In particular, RpT q is dense if and only if T ˚ is injective; and if RpT ˚q is dense, then T is injective. (See e.g. Proposition 5.46 in [Sc2].) The following theorem extends fundamental results on matrices known from linear algebra.

Theorem 2.8 (Riesz 1918, Schauder 1930). Let K P B0pXq and T “ I ´ K. Then the following assertions hold. (a) RpT q is closed. (b) dim NpT q ă 8 and codim RpT q :“ dim X{ RpT q ă 8. (c) T is bijective ô T is surjective ô T is injective ô T ˚ is bijective ô T ˚ is surjective ô T ˚ is injective. More precisely, we have dim NpT q “ codim RpT q “ dim NpT ˚q “ codim RpT ˚q. (2.2)

Corollary 2.9 (Fredholm alternative). Let L P B0pXq, λ P Czt0u, and y P X. Then one of the following alternatives holds. A) λx “ Lx has only the trivial solution x “ 0. Then for every y P X there is a unique solution x P X of λx ´ Lx “ y given by x “ Rpλ, Lqy. B) λx “ Lx has an n-dimensional solution space NpλI ´Lq for some n P N. ˚ ˚ ˚ ˚ Then there are n linearly independent solutions x1 , . . . , xn P X of λx “ L˚x˚. The equation λx ´ Lx “ y has a solution x P X if and only if ˚ xy, xky “ 0 for every k “ 1, . . . , n. Every z P X satisfying λz ´ Lz “ y is of the form z “ x ` x0, where λx ´ Lx “ y and x0 P NpλI ´ Lq. 1 Proof of Corollary 2.9. We set K “ λ L P B0pXq and note that 1 λx ´ Lx “ y is equivalent to pI ´ Kqx “ λ y. By Theorem 2.8(b), we have either dim NpI ´ Kq “ 0 (case A) or dim NpI ´ Kq “ n P N (case B). In the ﬁrst case, I ´K is bijective due to Theorem 2.8(c) which yields A). In the second case, Theorem 2.8(a) shows that RpI ´ Kq is closed so that RpI ´ Kq “ K NpI ´ K˚q by (2.1). We thus deduce the solvability condition from case B) noting that dim NpI ´ K˚q “ n due to (2.2). If x ´ Kx “ y and z ´ Kz “ y, then z ´ x belongs to NpI ´ Kq, as required in case B). t Example 2.10. Let X “ Cpr0, 1sq and V fptq “ 0 fpsqds for t P r0, 1s and f P X. Then ş RpV q “ tg P C1pr0, 1sq gp0q “ 0u, which is not closed in X. Moreover, NpVˇq “ t0u and V is compact by Examples 1.17 and 2.5, respectively. In thisˇ case, V f “ g can not be solved for all g P X. Summing up, the Fredholm alternative fails for λ “ 0. ♦ Proof of Theorem 2.8. 1) The space N :“ NpT q “ T ´1pt0uq is closed in X. For x P N we have Kx “ x P N, so that K leaves N invariant and its restriction KN to N coincides with the identity on N. On the other hand, KN is still compact so that dim N ă 8 by Remark 2.2d). 2.2. The Fredholm alternative 19

2) Since dim N ă 8, there is a closed subspace C Ď X such that N XC “ t0u and N ` C “ X; i.e., X “ N ‘ C, see e.g. Proposition 5.17 in [Sc2]. Let T˜ : C Ñ RpT q be the restriction of T to C and endow C and RpT q with the norm of X. (Observe that C is a Banach space, but we do not know yet whether RpT q is a Banach space.) If T˜ x “ 0 for some x P C, then x P N and so x “ 0. Let y P RpT q. Then there is an x P X such that T x “ y. We can write x “ x0 ` x1 with x0 P N and x1 P C. Hence, T˜ x1 “ T x1 ` T x0 “ y, and so T˜ is bijective. A corollary to the open mapping theorem (see e.g. Corollary 4.31 in [Sc2]) now yields that RpT q “ RpT˜q is closed if and only if T˜´1 :RpT q Ñ C is bounded. Assume that T˜´1 was unbounded. Then there would exist elementsy ˆn “ T˜xˆn of RpT q withx ˆn P C such thaty ˆn Ñ 0 as n Ñ 8 and ˜´1 ´1 kxˆnk “ kT yˆnk ě δ for some δ ą 0 and all n P N. We set xn “}xˆn} xˆn and note that }xn}“ 1 for all n P N and that 1 yn :“ T˜ xn “ yˆn kxˆnk still converges to 0 as n Ñ 8. Since K is compact, there exists a subsequence pxnj qj and a vector z P X such that Kxnj Ñ z as j Ñ 8. Consequently, xnj “ ynj ` Kxnj Ñ z as j Ñ 8 and so kzk “ 1. Since C is closed, z belongs to C. On the other hand, z is contained in N because of

T z “ z ´ Kz “ lim Kxnj ´ K lim xnj “ 0. jÑ8 jÑ8 Hence, z P C X N “ t0u, which contradicts kzk “ 1. Assertion (a) has been established. 3) Theorem 2.7 shows the compactness of K˚ so that dim NpI ´K˚q ă 8 by step 1). Using (2.1) and Proposition 5.23 in [Sc2], we further obtain NpT ˚q “ RpT qK – pX{ RpT qq˚. Since NpT ˚q is ﬁnite-dimensional, linear algebra yields that dim NpT ˚q “ dimpX{ RpT qq˚ “ dim X{ RpT q “ codim RpT q (2.3) and so codim RpT q ă 8; i.e., assertion (b) is true. We next prove in two steps the remaining equalities in (2.2) which then yield the ﬁrt part of (c).1 4) Claim A: There is a closed linear subspace Nˆ with dim Nˆ ă 8 and a closed linear subspace Rˆ of X such that ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ X “ N ‘ R,T N Ď N,T R Ď R and T2 :“ T|Rˆ : R Ñ R is bijective. ˆ Suppose for a moment that Claim A is true. Setting T1 :“ T|Nˆ P BpNq, we obtain the following properties. ˆ n (i) dim N{RpT1q “ dim NpT1q (by the dimension formula in C ). (ii) NpT q “ NpT1q. In fact, writing x “ x1 ` x2 for x P X, x1 P Nˆ and x2 P Rˆ we deduce that T x “ 0 if and only if T2x2 “ ´T1x1 P Nˆ X Rˆ “ t0u. As T2 is injective, the latter statement is equivalent to x2 “ 0 and T1x1 “ 0. Hence, x P NpT q if and only if x P Nˆ and T1x “ 0; i.e., NpT q “ NpT1q.

1In the lectures (2.2) was not shown. Instead of steps 4) and 5) below, only the (much easier) proof of the ﬁrst part of (c) was given. 2.2. The Fredholm alternative 20

(iii) We deﬁne the map

Φ: Nˆ{ RpT1q Ñ X{ RpT q; x ` RpT1q ÞÑ x ` RpT q, for x P Nˆ Ď X. Because of RpT1q Ď RpT q, the map Φ is well deﬁned. Of course, it is linear. We want to show that Φ is bijective, which leads to

dim Nˆ{ RpT1q “ dim X{ RpT q.

Proof of (iii). If Φpx ` RpT1qq “ 0 for some x P Nˆ, then there is a vector y P X with x “ T y. There are y1 P Nˆ and y2 P Rˆ with y “ y1 ` y2. Hence, T2y2 “ x ´ T1y1 belongs to Rˆ X Nˆ “ t0u. Since T2 is injective, we infer that y2 “ 0. Thus, Φ is injective. Take x P X. There are x1 P Nˆ and x2 P Rˆ “ T Rˆ with x “ x1 ` x2. We now conclude that x ´ x1 “ x2 P RpT q and so

Φpx1 ` RpT1qq “ x1 ` RpT q “ x ` RpT q. Hence, Φ is bijective. ♦ The properties (i)–(iii) yield

dim NpT q “ dim NpT1q “ dim Nˆ{RpT1q “ dim X{RpT q “ codim RpT q. (2.4) Since also K˚ is compact by Theorem 2.7, we further obtain dim NpT ˚q “ codim RpT ˚q. (2.5) Combining (2.3)–(2.5), we deduce assertion (c) assuming Claim A. k k 5) Proof of Claim A. We set Nk “ NpT q and Rk “ RpT q for k P N0. Observe that

t0u “ N0 Ď N1 Ď N2 Ď ...,X “ R0 Ě R1 Ě R2 Ě ...,

TNk Ď Nk´1 Ď Nk, and TRk “ Rk`1 Ď Rk (2.6) for all k P N0. We further have k k T k “ pI ´ Kqk “ I ´ p´1qj`1Kj “: I ´ C , j k j“1 ÿ ˆ ˙ where Ck is compact for each k P N due to Proposition 2.3. Assertions (a) and (b) now imply that

Nk,Rk are closed and dim Nk ă 8 (2.7) for every k P N. We need four more claims to establish Claim A. Claim 1: There is a minimal n P N0 such that Nn “ Nn`j for all j P N0. Indeed, if it was true that Nj Ř Nj`1 for all j P N0, then Riesz’ Lemma (see e.g. Lemma 1.43 in [Sc2]) would give xj P Nj with kxjk “ 1 and 1 dpxj,Nj´1q ě {2 for every j P N0. (Here we use that Nj´1 is closed.) Take l ą k ě 0. Since T xl ` xk ´ T xk P Nl´1 by (2.6), we deduce that

kKxl ´ Kxkk “ kxl ´ pT xl ` xk ´ T xkqk ě 1{2.

As a result, pKxkqk has no converging subsequence, which contradicts the compactness of K. So there is a minimal n P N0 with Nn “ Nn`1. Let x P Nn`2. Then, T x P Nn`1 “ Nn so that x P Nn`1. This means that Nn`1 “ Nn`2, and Claim 1 follows by induction. Claim 2: There is a minimal m P N0 such that Rm “ Rm`j for all j P N0. 2.2. The Fredholm alternative 21

Indeed, if it was true that Rj`1 Ř Rj for all j P N0, then Riesz’ Lemma (see e.g. Lemma 1.44 in [Sc2]) would give xj P Rj with kxjk “ 1 and 1 dpxj,Rj`1q ě {2 for every j P N0. (Here we use that Rj`1 is closed.) Take l ą k ě 0. Since T xk ` xl ´ T xl P Rk`1 by (2.6), we deduce that

kKxk ´ Kxlk “ kxk ´ pT xk ` xl ´ T xlqk ě 1{2.

As a result, pKxkqk has no converging subsequence, which contradicts the compactness of K. So there is a minimal m P N0 with Rm “ Rm`1. Let m`1 m y P Rm`1. Then there is a vector x P X such that y “ T x “ TT x and hence y P TRm “ TRm`1 “ Rm`2 by (2.6). This means that Rm`1 “ Rm`2, and Claim 2 follows by induction.

Claim 3: Nn X Rn “ t0u and Nm ` Rm “ X. n Indeed, for the ﬁrst part, let x P Nn X Rn. Then T x “ 0 and there is a n 2n vector y P X such that T y “ x. Hence, T y “ 0 and so y P N2n “ Nn by Claim 1. Consequently, x “ T ny “ 0. m For the second part, let x P X. Claim 2 yields that T x P Rm “ R2m, and thus there exists a vector y P X with T mx “ T 2my. Therefore, x “ m m px ´ T yq ` T y belongs to Nm ` Rm. Claim 4: n “ m. Indeed, suppose that n ą m. Due to Claim 1 and Claim 2, there is an element x P NnzNm and we have Rn “ Rm. Claim 3 further gives y P Nm Ď Nn and z P Rm “ Rn such that x “ y ` z. Therefore, z “ x ´ y P Nn so that z “ 0 by Claim 3. As a result, x “ y P Nm, which is impossible. Second suppose that n ă m. Due to Claim 1 and Claim 2, we have Nn “ Nm and there is an element x P RnzRm. Claim 3 further gives y P Nm “ Nn and z P Rm Ď Rn such that x “ y ` z. Therefore, y “ x ´ z P Rn so that y “ 0 by Claim 3. As a result, x “ z P Rm, which is impossible.

We can now ﬁnish the proof of Claim A, setting Nˆ :“ Nn and Rˆ “ Rn. By (2.7), the spaces Nn and Rn are closed and dim Nn ă 8. From Claims 3 and 4 we then infer that X “ Nˆ ‘ Rˆ. Moreover, (2.6) and Claim 3 yield T Nˆ Ď Nˆ and T Rˆ “ Rˆ. If T x “ 0 for some x “ T ny P Rˆ and y P X, then y P Nn`1 “ Nn by Claim 1. Therefore, x “ 0 and T|Rˆ is bijective. We next reformulate the Riesz-Schauder theorem in terms of spectral theory. Observe that the Voltera operator in Example 2.10 is compact and has the spectrum σpV q “ t0u with σppV q “ H. Theorem 2.11. Let dim X “ 8 and K P BpXq be compact. Then the following assertions hold. (a) σpKq “ t0u Y9 tλj j P Ju, where J P tH, N, t1, . . . , nu n P Nu. (b) σpKqzt0u “ σppKqzt0u. For all λ P σpKqzt0u the range of λI ´ K is ˇ ˇ closed and ˇ ˇ dim NpλI ´ Kq “ codim RpλI ´ Kq ă 8.

(c) For all ε ą 0 the set σpKqzBp0, εq is ﬁnite. Hence, λj Ñ 0 as j Ñ 8 if J “ N. Proof. If 0 P ρpKq, then K would be invertible. Proposition 2.3 now shows that I “ K´1K would be compact which contradicts dim X “ 8. 2.3. The Dirichlet problem and boundary integrals 22

Hence, 0 always belongs to σpKq. Observe that thus assertion (a) follows from (c) by taking ε “ 1{n for n P N. 1 1 For λ P Czt0u we have λI ´K “ λpI ´ λ Kq. Since λ K P B0pXq, Theorem 2.8 implies either λ P ρpKq or λ P σppKq with 1 1 dim NpλI´Kq“ dim NpI´ λ Kq “ codim RpI´ λ Kq “ codim RpλI´Kqă8. So we have established (b). To prove assertion (c), we suppose that for some ε0 ą 0 we have points λn in σpKqzBp0, ε0q with λn ‰ λm for all n ‰ m in N and vectors xn in Xzt0u with Kxn “ λnxn. In Linear Algebra it is shown that eigenvectors to diﬀerent eigenvalues are linearly independent. Hence, the subspaces

Xn :“ lintx1, . . . , xnu satisfy Xn Ř Xn`1 for every n P N. Moreover, KXn Ď Xn and Xn is closed for each n P N (since dim Xn ă 8). Riesz’ lemma (see e.g. Lemma 1.44 in 1 [Sc2]) gives vectors yn in Xn such that kynk “ 1 and dpyn,Xn´1q ě {2 for each n P N. There are αn,j P C with yn “ αn,1x1 ` ¨ ¨ ¨ ` αn,nxn, and hence n n´1 λnyn ´ Kyn “ pλn ´ λjqαn,jxj “ pλn ´ λjqαn,jxj j“1 j“1 ÿ ÿ belongs to Xn´1. For n ą m, the vector λnyn ´Kyn `Kym is thus contained in Xn´1 so that

1 |λn| ε0 kKyn ´ Kymk “ |λn| yn ´ pλnyn ´ Kyn ` Kymq ě ě . λn 2 2 › › This fact contradicts the compactness› of K. › › › Example 2.12. The following bounded operators are not compact since their spectra are not ﬁnite or a null sequence. p a) the left and right shifts L and R on c0 or ` , 1 ď p ď 8, see Exam- ples 1.17 and 1.25; p b) the translation T ptqf “ fp¨ ` tq for t P Rzt0u, on L pRq, 1 ď p ď 8, see Example 1.21; d c) multiplication operators T f “ mf on X “ CbpSq for S Ď R and a given function m P CbpSq if mpSq is not ﬁnite or a null sequence, see Proposition 1.14. ♦

2.3. The Dirichlet problem and boundary integrals In this subsection, we discuss a principal application of Theorem 2.8 to partial diﬀerential equations. Here we work with real-valued functions for simplicity. We observe that Theorem 2.8 is still true for real scalars by the same proof. 3 2 Let D Ď R be open, bounded and connected with BD P C and outer unit normal ν at BD (see part 3) below). Let ϕ P CpBDq be given. 2 Claim. There is a unique solution u in C pDqXCpDq :“ tu P CpDq u|D P C2pDqu of the Dirichlet problem ˇ ∆upxq “ 0, x P D, ˇ (2.8) upxq “ ϕpxq, x PBD. 2.3. The Dirichlet problem and boundary integrals 23

1) Tools from partial differential equations and uniqueness. We first state the strong maximum principle for the Laplacian, see The- orem 2.2.4 in [Ev]. 2 (MP) Let u P C pDq X CpDq satisfy ∆u “ 0 on D. Then maxD u “ maxBD u. If there is a point x0 P D such that upx0q “ maxD u, then u is constant. Hence, if u, v P C2pDqXCpDq solve (2.8), then w “ u´v P C2pDqXCpDq satisfies ∆w “ 0 on D and w “ 0 on BD. The maximum principle (MP) thus yields that maxD w “ 0. Similarly, the maximum of ´w is 0, so that w “ 0. This means that the problem (2.8) has at most one solution. Theorem 2.7 of [PW] implies the following version of Hopf’s lemma. (HL) Let u P C2pDq X CpDq satisfy ∆u ě 0 on D. Assume that there is a point x0 PBD such that upx0q “ maxD u, Bνupx0q exists, and Bνu is continuous at x0. Then either u is constant or Bνupx0q ą 0.

2) The double layer potential. We want to reformulate (2.8) using an integral operator. To this aim, we first consider the “Newton potential” on 3 given by γpxq “ 1 for x P 3zt0u. It satisfies ∆γ “ 0 on 3zt0u. R 4π|x|2 R R We define B px ´ yq ¨ νpyq kpx, yq “ γpx ´ yq “ ´p∇γqpx ´ yq ¨ νpyq “ 3 (2.9) Bνpyq 4π|x ´ y|2 3 for all x P R and y PBD with x ‰ y, where the dot denotes the Euclidean 3 scalar product in R . One introduces the “double layer potential” by setting

Sgpxq “ kpx, yqgpyq dσpyq (2.10) żBD 3 for all x P R zBD and g P CpBDq, where the surface integral is recalled below in part 3). Standard results from Analysis 2 or 3 then imply that 3 3 Sg P C8pR zBDq and ∆Sg “ 0 on R zBD. For each ϕ P CpBDq, one thus obtains the solution u “ Sg|D of (2.8) if one can find a map g P CpBDq satisfying lim Sgpxq “ ϕpzq for all z PBD. (2.11) xÑz xPD 3) The surface integral. A compact boundary BΩ of an open subset d k ˜ ˜ d Ω Ď R belongs to C , k P N, if there are open subsets Uj and Vj of R and k C –diffeomorphisms Ψj : V˜j Ñ U˜j, j P t1, . . . , mu, such that the functions ´1 Ψj and Ψj and their derivatives up to order k have continuous extensions to BV˜j and BU˜j, respectively, BΩ Ď V˜1 Y¨ ¨ ŸV˜m, and Ψj maps Vj :“ V˜j XBΩ ˜ d´1 ´1 onto Uj :“ Uj X pR ˆ t0uq. We set Fj “ Ψj |Uj . Below we use these 2 notions for k “ 2 and D “ Ω. We also identify Uj with a subset of R 2 3 writing t P R instead of pt, 0q P R . We recall that the surface integral for a (Borel) measurable function h : BD Ñ R is given by m 1 T 1 hpyq dσpyq “ ϕjpFjptqqhpFjptqq det F ptq F ptqdt, BD j“1 Uj ż ÿ ż b 2.3. The Dirichlet problem and boundary integrals 24

8 3 ˜ if the right hand side exists. Here, 0 ď ϕj P Cc pR q satisfy supp ϕj Ď Vj and m ˜ ˜ j“1 ϕj “ 1. This definition does not depend on the choice of Ψj : Vj Ñ Uj and ϕ . Moreover, σpBq “ 1 dσ defines a (finite) measure on the Borel ř j BD B sets of BD. In particular, the above integral has the usual properties of integrals. We mostly omit theş index j P t1, . . . , mu. 2 Recall that for y “ F ptq P V and t P U Ď R , the tangent plane of BD 2 at y is spanned by B1F ptq and B2F ptq, where t P U Ď R . Taylor’s formula ´1 2 ˜ applied to Ψ P Cb pUq at pt, 0q yields that s ´ t x :“ Ψ´1ps, 0q “ y ` pΨ´1q1pt, 0q ` Op|s ´ t|2q 0 2 ˆ ˙ 1 2 “ y ` F ptqps ´ tq ` Op|s ´ t|2q. for s P U. Using that νpyq is orthogonal to BjF ptq, we deduce that T 1 2 2 px ´ yq ¨ νpyq “ νpyq F ptqps ´ tq ` Op|s ´ t|2q “ Op|s ´ t|2q. (2.12) On the other hand, Ψ´1 and Ψ are globally Lipschitz so that

c |s ´ t|2 ď |x ´ y|2 ď C |s ´ t|2 (2.13) for all x “ F psq P V , y “ F ptq P V with s, t P U and some constants C, c ą 0. In the following we denote by c various, possibly diﬀering constants.

4) Compactness of a version of S on BD. Using (2.9), (2.12) and (2.13), we obtain |px ´ yq ¨ νpyq| c c |kpx, yq| “ 3 ď ď (2.14) 4π |x ´ y|2 |x ´ y|2 |s ´ t|2 for all x “ F psq and y “ F ptq in V with x ‰ y. As a result, the integrands

ϕpF ptqqkpF psq,F ptqqgpF ptqq det F 1ptqT F 1ptq

b ´1 of Sg are bounded by a constant times |s ´ t|2 }g}8 for all x “ F psq and y “ F ptq in BD with x ‰ y. We next set kpx, xq “ 0 for x PBD and

kpx, yq, |x ´ y| ą 1{n, k x, y 2 np q “ 3 ´1 #n p4πq px ´ yq ¨ νpyq, |x ´ y|2 ď 1{n, for n P N. By means of (2.12) and (2.13), we estimate 3 2 ´1 |knpx, yq| ď cn |s ´ t|2 ď c|s ´ t|2 (2.15) if |x ´ y|2 ď 1{n because then |s ´ t|2 ď c{n. Since kn is continuous on BD ˆ BD, we can deﬁne an operator Tn P BpCpBDqq by

Tngpxq “ knpx, yqgpyq dσ żBD for x PBD and g P CpBDq. As in Example 2.5a), one shows that Tn is compact thanks to the Arzela-Ascoli theorem. For g P CpBDq and x PBD, 1 we set Dpx, nq “ D X Bpx, n q and calculate

ď }g}8 p|kpx, yq| ` |knpx, yq|q dσpyq żDpx,nq m dt ď c kgk8 c |s ´ t|2 j“1 Uj XBps, n q ÿ ż dv ď c kgk8 Bp0, c q |v|2 ż n c n rdr c kgk ď c kgk ď 8 , (2.16) 8 r n ż0 2 employing (2.13), (2.14), (2.15), and polar coordinates in R . Hence, for each x PBD the function y ÞÑ kpx, yqgpyq is integrable for the surface measure σ on BD. So we can deﬁne T gpxq :“ kpx, yqgpyq dσpyq, żBD for x PBD and g P CpBDq. By (2.16), the functions Tng converge uniformly on BD to T g as n Ñ 8 so that T g P CpBDq. Estimate (2.16) actually implies that the diﬀerences Tn ´ T belong to BpCpBDqq and converge to 0 in this space. Hence, T is contained in BpCpBDqq and it is compact by Proposition 2.3 since all Tn are compact. 5) Facts from potential theory. In Theorems VIII and IX in Chap- ter VI of [Ke] it is shown that

lim Sgpxq “ T gpzq ´ 1 gpzq and lim Sgpxq “ T gpzq ` 1 gpzq, (2.17) xÑz 2 xÑz 2 3 xPD xPR zD for all z PBD and g P CpBDq. We set 3 Sgpxq, x P R zD, vpxq “ (2.18) #gpxq, x PBD.

If v “ 0 on D, then there exists Bνvpyq “ 0 for y PBD due to Theorem X in Chapter VI of [Ke]. 6) Conclusion. Let ϕ P pBDq. In view of (2.11) and (2.17), the function Sg P C2pDq has an extension u P C2pDq X CpDq solving (2.8) provided that 1 g P CpBDq satisﬁes 2 g ´ T g “ ´ϕ. Since T is compact, thanks to the Fredholm alternative Corollary 2.9 it 1 remains to establish the injectivity of 2 I ´ T . So let g0 P CpBDq satisfy 1 2 g0 “ T g0. By the previous paragraph, the extension of Sg0 to D then solves (2.8) with ϕ “ 0. This problem has also the trivial solution u “ 0. The uniqueness of (2.8) thus yields Sg0 “ 0 on D. Deﬁne v0 by (2.18) with g0 instead of g. Then Bνv0 “ 0 on BD due to the result mentioned after (2.18). We are now looking for a contradiction 3 with Hopf’s lemma (HL), employing Sg0 on R zD. Fix r0 ą 0 such that D Ď Bp0, r0q. For r ě r0 ` 1, x PBBp0, rq and y PBD, from (2.10) and (2.9) we deduce that c c c |kpx, yq| ď 2 ď 2 ď 2 , |x ´ y|2 pr ´ r0q r 2.4. Closed operators with compact resolvent 26

c c |Sg pxq| ď kg k dσ ď . 0 r2 0 8 r2 żBD Suppose that g0 ‰ 0. We can thus ﬁx a radius r ě r0 ` 1 such that

kg0k8 ą max |Sg0pxq|. xPBBprq

In particular, v0 is not constant on BprqzD. Since ∆v0 “ 0 on BprqzD, the strong maximum principle (MP) says that v0 does not attain its maximum on BprqzD. Since the maximum exists on BprqzD, it must be attained at a point y0 PBD, and hence v0py0q ą v0pxq for all x P BprqzD. Noting that ´νpy0q is the outer unit normal of BprqzD at y0, we infer from (HL) that 1 Bνv0py0q ă 0 contradicting Bνv0 “ 0 on BD. As a result, 2 I ´ T is injective and we have established the claim.

2.4. Closed operators with compact resolvent We now extend the results of Section 2.2 to a class of closed operators introduced in the next deﬁnition.

Definition 2.13. A closed operator A on X has a compact resolvent if there exists a λ P ρpAq such that Rpλ, Aq P BpXq is compact. Remark 2.14. a) Let A have a compact resolvent Rpλ, Aq and µ P ρpAq. The resolvent equation then yields

Rpµ, Aq “ Rpλ, Aq ` pλ ´ µqRpλ, AqRpµ, Aq, so that Rpµ, Aq is also compact due to Proposition 2.3.

b) Let A be closed on X with λ P ρpAq. Recall that rDpAqs “ pDpAq, k¨kAq. The following assertions are equivalent. (i) A has a compact resolvent. (ii) Each bounded sequence in rDpAqs has a subsequence that converges in X. (iii) The inclusion map J : rDpAqs Ñ X is compact.

Proof. Let (i) hold. Take xn P DpAq with kxnkA ď c for n P N. Set yn “ λxn ´ Axn. Then kynk ď p|λ| ` 1qc for every n P N so that xn “ Rpλ, Aqyn has a subsequence which converges in X by (i), and (ii) is true. The implication ‘(ii) ñ (iii)’ follows from Remark 2.2. Let (iii) be valid. Define Rλ P BpX, rDpAqsq by Rλx “ Rpλ, Aqx for x P X. The operator Rpλ, Aq “ JRλ : X Ñ X is then compact due to Proposition 2.3. l c) If T P BpXq has a compact resolvent, then dim X ă 8. In fact, the identity I “ pλI´T qRpλ, T q would be compact by Proposition 2.3, if Rpλ, T q was compact for some λ P ρpT q. d) Let A be closed and densely defined on X with λ P ρpAq. Then, A has a compact resolvent if and only if A˚ has a compact resolvent. Indeed, first note that λ P ρpA˚q and Rpλ, Aq˚ “ Rpλ, A˚q by Theorem 1.24. Theorem 2.7 then yields that the compactness of Rpλ, Aq and Rpλ, Aq˚ “ Rpλ, A˚q are equivalent. ♦ 2.4. Closed operators with compact resolvent 27

In the next example we use the H¨olderspace

α |fpyq ´ fpxq| C pSq “ f P CbpSq rfsα “ sup α ă 8 x‰y |x ´ y| ! ) d ˇ for α P p0, 1q and S Ď R whichˇ is a Banach space with norm }f}Cα “ 1´ }f}8 ` rfsα. For α “ 1 the corresponding space is denoted by C pSq. We 1´ α β have C pSq ãÑ C pSq ãÑ C pSq ãÑ CbpSq if 0 ă β ď α ă 1 where the embeddings are given by the inclusion map. d Example 2.15. a) Let U Ď R be open and bounded, E be a closed subspace of CpUq, and A be closed on E with λ P ρpAq. Assume that α DpAq ãÑ C pUq for some α P p0, 1q. A bounded sequence pfnq in rDpAqs is thus bounded in CαpUq. The theorem of Arzela-Ascoli (see e.g. Corol- lary 1.48 in [Sc2]) yields a subsequence that converges in CpUq, and hence in E. By Remark 2.14, A has a compact resolvent in E. b) Let X “ Cpr0, 1sq, Au “ u1 and DpAq “ tu P C1pr0, 1sq up0q “ 0u. The spectrum of A is empty by Example 1.12. So part a) implies that A ˇ has compact resolvent. ˇ c) Let X “ Cpr0, 1sq, Au “ u1 and DpAq “ C1pr0, 1sq. The resolvent set of A is empty by Example 1.12. Therefore A has no (compact) resolvent although rDpAqs is compactly embedded in X by a). ♦ Theorem 2.16. Let dim X “ 8 and A be a closed operator with compact resolvent. Then the following assertions are true (a) σpAq is either empty or σpAq “ σppAq contains at most countably many eigenvalues λj. (b) If σpAq is infinite, then |λj| Ñ 8 as j Ñ 8. (c) For all λj P σpAq, the range of λjI Á is closed and dim NpλjI Áq “ codim RpλjI ´ Aq ă 8. Proof. Fix µ P ρpAq. By Theorem 2.11, the spectrum σpRpµ, Aqq only contains 0 and either no or finitely many eigenvalues µj or a nullse- quence of eigenvalues µj. Moreover, the range of µjI ´ Rpµ, Aq is closed and dim NpµjI ´ Rpµ, Aqq “ codim RpµjI ´ Rpµ, Aqq ă 8 for all j. Propo- ´1 ´1 sition 1.20 yields pµ ´ σpAqq “ σpRpµ, Aqqzt0u and pµ ´ σppAqq “ σppRpµ, Aqqzt0u. These facts imply assertions (a) and (b), where λj “ ´1 µ ´ µj . Observe that ´1 λjI ´ A “ µj pµjI ´ Rpµ, AqqpµI ´ Aq. Hence, also (c) follows from the results of Theorem 2.11 stated above. Example 2.17. a) Let X “ Cpr0, 1sq and Au “ u1 with DpAq “ tu P C1pr0, 1sq up0q “ up1qu. Then A is closed, has a compact resolvent and σpAq “ σppAq “ 2πiZ. ˇ Proof.ˇ The closedness is shown as in Example 1.2. Let f P X.A function u belongs to DpAq and satisfies u ´ Au “ f if and only if u P C1pr0, 1sq, up0q “ up1q and u1 “ u ´ f. These properties are equivalent to t t t´s uptq “ ce ´ e fpsq ds “: Rcfptq, t P r0, 1s, and up0q “ up1q, ż0 2.5. Fredholm operators 28 for some c “ cpfq, which means that 1 ´s c “ Rcfp0q “ Rcfp1q “ ce ´ e e fpsq ds, ż0 e 1 c “ e´sfpsq ds. e ´ 1 ż0 We thus derive 1 P ρpAq and et`1 1 t Rp1,Aqfptq “ e´sfpsq ds ´ et´sfpsq ds e ´ 1 ż0 ż0 for t P r0, 1s. Due to Example 2.15a), the operator A thus has a compact resolvent and hence σpAq “ σppAq by Theorem 2.16. Finally, λ P C belongs 1 to σppAq if and only if there is a non-zero u P C pr0, 1sq with up0q “ up1q 1 λ and u “ λu which is equivalent to u “ eλup0q and 1 “ eλp0q “ eλp1q “ e ,; i.e., λ P 2πiZ. l b) Let X “ Cpr0, 1sq and Au “ u2 with DpAq “ tu P C2pr0, 1sq up0q “ up1q “ 0u. Then A is closed, has a compact resolvent and σpAq “ σppAq “ 2 2 ˇ t´π k k P Nu. ˇ Proof. We first note that uptq “ sinpπktq is an eigenfunction for A and ˇ 2 2 the eigenvalueˇ λ “ ´π k , where k P N. Conversely, if λ P σppAq, we have a map u P C2pr0, 1sq with up0q “ up1q “ 0 and u2 “ λu. There thus exist 2 a, b, µ P C with µ “ λ and u “ aeµ ` be´µ ‰ 0. The conditions up0q “ 0 and up1q “ 0 then yield a ` b “ 0 and aeµ ` be´µ “ 0, respectively. Hence, 2µ 2 2 e “ 1 and µ ‰ 0; i.e., µ “ iπk and λ “ ´π k for some k P Zzt0u. As in a), it remains to find a point in ρpAq. To this aim, set 1 1 1 t 1 1 Rfptq “ |t ´ s| fpsq ds “ pt ´ sqfpsq ds ` ps ´ tqfpsq ds, 2 2 2 ż0 ż0 żt for t P r0, 1s and f P X. Clearly, R is a bounded operator on X and d 1 t 1 1 Rfptq “ fpsq ds ´ fpsq ds, dt 2 2 ż0 żt so that Rf P C2pr0, 1sq with pRfq2 “ f. Then the function

R0fptq “ Rfptq ´ p1 ´ tqRfp0q ´ tRfp1q, t P r0, 1s, belongs to DpAq and AR0f “ f. Since A is injective, we have shown that A has the inverse R0. l We note that in the above simple examples one could compute the resolvent for all λ P ρpAq, cf. Example 5.9. In this sense the power of Theo- rem 2.16 is not really needed here.

2.5. Fredholm operators In this section we study a class of operators which satisfy most of the assertions of the Riesz–Schauder Theorems 2.8 or 2.16. This class turns out to be stable under compact perturbations, and it arises in many applications as we indicate in Example 2.26. 2.5. Fredholm operators 29

Definition 2.18. A map T P BpX,Y q is called a Fredholm operator if a) its range RpT q is closed in Y , b) dim NpT q ă 8. c) codim RpT q “ dim Y { RpT q ă 8. In this case the index of T is the integer indpT q “ dim NpT qćodim RpT q. For a closed operator A in Y we use the above definition with X “ rDpAqs. One sometimes calls dim NpT q the nullity and codim RpT q the defect of T . Remark 2.19. a) An invertible operator T P BpX,Y q is Fredholm. Loosely speaking, Fredholmity means ‘invertibility except for finite dimensional spaces’, cf. Proposition 2.20 and the exercises. b) Theorem 2.8 says that the operator λI ´ K is Fredholm with index 0 if K P BpXq is compact and λ P Czt0u. The same is true for λI ´ A if λ P C and A is closed in X with compact resolvent, by Theorem 2.16. c) Each integer can occur as an index of a Fredholm operator. For in- n p stance, let T “ L for the left shift L on ` , 1 ď p ď 8, and some n P N; p i.e., T x “ pxn`kqk. Because of RpT q “ ` and NpT q “ linte1,..., enu, the operator T is Fredholm with index n. Moreover, S “ Rn has index ń p for the right shift R on ` and n P N since Sx “ p0, ¨ ¨ ¨ , 0, x1, x2, ¨ ¨ ¨ q with p n zeros, S is injective, and ` { RpSq – linte1,..., enu (cf. Example 2.20 of [Sc2]). d) If dim X “ 8 then the Fredholm operators are not a linear subspace of BpXq, since e.g. the identity I is Fredholm, but I ´ I “ 0 not. e) One can omit condition a) in Definition 2.18. Proof. [Kato (1958)] Let T P BpX,Y q satisfy codim RpT q “ n P N0. If n “ 0, then RpT q “ Y and we are done. Hence, take n P N. The operator Q : X Ñ X{ NpT q; Qx “ x ` NpT q “ xˆ, is a surjective contraction with kernel NpT q, see e.g. Proposition 2.19 in [Sc2]. By Tˆpx`NpT qq “ Tˆxˆ :“ T x, we define a bijective operator Tˆ P BpX{ NpT q, RpT qq such that T “ TQˆ , cf. Proposition A.1.3 of [Co2]. Further, there are y1, ¨ ¨ ¨ , yn P Y such that Y “ RpT q ` linty1, ¨ ¨ ¨ , ynu and RpT q X linty1, ¨ ¨ ¨ , ynu “ t0u. On the n Banach space E “ pX{ NpT qq ˆ C we define the operator n n S : E Ñ Y ; Spx,ˆ pλ1, ¨ ¨ ¨ , λnqq “ Tˆxˆ ` λjyj “ T x ` λjyj. j“1 j“1 ÿ ÿ It straightforward to check that S is linear, bounded and bijective. The open mapping theorem (see e.g. Theorem 4.28 in [Sc2]) then says that S is an isomorphism. Consequently, RpT q “ SppX{ NpT qq ˆ t0uq is closed. l In context of part e) of the above remark, we stress that for each infinite dimensional Banach space X there are non-closed subspaces Z of X with codimension 1. To see this, take a countable subset B0 “ tbk k P Nu of an algebraic basis B of X as in Remark 1.6. Set ϕpb q “ k and ϕpbq “ 0 for k ˇ b P BzB0. Then ϕ extends to an unbounded linear map ϕ : XˇÑ C. Define ϕˆ as in the proof of Remark 2.19e). It is a linear bijection from X{ Npϕq to Rpϕq “ C; i.e., codim Npϕq “ 1. However, Z “ Npϕq is not closed by Proposition III.5.3 of [Co2]. 2.5. Fredholm operators 30

The analysis in this section is based on the properties of Fredholm operators established in the next result. (See the exercises for a related charac- terization of Fredholm operators of any index.) Proposition 2.20. Let T P BpX,Y q be a Fredholm operator with indpT q “ 0. Then there exists an invertible operator J P BpY,Xq and a ﬁnite rank operator K P B00pXq such that JT “ IX ´ K.

Proof. By the assumptions, there are closed subspaces X1 of X and Y0 of Y such that X “ NpT q ‘ X1, Y “ Y0 ‘ RpT q and dim Y0 “ codim RpT q “ dim NpT q. (Use e.g. Proposition 5.17 of [Sc2].) We set

T1 : X1 ˆ Y0 Ñ Y ; T1px1, y0q “ T x1 ` y0.

Clearly, T1 is linear and continuous. If T1px1, y0q “ 0 for some px1, y0q in X1 ˆ Y0, then y0 “ ´T x1 P RpT q so that y0 “ 0. Hence, x1 belongs to NpT q X X1; i.e., x1 “ 0 and T1 is injective. Let y P Y . Then y “ y0 ` y1 for some y0 P Y0 and y1 “ T x1 with x1 P X. As a result, y “ y0 ` T x1 “ ´1 T1px1, y0q and T1 is bijective. The inverse T1 : Y Ñ X1 ˆ Y0 is then bounded by the open mapping theorem, see e.g. Theorem 4.28 in [Sc2]. ´1 Observe that T1 y1 “ px1, 0q if T x1 “ y1 for some x1 P X1. There exists an isomorphism S : Y0 Ñ NpT q since these spaces have the same ﬁnite dimension. Deﬁne

S1 : X1 ˆ Y0 Ñ Y ; S1px1, y0q “ x1 ` Sy0.

As above, one checks that S1 is invertible. We now introduce the invertible ´1 operator J “ S1T1 : Y Ñ X and the map K :“ IX ´ JT P BpXq. For x1 P X1 we compute ´1 Kx1 “ x1 ´ S1T1 T x1 “ x1 ´ S1px1, 0q “ x1 ´ x1 “ 0.

Since X “ NpT q ‘ X1, we derive KX Ď K NpT q, and hence dim RpKq ď dim NpT q ă 8 as asserted. We can now show an important perturbation result for Fredholmity and the index. The quantitative smallness condition from Theorem 1.27 (on invertibility) is replaced by compactness. Theorem 2.21. Let T P BpX,Y q be Fredholm and K P BpX,Y q be compact. Then T ` K P BpX,Y q is Fredholm with indpT ` Kq “ indpT q.

Proof. Set n “ indpT q P Z. 1) Let n “ 0. Proposition 2.20 yields an invertible operator J P BpY,Xq and a map K1 P B00pXq such that JT “ IX ´ K1. The product JK is compact by Proposition 2.3. We thus deduce that JpT ` Kq “ IX ´ pK1 ´ JKq is Fredholm with index 0 from Theorem 2.8. The invertibility of J easily implies that also T ` K is Fredholm with index 0. n 2) Let n ą 0. Set Y˜ “ Y ˆ C , and deﬁne T˜ : X Ñ Y˜ ; T˜ x “ pT x, 0q, K˜ : X Ñ Y˜ ; Kx˜ “ pKx, 0q. It is straightforward to check that RpT˜q is closed, NpT q “ NpT˜q, Y˜ { RpT˜q – pY { RpT qq ˆ t0u. In particular, codim RpT˜q “ codim RpT q ` n and so T˜ is Fredholm with index 0. Since K˜ is still compact, by part 1) the operator 2.5. Fredholm operators 31

T˜ `K˜ is also Fredholm with index 0. Noting that pT˜ `K˜ qx “ pT x`Kx, 0q, we infer that T ` K is Fredholm with index n. n 3) Let n ă 0. Set Xˆ “ X ˆ C| |, and define Tˆ : Xˆ Ñ Y ; Tˆpx, ξq “ T x, Kˆ : Xˆ Ñ Y ; Kˆ px, ξq “ Kx. Starting from dim NpTˆq “ dim NpT q`|n| and RpTˆq “ RpT q, one derives the assertion as in part 2). To exploit the above result in spectral theory, we need another definition. Definition 2.22. For T P BpXq we define the essential spectrum by

σesspT q “ tλ P C λI ´ T is not Fredholmu. We also set ˇ 0 ˇ σesspT q “ tλ P C λI ´ T is not Fredholm of index 0u. For a closed operator A on ˇX we define analogously ˇ σesspAq “ tλ P C λI ´ A : rDpAqs Ñ X is not Fredholmu, 0 σesspAq “ tλ P C ˇ λI ´ A : rDpAqs Ñ X is not Fredholm of index 0u. ˇ Observe that σ pˇBq Ď σ0 pBq Ď σpBq and that λ P σpBqzσ0 pBq is an ess ˇ ess ess eigenvalue with finite dimensional eigenspace since λI ´ B is then Fredholm with dim NpBq “ codim RpBq ă 8. (Here B P BpXq or B is closed on X.) There are various differing concepts of the essential spectrum in the literature. Typically, they lead to the same essential spectral radius

resspT q “ supt|λ| λ P σesspT qu, if T P BpXq. The next concept is usedˇ below in our perturbation result ˇ Theorem 2.25 for σesspAq. Definition 2.23. Let A and B be linear from X to Y with DpAq Ď DpBq. Then B is called A–compact (or, relatively compact with respect to A) if B : rDpAqs Ñ Y is compact. Note that an A–compact operator is automatically A–bounded. Relative compactness is further discussed in the excercises. Lemma 2.24. Let A be closed from X to Y and B be A–compact. Then A ` B with DpA ` Bq “ DpAq is closed and B is relatively compact with respect to A ` B.

Proof. 1) We first check that B is pA ` Bq–compact. Let pxnq Ď DpAq be bounded for } ¨ }A`B. In particular, pxnq is bounded in X. ´1 Suppose that αn :“}Axn} tends to infinity as n Ñ 8. Setx ñ “ αn xn for n P N. (We may assume that Axn ‰ 0 for all n.) Thenx ñ Ñ 0 in X and ´1 pA ` Bqxñ “ αn pA ` Bqxn Ñ 0 in Y as n Ñ 8, whereas }Axñ}“ 1 for all n. Since B is A–compact, there is a subsequence such that Bxñj converges to some y in Y as j Ñ 8. Hence, Axñj tends to ý. The closedness of A then yields y “ 0, which is impossible since 1 “}Axñj }Ñ}y} as j Ñ 8.

We conclude that there exists a subsequence such that pAxnk qk is bounded in Y . Using again the A–compactness of B, we obtain another subsequence pBxnkl ql with a limit in Y ; i.e., B is pA ` Bq–compact. 2.5. Fredholm operators 32

2) Let xn P DpAq tend to some x in X and pA ` Bqxn to some y in Y as n Ñ 8. By part 1), there is subsequence and a vector z P Y such that

Bxnj Ñ z in Y as j Ñ 8. As a result, Axnj tends to y ´ z. Since A is closed, we infer that x P DpAq “ DpA ` Bq and Ax “ y ´ z. This means that xnj Ñ x in rDpAqs, and hence Bxnj Ñ Bx “ z by continuity. It follows pA ` Bqx “ y and so A ` B is closed. Theorem 2.25. Let A be closed on X and B be A–compact. Then 0 0 σesspA ` Bq “ σesspAq and σesspA ` Bq “ σesspAq. Proof. We apply Theorem 2.21 to the operators λI ´ A and ´B from 0 rDpAqs to X, where λ P σesspAq or λ P σesspAq, and to the operators λI Á´ 0 B and B from rDpA ` Bqs to X, where λ P σesspA ` Bq or λ P σesspA ` Bq. In the second part we use that B is pA ` Bq–compact by Lemma 2.24. Consequently, λI ´ A is Fredholm (with index 0) if and only if λI ´ A ´ B is Fredholm (with index 0), as asserted. We apply the above result to a typical situation arising in partial differential equations. However, we can only give a rough sketch. Example 2.26. We study the asymptotic stability of stationary solutions to the reaction-convection-diffusion equation Btupt, xq “ dBxxupt, xq ` cBxupt, xq ` fpupt, xqq, t ě 0, x P R, (2.19) up0, xq “ u0pxq, x P R, for diffusion and convection constants d ą 0 and c P R, a given initial state u0 P CubpRq “ tv P CbpRq v is uniformly continuousu, and a function f P C2p q with fp q Ď describing (auto-)reaction (where we mean real C R R ˇ differentiability). We interpret uˇpt, xq as the density of a species. It is known that there is a maximal existence time t “ tpu0q P p0, 8s and a 1 unique solution u of (2.19) in the space Cpr0, tq,CbpRqqXC pp0, tq,CbpRqqX 2 2 Cpp0, tq,Cb pRqq. Moreover, if u0 ě 0 and fp0q ě 0, then u ě 0. (See e.g. Proposition 7.3.1 in [Lu] or Theorem 3.8 in [Sc3].) 2 Let u˚ P Cb pR, Rq be a stationary solution of (2.19); i.e., upt, xq “ u˚pxq solves (2.19) which is equivalent to 2 1 0 “ du˚ ` cu˚ ` fpu˚q on R. One now asks whether such special solutions describe well the behavior of (2.19), at least locally near u˚. One possible answer is the principle of linearized stability. Here, one proceeds similar as for ordinary differential equations. Define in X “ CbpRq the maps A and F by 2 1 2 Av “ dv ` cv with DpAq “ Cb pRq,F pvq “ f ˝ v. One can then check that F P C1pX,Xq with derivative F 1pvq P BpXq at v P X given by F 1pvqw “ f 1pvqw for w P X. (One defines differentiablity in n Banach spaces analogously as in R .) We introduce the linearized operator at u˚ by setting 1 2 1 1 2 A˚v “ Av ` F pu˚qv “ dv ` cv ` f pu˚qv with DpA˚q “ Cb pRq. 2 We note that one needs the uniform continuity of u0 to obtain the stated continuity of u at t “ 0. 2.5. Fredholm operators 33

The principle of linearized stability now says the following. If spA˚q :“ suptRe λ λ P σpAqu ă ´δ ă 0, then there are constants c, r ą 0 such that ´δt @ u0 P BXˇ pu˚, rqXCubpRq : tpu0q “ 8 and }uptqú˚}8 ď ce }u0 ú˚}8 ˇ for all t ě 0, where u solves (2.19). (See e.g. Theorem 3.8 in [Sc3].) We note that such results fail for certain partial differential equations. Here it works since (2.19) is of ‘parabolic type’. Of course, one now has to compute the sign of the spectral bound spA˚q (or different properties of σpA˚q for more refined versions of the above result). We sketch a partial answer for the important special case that u˚psq has limits ξ˘ in R as s Ñ ˘8. Then the limit operators 1 1 2 A˘ “ A ` F pξ˘ q with DpA˘q “ Cb pRq have constant coefficients which simplifies the computation of their spectral properties. We now follow the survey article [Sa]. Let λ P C. We rewrite A˘u ´ λu “ g as a first order system by 1 v1 v 0 1 v1 0 Lpλq :“ 1 ´ “ , v v1 1 pλ ´ f 1pξ qq ´ c v 1 g ˆ 2˙ ˆ 2˙ ˆ d ˘ d ˙ ˆ 2˙ ˆ d ˙ 1 where pv1, v2q fl pu, u q. We set 0 1 M pλq “ . ˘ 1 pλ ´ f 1pξ qq ´ c ˆ d ˘ d ˙ u We denote by X˘pλq the linear span of all (generalized) eigenvectors of M˘pλq for eigenvalues µ with Re µ ą 0. Theorems 3.2 and 3.3 and Re- mark 3.3 in [Sa] then yield

λI ´ A˚ is Fredholm ðñ λ R σesspA˚q ðñ σpM˘pλqq X iR “H, u u indpλI ´ A˚q “ dim X´pλq ´ dim X`pλq, 0 u u λ R σesspA˚q ðñ σpM˘pλqq X iR “H and dim X´pλq “ dim X`pλq. Note that here non-zero indices naturally occur. The proofs of these results use Theorems 2.21 and 2.25 and properties of the ordinary diﬀerential equation governed by the matrices 0 1 M psq “ , s P . λ 1 pλ ´ f 1pu psqqq ´ c R ˆ d ˚ d ˙ One thus has to study the eigenvalues of M˘pλq (which is easy) to determine the location of the essential spectrum. It then remains the (diﬃcult) task to locate the eigenvalues of A˚. In particular for one space dimension, corresponding tools are discussed in [Sa]. ♦ CHAPTER 3

Sobolev spaces and weak derivatives

d Throughout, U Ď R is open and non-empty. 3.1. Basic properties We are looking for properties of C1–functions which can be generalized to a theory of derivatives suited to Lp spaces. Looking at the theorem of dominated convergence for instance, one sees that here the basic concepts should not be based on pointwise limits. It turns out that integration by parts is an excellent starting point for such a theory. 1 8 Let f P C pUq and ϕ P Cc pUq. Set g0 “ ϕf on supp ϕ and extend it by 1 d 0 to a function g P Cc pR q. Then B1g “ pB1ϕqf ` ϕB1f on U since both sides are trivially equal to 0 on Uz supp ϕ. Take a number a ą 0 such that d 1 supp g Ď p´a, aq “: Cd and write x “ px1, x q. We then derive

pB1fqϕ dx “ ´ fB1ϕ dx ` B1g dx U U U ż ż ż a 1 1 “ ´ fB1ϕ dx ` B1gpx1, x q dx1 dx żU żCd´1 ż´a 1 1 1 “ ´ fB1ϕ dx ` pgpa, x q ´ gp´a, x qq dx żU żCd´1

“ ´ fB1ϕ dx. żU Inductively one shows that

pBαfqϕ dx “ p´1q|α| fBαϕ dx, (3.1) żU żU k 8 d for all f P C pUq, ϕ P Cc pUq and α P N0 with |α| ď k, where α α1 αd B :“B1 ¨ ¨ ¨ Bd and |α| :“ α1 ` ¨ ¨ ¨ ` αd. To imitate (3.1) in a deﬁnition, we set p p LlocpUq “ tf : U Ñ C f measurable, f|K P L pKq for all compact K Ď Uu p for p P r1, 8s. We extendˇ f P LlocpUq by 0 to a measurable function f : d ˇ p R Ñ C without further notice. Convergence in LlocpUq means that the restrictions to K converge in LppKq for all compact K Ď U. Observe that p p 1 L pUq Ď LlocpUq Ď LlocpUq for all 1 ď p ď 8. 1 d Definition 3.1. Let f P LlocpUq, α P N0, and p P r1, 8s. If there is a 1 function g P LlocpUq such that gϕ dx “ p´1q|α| fBαϕ dx (3.2) żU żU 34 3.1. Basic properties 35

8 α for all ϕ P Cc pUq, then g “: B f is called weak derivative of f. We set α,p p α p D pUq “ tf P LlocpUq DB f P LlocpUqu. Moreover, one deﬁnes the Sobolev spacesˇ by ˇ W k,ppUq “ tf P LppUq f P Dα,ppUq, Bαf P LppUq for all |α| ď ku for k P and endows themˇ with N ˇ 1{p α p kB fkp , 1 ď p ă 8, $ˆ 0ď|α|ďk ˙ kfkk,p “ ’ ÿ ’ &’ α max kB fk8, p “ 8, 0ď|α|ďk ’ ’ 0 ’ 0,p p ej where B f :“ f. We write%’ W pUq “ L pUq, B “Bj, B“B1 if d “ 1, and k,p α,p D “ 0ď|α|ďk D . Ş p α,p k,p As usually, LlocpUq, D pUq and W pUq are spaces of equivalence classes modulo the subspace N “ tf : U Ñ C f is measurable, f “ 0 a.e.u. d ˇ Remark 3.2. Let α, β P N0, p P r1, 8sˇ , and k P N. a) We will see in Lemma 3.4 that Bαf is uniquely determined for a.e. x P U. Formula (3.1) thus implies that CkpUq`N Ď Dα,ppUq for all |α| ď k and that weak and classical derivatives coincide for f P CkpUq. α,p α α,p p b) D pUq is a vector space and the map B : D pUq Ñ LlocpUq is linear. c) Let f P Dα,ppUq X Dα`β,ppUq. Then Bαf belongs to Dβ,ppUq and BβBαf “Bα`βf. Hence, if also f P Dβ,ppUq, then Bβf P Dα,ppUq and BαBβf “Bα`βf “BβBαf. 8 α,p α`β,p p Proof. Let ϕ P Cc pUq and f P pD pUqXD pUqq Ď LlocpUq. Using (3.2) and Schwarz’ theorem from Analysis 2, we then compute

p´1q|β| pBαfq Bβϕ dx “ p´1q|α|`|β| f BαBβϕ dx żU żU “ p´1q|α`β| f Bα`βϕ dx żU “ pBα`βfq ϕ dx; żU i.e., Bαf P Dβ,ppUq and BβBαf “Bα`βf. l k,p d) pW pUq, k¨kk,pq is a normed vector space. A sequence pfnqn converges k,p α p in W pUq if and only if pB fnqn converges in L pUq for each α with |α| ď k. p p p 1,p Note that }f}1,p “}f}p ` } |∇f|p }p for f P W pUq and p ă 8. e) The map k,p p m α J : W pUq Ñ L pUq ; f ÞÑ pB fq|α|ďk , d is a linear isometry, where m is the number of α in N0 with |α| ď k and p m L pUq has the norm }pfjqj} “ |p}fj}pqj|p. Since the p-norm and the 1-norm 3.1. Basic properties 36

m on R are equivalent, there are constants Ck, ck ą 0 such that α α ck kB fkp ď kfkk,p ď Ck kB fkp 0 |α| k 0 |α| k ďÿď ďÿď k,p for all f P W pUq. ♦ We start with simple, but instructive one-dimensional examples. 3.3 a) Let f C be such that f : f belong to C1 . Example . P pRq ˘ “ |R˘ pR˘q 1,1 We then have f P D pRq with f 1 on r0, 8q Bf “ ` “: g. f 1 on p´8, 0q " ´ * 1 1 For fpxq “ |x|, we thus obtain Bf “ R` ´ p´8,0q, for instance. 8 Proof. For every ϕ P Cc pRq, we compute 0 8 1 1 1 fϕ dt “ f´ϕ dt ` f`ϕ dt żR ż´8 ż0 0 8 1 0 1 1 8 “ ´ f´ϕ dt ` f´ϕ|´8 ´ f`ϕdt ` f`ϕ |0 ż´8 ż0 “ ´ gϕ dt, żR since f`p0q “ f´p0q by the continuity of f. l 1 1,1 b) The function f “ R` does not belong to D pRq. 1 Proof. Assume there would exist g “Bf P LlocpRq. Then we would 8 obtain for every ϕ P Cc pRq that 8 1 1 1 gϕ dt “ ´ R` ϕ dt “ ´ ϕ ptq dt “ ϕp0q. żR żR ż0 Taking ϕ with supp ϕ Ď p0, 8q, we deduce from Lemma 3.4 below that g “ 0 on p0, 8q. Similarly, it follows that g “ 0 on p´8, 0q. Hence, g “ 0 and so 8 ϕp0q “ 0 for all ϕ P Cc pRq, which is false. l 1 2 c) Set fpx, yq “ R` pxq for px, yq P R . Then there exists the weak derivatives B2f “ 0, and thus B1B2f “B2B2f “ 0. However, the weak derivative B1f does not exist. (See exercises.) ♦ So far we have just used the definition of weak derivatives by duality. For further examples and deeper results one needs mollifiers which we recall and discuss next. d Fix a function 0 ď χ P C8pR q with support Bp0, 1q and χ ą 0 on Bp0, 1q. d For x P R and ε ą 0, we set 1 ´d 1 kpxq “ χpxq and kεpxq “ ε kp ε xq. kχk1 8 d Note that 0 ď kε P C pR q, kεpxq ą 0 if and only if |x|2 ă ε, and kkεk1 “ 1. 1 d Let f P LlocpR q and ε ą 0. We now introduce the mollifier Gε by

Gεfpxq “ pkε ˚ fq pxq “ kεpx ´ yqfpyq dy “ kεpzqfpx ´ zq dz, żBpx,εq żBp0,εq (3.3) 3.1. Basic properties 37

d d for x P R or x P U, where we have set fpxq “ 0 for x P R zU. For a subset S of Banach space and ε ą 0, we deﬁne

Sε “ S ` Bp0, εq. From e.g. Proposition 4.13 in [Sc2] and its proof we recall that 8 d Gεf P C pR q, (3.4) supp Gεf Ď Sε for S :“ supp f, Sε is compact if S is compact, (3.5) p kGεfkLppUq ď kGεfkLppRdq ď kfkp if f P L pUq and 1 ď p ď 8, (3.6) p p Gεf Ñ f in L pUq as ε Ñ 0 if f P L pUq and 1 ď p ă 8 (3.7) or if p “ 8 and f is uniformly continuous. 8 Lemma 3.4. Let K Ď U be compact. Then there is a function ψ P Cc pUq 1 such that 0 ď ψ ď 1 on U and ψ “ 1 on K. Let g P LlocpUq satisfy

gϕ dx “ 0 żU 8 for all ϕ P Cc pUq. Then g “ 0 a.e.. In particular, weak derivatives are uniquely deﬁned. 1 Proof. 1) Let 0 ă δ ă 2 distpBK, BUq. Then K2δ “ K ` Bp0, 2δq is 1 8 compact and K2δ Ď U. The function ψ :“ Gδ Kδ thus belongs to Cc pUq by (3.4) and (3.5). Moreover, (3.3) and (3.6) imply that 0 ď ψpxq ď kψk8 ď 1 k Kδ k8 “ 1 for all x P U and 1 ψpxq “ kδpx ´ yq Kδ pyq dy “ kkδk1 “ 1 żBpx,δq for all x P K. The ﬁrst claim is shown. 2) Assume that g ‰ 0 on a Borel set B Ď U with λpBq ą 0. Theorem 2.20 of [Ru1] yields a compact set K Ď B Ď U with λpKq ą 0. Since ψg P L1pUq, 1 the functions Gεpψgq converge to ψg in L pUq as ε Ñ 0 due to (3.7). Hence, there is a nullset N and a subsequence εj Ñ 0 with εj ď δ such that pGεj pψgqqpxq Ñ gpxq ‰ 0 as j Ñ 8 for each x P KzN. For every x P KzN and j P N, we also deduce

pGεj pψgqqpxq “ kεj px ´ yqψpyq gpyq dy “ 0 żU from the assumption, since the function y ÞÑ kεj px ´ yqψpyq belongs to 8 Cc pUq. This is a contradiction. Recall that H¨older’sinequality implies that the map

p p1 L pBq ˆ L pBq Ñ C; pf, gq ÞÑ fg dx, (3.8) żB d is continuous for all 1 ď p ď 8 and Borel sets B Ď R . The next lemma is the key to many properties of weak derivatives. d Lemma 3.5. Let α P N0, p P r1, 8s, and ε ą 0. α,p 8 (a) Let f P D pUq and p ă 8. Then the functions Gεf P C pUq α α p converge to f and B pGεfq tend to B f in LlocpUq as ε Ñ 0. Moreover, α α B pGεfqpxq “ GεpB fqpxq if x P U, ε ă dpx, BUq. 3.1. Basic properties 38

For all εj Ñ 0 we obtain a subsequence εn :“ εjn Ñ 0 such that Gεn f Ñ f α α β,p and B pGεn fq Ñ B f a.e. on U as n Ñ 8. If f also belongs to D pUq for d some β P N0, we can take the same εn for all such β. p α,p α (b) Let f, g P LlocpUq and fn P D pUq such that fn Ñ f and B fn Ñ g p α,p α in LlocpUq as n Ñ 8. Then f is contained in D pUq and B f “ g. If these limits exist in LppUq and for all α with |α| ď k, then f is an element of W k,ppUq. Proof. (a) 1) Let ε ą 0 and x P U. If ε ă dpx, BUq, then the function 8 y ÞÑ ϕε,xpyq :“ kεpx ´ yq belongs to Cc pUq since supp ϕε,x “ Bpx, εq. Using a corollary to Lebesgue’s theorem and (3.2), we can thus deduce

α α |α| α B Gεfpxq “ Bx kεpx ´ yqfpyq dy “ p´1q pB ϕε,xqpyqfpyq dy żU żU α α “ ϕε,xpyqpB fqpyq dy “ pGεB fqpxq. żU 2) Choose a compact subset K Ď U and ﬁx δ ą 0 with Kδ Ď U. Take ε P p0, δs. Note that the integrand of pGεgqpxq is then supported in Kδ for 1 every x P K and g P LlocpUq, see (3.3). Hence, (3.7) and part 1) imply that 1 α 1 α 1 1 α 1 1 α 1 α K B pGεfq “ K GεpB fq “ K Gεp Kδ B fq ÝÑ K Kδ B f “ K B f p p converge in L pKq as ε Ñ 0. So the asserted convergence in LlocpUq is true. 3) For m P N, we deﬁne 1 Km “ tx P U dpx, BUq ě m and |x|2 ď mu

These sets are compact and ˇ mP Km “ U. Let εj Ñ 0. Then, for each ˇ N m P N there is a null set Nm Ď Km and a subsequence νm Ď νm´1 such that αG f x tends to Ťαf x and G f x tends to f x for all B ενmpkq p q B p q ενmpkq p q p q x P KmzNm as k Ñ 8. By means of a diagonal sequence, one obtains a sub- α α sequence εn “ εjn Ñ 0 such that B Gεn fpxq Ñ B fpxq and Gεn fpxq Ñ fpxq for x U N as n , where N is a null set. Employing P zp mPN mq Ñ 8 mPN m another diagonal sequence, we can achieve this for countably many Bβf at the same time.Ť Ť 8 (b) Let ϕ P Cc pUq and S “ supp ϕ. Since S is compact, we have fn Ñ f α p and B fn Ñ g in L pSq as n Ñ 8. From (3.8) on S we deduce that

α α |α| α fB ϕ dx “ lim fnB ϕ dx “ p´1q lim pB fnqϕ dx nÑ8 nÑ8 żU żU żU “ p´1q|α| gϕ dx. U α,p α ż Hence, f P D pUq and B f “ g. The last assertion then easily follows. In the next examples we also argue by approximation, but using a diﬀer- ent, more explicit regularisation method. Example 3.6. Let U “ Bp0, 1q. a) Let d ě 2, 1 ď p ă d, and fpxq “ ln|x|2 for x P Uzt0u. Then f belongs to W 1,ppUq with xj Bjfpxq “ 2 “: gjpxq, |x|2 for x ‰ 0 and j P t1, ¨ ¨ ¨ , du. Moreover, f P LqpUqzL8pUq for all q P r1, 8q. 3.1. Basic properties 39

Proof. Using polar coordinates and |xj| ď r, we obtain 1 q q d´1 kfkq “ c |ln r| r dr ă 8, ż0 1 rp 1 kg kp ď c rd´1 dr “ c rd´p´1 dr ă 8, j p r2p ż0 ż0 q p 8 since p ă d. Hence, f P L pUq and gj P L pUq. Deﬁne un P C pUq Ď 1,p ´2 2 1{2 W pUq by unpxq “ lnpn ` |x|2q for n P N. Observe that Bjunpxq “ ´2 2 ´1 pn ` |x|2q xj, unpxq Ñ fpxq and Bjunpxq Ñ gjpxq as n Ñ 8 for all x P Uzt0u. We have the pointwise bounds |fpxq|, n´2 ` |x|2 ď 1, u x ? 2 , u x g x , x U. | np q| ď ´2 2 |Bj np q| ď jp q P #ln 2, n ` |x|2 ą 1, p Lebesgue’s theorem thus yields that un Ñ f and Bjun Ñ fj in L pUq as n Ñ 8. The assertion then follows from Lemma 3.5(b). l d β β´2 (b) Let p P r1, 8q and β P p1´ p , 1s. Set upxq “ |x|2 and fjpxq “ βxj|x|2 1,p for x P Uzt0u and j P t1, . . . , du. Then u P W pUq and Bju “ fj. (See Example 4.18 in [Sc2].) ♦ k,p Proposition 3.7. For 1 ď p ď 8 and k P N. Then W pUq is a Banach space which is isometrically isomorphic to a closed subspace of a LppUqm for some m P N. It is separable if p ă 8 and reﬂexive if 1 ă p ă 8. Moreover, W k,2pUq is a Hilbert space endowed with the scalar product

α α |α| α fB ϕ dx “ lim fnB ϕ dx “ lim p´1q pB fnqϕ dx nÑ8 nÑ8 żU żU żU |α| “ p´1q gαϕ dx. żU α k,p This means that gα is the weak derivative B f so that f P W pUq and k,p k,p fn Ñ f in W pUq. Hence, W pUq is a Banach space. We then deduce from Remark 3.2e) that W k,ppUq is isometrically isomorphic to a subspace of LppUqm which is closed by e.g. Remark 2.11 in [Sc2]. The remaining claims now follow by isomorphy from known results of functional analysis. We next establish product and chain rules. One can derive many variants by modiﬁcations of the proofs. Proposition 3.8. (a) Let f, g P D1,1pUq X L8pUq. Then, fg belongs to D1,1pUq X L8pUq and has the derivatives

Bjpfgq “ pBjfqg ` fpBjgq, j P t1, . . . , du. (3.9) 3.1. Basic properties 40

1 (b) Let 1 ď p ď 8, f P W 1,ppUq and g P W 1,p pUq. Then, fg is contained in W 1,1pUq and satisﬁes (3.9). 1,1 8 Proof. 1) Let f, g P D pUq. Set fn “ Gεn f P C pUq and gn “ 8 Gεn g P C pUq with εn Ñ 0 as in Lemma 3.5(a). Fix m P N and take 8 ϕ P Cc pUq and j P t1, . . . , du. Choose an open and bounded set V such 1 that supp ϕ Ď V Ď V Ď U. Since fn Ñ f and Bjfn ÑBjf on L pV q by Lemma 3.5(a), the formulas (3.8) and (3.1) yield

fgmBjϕ dx “ lim fngmBjϕ dx nÑ8 żU żV “ ´ lim ppBjfnqgm ` fnpBjgmqq ϕ dx nÑ8 żV “ ´ ppBjfqgm ` fpBjgmqq ϕ dx, żU 1,1 so that fgm P D pUq and Bjpfgmq “ pBjfqgm ` fpBjgmq. 1,1 8 2) Let f, g P D pUq X L pUq and gm as in 1). Note that gm Ñ g and 1 Bjgm ÑBjg in LlocpUq as m Ñ 8. Since f is bounded, we obtain

fgBjϕ dx “ lim fgmBjϕ dx mÑ8 żU żU “ lim ´ pBjfqgmϕ dx ` fpBjgmqϕ dx mÑ8 „żU żU  using step 1). In the last line, the second integral converges to U fpBjgqϕ dx, 8 again because of f P L pUq. For the ﬁrst integral we use that gm Ñ g a.e. ş by Lemma 3.5(a) and that kgmk8 ď kgk8 by (3.6). Lebesgue’s theorem (with the majorant |Bjf| kgk8kϕk81supp ϕ) then implies

fgBjϕ dx “ ´ ppBjfqg ` fpBjgqqϕ dx. żU żU 1 Note that pBjfqg ` fpBjgq P LlocpUq by our assumptions. Part (a) is shown. 1 3) Let f P W 1,ppUq and g P W 1,p pUq. If p P p1, 8s we show (3.9) as in p1 step 2), using (3.8) and that gm, Bjgm converge in LlocpUq by Lemma 3.5(a). If p “ 1, we replace the roles of f and g. Hölder’sinequality and (3.9) finally 1 yield that Bjpfgq is contained in L pUq. Proposition 3.9. Let 1 ď p ď 8, j P t1, . . . , du, and f P D1,ppUq. 1 1 (a) Let f be real-valued and h P C pRq with h P CbpRq. Then h˝f belongs to D1,ppUq with derivatives 1 Bjph ˝ fq “ ph ˝ fqBjf. d (b) Let V Ď R be open and Φ “ pΦ1, ¨ ¨ ¨ , Φdq : V Ñ U be a diffeomorphism such that Φ1 and pΦ´1q1 are bounded. Then f ˝ Φ belongs to D1,ppV q with derivatives d Bjpf ˝ Φq “ ppBkfq ˝ Φq BjΦk. k“1 ÿ In both results we can replace D1,ppUq by W 1,ppUq, where in (a) we then also assume that hp0q “ 0 if λpUq “ 8 and p ă 8. 3.1. Basic properties 41

1 8 Proof. By Lemma 3.5(a), there are fn P C pUq such that fn Ñ f and p Bjfn ÑBjf in LlocpUq and a.e. as n Ñ 8. p (a) The function h ˝ f belongs to LlocpUq since 1 |hpfpxqq| ď |hpfpxqq ´ hp0q| ` |hp0q| ď kh k8|fpxq| ` |hp0q| for all x P U. It is contained in LppUq if f P LppUq and if hp0q “ 0 in the case that λpUq “ 8 and p ‰ 8. Let K Ď U be compact. We obtain that

1 |hpfnpxqq ´ hpfpxqq| dx ď kh k8 |fnpxq ´ fpxq| dx ÝÑ 0, żK żK 1 1 |h pfnpxqqBjfnpxq ´ h pfpxqqBjfpxq| dx żK 1 1 1 ď kh k8 |Bjfnpxq ´ Bjfpxq| dx ` |h pfnpxqq ´ h pfpxqq||Bjfpxq| dx Ñ 0 żK żK as n Ñ 8 where we also used Lebesgue’s theorem and the majorant 1 1 1 2kh k8|Bjf| in the last integral. Since h˝fn P C pUq, Bjph˝fnq “ ph ˝fnqBjfn 1 p and ph ˝ fqBjf belongs to LlocpUq, Lemma 3.5(b) yields assertion (a). If 1,p 1 p 1,p f P W pUq, then ph ˝ fqBjf P L pUq and so h ˝ f P W pUq. (b) Let B “ Φ´1pAq for an open set A Ď U (or the closure of an open set whose boundary has measure 0) and g P LppAq. For p ă 8, the transformation rule yields

p p ´1 1 p |gpΦpxqq| dx “ |gpyq| | detrpΦ q pxqs| dx ď c }f}LppAq. żB żA An analogous estimate is also true for p “ 8. Hence, f ˝ Φ belongs to p p LlocpV q and fn ˝ Φ tends to f ˝ Φ in LlocpV q. We further have d Bjpfn ˝ Φq “ ppBkfnq ˝ Φq BjΦk. k“1 ÿ p The above estimate also implies that pBkfnq˝Φ tends to pBkfq˝Φ in LlocpV q as n Ñ 8. Since BjΦk is uniformly bounded, Lemma 3.5(b) yields that f ˝Φ in contained in D1,ppUq and that it has the asserted derivative. If f P LppUq p p we can replace throughout LlocpV q by L pV q. 1,1 Corollary 3.10. Let f P D pUq be real-valued. Then the functions f`, 1,1 f´ and |f| belong to D pUq with 1 1 1 Bjf˘ “ ˘ tfż0uBjf and Bj|f| “ tfą0u ´ tfă0u Bjf for all j P t1, . . . , du. Here one can replace D`1,1 by W 1,p for all˘ 1 ď p ď 8. ? 2 1 2 2 Proof. We employ the map hε P C pRq given by hεptq :“ t ` ε ´ 1 ε ď t for t ě 0 and hεptq :“ 0 for t ă 0, where ε ą 0. Observe that khεk8 “ 1 1 for ε ą 0 and that hεptq Ñ r0,8qptqt for t P R as ε Ñ 0. Proposition 3.9 1,1 shows that hε ˝ f P D pUq and

1 f hεpfqBjϕ dx “ ´ h pfqpBjfqϕ dx “ ´ pBjfqϕ dx ε 2 2 U U tfą0u f ` ε ż ż ż 1Not shown in the lectures. a 2Not shown in the lectures. 3.1. Basic properties 42

8 1 1 for each ϕ P Cc pUq. Using the majorants kBjϕk8 B|f| and kϕk8 B|Bjf| with B “ supp ϕ, we deduce from Lebesgue’s convergence theorem that f f B ϕ dx “ ´ pB fqϕ dx “ ´ 1 pB fqϕ dx. ` j |f| j tfą0u j żU żtfą0u żU 1 1 p There thus exists Bjf` “ tfą0uBjf P LlocpUq. Clearly, Bjf` P L pUq if f P 1,p W pUq. The other claims follow from f´ “ p´fq` and |f| “ f` ` f´. We brieﬂy discuss two special cases, namely d “ 1 and p “ 8.

Theorem 3.11. Let J Ď R be an open interval, 1 ď p ă 8, and f P p 1,p LlocpJq. Then f belongs to D pJq if and only if there is a function g P p LlocpJq and a representative of f which is continuous on J and satisﬁes t fptq “ fpsq ` gpτq dτ (3.10) żs for all s, t P J. In this case, g “Bf a.e..

1,p 8 Proof. 1) Let f P D pJq. Take the functions fn “ Gεn f P C pJq from Lemma 3.5(a). Then for a.e. t P J and for a.e. t0 P J we have t t 1 fptq ´ fpt0q “ lim pfnptq ´ fnpt0qq “ lim fnpτq dτ “ Bfpτq dτ. nÑ8 nÑ8 żt0 żt0 Fixing one t and noting that t ÞÑ t Bfpτqdτ is continuous, we obtain a 0 t0 continuous representative of f on J which fulfills (3.10) for t J, s t and ş P “ 0 g “Bf. Subtracting two such equations for any given t, s P J and the fixed t0, we deduce (3.10) with g “Bf for all t, s P J. p 8 2) If (3.10) is satisfied by some g P LlocpJq, take gn P C pJq such that p gn Ñ g in LlocpJq as n Ñ 8. For any s P J and n P N, the function t 8 1 fnptq :“ fpsq ` s gnpτq dτ, t P J, belongs to C pJq with fn “ gn. For ra, bs Ď J with s P ra, bs, we estimate ş b t p p kfn ´ fkLppra,bsq “ pgnpτq ´ gpτqq dτ dt ża ˇżs ˇ b ˇ b ˇ p{p ˇ p{p1 ˇ p ď |ˇt ´ s| |gnpτqˇ ´ gpτq| dτ dt ża ˆża ˙ 1`p{p1 p ď pb ´ aq kgn ´ gkLppra,bsq, p using (3.10) and Hölder’s inequality. Hence, fn tends to f in LlocpJq as 1,p n Ñ 8. Lemma 3.5(b) then yields f P D pJq and Bf “ g. Remark 3.12. a) Let J “ pa, bq for some a ă b in R and f : J Ñ C. We then have f P W 1,1pJq if and only if f is absolutely continuous; i.e., for all ε ą 0 there is a δ ą 0 such that for all subdivisions a ă α1 ă β1 ă ¨ ¨ ¨ ă n αn ă βn ă b with n P N and j“1pβj ´ αjq ď δ we obtain n ř |fpβjq ´ fpαjq| ď ε. j“1 ÿ 3.1. Basic properties 43

In this case, f is diﬀerentiable for a.e. t P J and the pointwise derivative f 1 is equal a.e. to the weak derivative Bf P L1pJq.3 Proof. The implication “ñ” is true since (3.10) yields that

n n βj |fpβjq ´ fpαjq| “ Bfpτq dτ ď |Bfpτq| dτ “: S, n ˇ αj ˇ pαj ,βj q j“1 j“1 ˇż ˇ ż j“1 ÿ ÿ ˇ ˇ where S Ñ 0 as λp n pα ,ˇ β qq Ñ 0. ˇ Ť j“1 j ˇ j ˇ The other implication “ ð2 and the last assertion is shown in Theo- Ť rem 7.20 of [Ru1] (combined with our Theorem 3.11). l b) There is a continuous increasing function f : r0, 1s Ñ R with fp0q “ 0 and fp1q “ 1 such that f 1ptq “ 0 exists for a.e. t P r0, 1s. Consequently, 1 1 “ fp1q ‰ fp0q ` f 1pτq dτ “ 0, ż0 and so f is not absolutely continuous, does not belong to W 1,1p0, 1q and violates (3.10). (See §7.16 in [Ru1].) c) Let f P D1,1pJq and Bf be continuous. Then f is continuously diﬀer- entiable, since then fptq ´ fpsq 1 t “ Bfpτq dτ ÝÑ Bfpsq t ´ s t ´ s żs as t Ñ s, thanks to Theorem 3.11. ♦. Proposition 3.13. Let U be convex. Then W 1,8pUq is isomorphic to 1´ Cb pUq “ tf P CbpUq f is Lipschitzu, and the norm of W 1,8pUq is equivalent toˇ ˇ kfk 1´ “ kfk8 ` rfsLip, Cb where rfsLip is the Lipschitz constant of f.

4 1,8 Proof. 1) Let f P W pUq. Take εn Ñ 0 from Lemma 3.5(a). Let K Ď U be compact. For suﬃciently large n P N, Lemma 3.5 and (3.6) yield

|BjGεn fpzq| “ |Gεn Bjfpzq| ď kBjfk8 ď kfk1,8, for all j P t1, . . . , du and z P K. Using that Gεn fpxq Ñ fpxq as n Ñ 8 for all x P UzN and a null set N, for all x, y P UzN we thus estimate

3Note that a Lipschitz continuous function is absolutely continuous and that an absolutely continuous function is uniformly continuous. 4Not shown in the lectures. 3.2. Density and embedding theorems 44

1´ 8 2) Let f P Cb pUq. Take ϕ P Cc pUq, j P t1, . . . , du, and δ ą 0 such that 1 psupp ϕqδ Ď U. For ε P p0, δs the diﬀerence quotient ε pϕpx ` εejq ´ ϕpxqq converges uniformly on supp ϕ as ε Ñ 0, and hence 1 fBjϕ dx “ lim fpxq pϕpx ` εejq ´ ϕpxqq dx εÑ0 ε ˇżU ˇ ˇżsupp ϕ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ 1 ˇ ď lim |fpy ´ εejq ´ fpyq| |ϕpyq| dy ˇ ˇ εÑ0 ˇ ε ˇ żsupp ϕ ď rfsLipkϕk1. 8 1 Since Cc pUq is dense in L pUq, the map ϕ ÞÑ ´ U fBjϕ dx has a continuous 1 8 linear extension Fj : L pUq Ñ C. There thus exists a function gj P L pUq “ 1 ˚ ş L pUq with kgjk8 “ kFjk ď rfsLip such that

´ fBjϕ dx “ Fjpϕq “ gjϕ dx żU żU 8 for all ϕ P Cc pUq. This means that f has the weak derivative Bjf “ gj P 8 1,8 L pUq. As a result, f P W pUq and kfk1,8 ď kfk8 ` rfsLip. In the above proof convexity is only used in part 1). One can extend the result to other classes of domains, see Proposition 3.28. In the spirit of Remark 3.12, we mention Rademacher’s theorem (Theorem 5.8.5 in [Ev]) which says that a Lipschitz continuous function f is diﬀerentiable for a.e. x P U and that the weak derivative Bjf coincides with the pointwise one. 3.2. Density and embedding theorems In this and the next section we discuss some of the most important theorems on Sobolev spaces. 8 Definition 3.14. For k P N and 1 ď p ă 8, the closure of Cc pUq in k,p k,p W pUq is denoted by W0 pUq. Theorem 3.15. Let k P N and p P r1, 8q. We then have k,p d k,p d W0 pR q “ W pR q. Moreover, the set C8pUq X W k,ppUq is dense in W k,ppUq. Proof.5 We prove the theorem only for k “ 1, the general case can be treated similarly. 1,p d 1) Let f P W pR q. Take any φ P C8pRq such that 0 ď φ ď 1, φ “ 1 on r0, 1s and φ “ 0 on r2, 8q. Set 1 ϕnpxq “ φ n |x|2 (“cut-oﬀ function”) d 8 d for n P N and x P R . We` then˘ have ϕn P Cc pR q, 0 ď ϕn ď 1 and 1 1 d kBjϕnk8 ď kφ k8 n for all n P N, as well as ϕnpxq Ñ 1 for all x P R as n Ñ 8. Thus kϕnf ´ fkp Ñ 0 as n Ñ 8 by Lebesgue’s convergence theorem. Further, Proposition 3.8 implies that

kBjpϕnf ´ fqkp “ kpϕnBjf ´ Bjfq ` pBjϕnqfkp 1 1 ď kϕnBjf ´ Bjfkp ` n kφ k8kfkp ,

5Not shown in the lectures. The ﬁrst part is Theorem 4.21 in [Sc2] for k “ 1. 3.2. Density and embedding theorems 45 and the right hand side tends to 0 as n Ñ 8 for each j P t1, . . . , du. Given ε ą 0, we can thus ﬁx an index m P N such that kϕmf ´ fk1,p ď ε. Due 8 d to (3.4) and (3.5), the functions G 1 pϕmfq belong to Cc pR q for all n P N. n Equation (3.7) and Lemma 3.5 further yield that

G 1 pϕmfq Ñ ϕmf and BjG 1 pϕmfq “ G 1 Bjpϕmfq Ñ Bjpϕmfq n n n p d in L pR q as n Ñ 8, for j P t1, . . . , du. So there is an index n P N with

kG 1 pϕmfq ´ ϕmfk1,p ď ε, n and thus

kG 1 pϕmfq ´ fk1,p ď 2ε. n 2) For the second assertion, we only have to consider the case BU ‰H. Let f P W 1,ppUq. Set 1 Un “ x P U |x|2 ă n and dpx, BUq ą n 8 for all n P N. Then Un Ď Un Ďˇ Un`1 Ď U, Un is compact and( n“1 Un “ U. 8 ˇ Observe that U “ n“1 Un`1zUn´1, where U0, U´1 :“H. There are 8 Ť functions ϕn in Cc pUq such that supp ϕn Ď Un`1zUn´1, ϕn ě 0, and 8 Ť n“1 ϕnpxq “ 1 for all x P U (see e.g. Theorem 5.6 in [RR]). Fix ε ą 0. As in step 1), for each n P N there is a number δn ą 0 such ř 8 that gn :“ Gδn pϕnfq P Cc pUq, supp gn Ď psupp ϕnfqδn Ď Un`1zUn´1 and ń 8 kgn ´ ϕnfk1,p ď 2 ε. Define gpxq “ n“1 gnpxq for all x P U. Observe that on each ball Bpx, rq Ď U this sum is finite. Hence, g P C8pUq. Since 8 ř f “ n“1 ϕnf, we further have ř 8 gpxq ´ fpxq “ pgnpxq ´ ϕnpxqfpxqq, n“1 ÿ ń for all x P U and n P N. Due to kgn ´ ϕnfk1,p ď 2 ε, this series converges absolutely in W 1,ppUq, and 8 kf ´ gk1,p ď kgn ´ ϕnfk1,p ď ε. n“1 ÿ k,p k,p Remark 3.16. a) If U is bounded, then W0 pUq ‰ W pUq, see Lemma 6.67 in [RR]. b) For ‘not too bad’ BU one can replace in C8pUq by C8pUq in Theo- rem 3.15, see Theorem 3.27 below. We now want to study embeddings of Sobolev spaces. We clearly have W k,ppUq ãÑ W j,ppUq, (3.11) W k,ppUq ãÑ W j,qpUq if λpUq ă 8, (3.12) for k ě j ě 0, 1 ď q ď p ď 8. (Recall that we have set W 0,ppUq “ LppUq for 1 ď p ď 8.) The embedding X ãÑ Y means that there is an injective map J P BpX,Y q. Writing c “ kJk, one obtains kfkY ď c kfkX if one identifies Jf with f. 3.2. Density and embedding theorems 46

Theorem 3.17 (Sobolev, Morrey). Let k P N and 1 ď p ă 8. We have the following embeddings. (a) If kp ă d, then

˚ pd k,p d q d ˚ p :“ P pp, 8q and W pR q ãÑ L pR q for all q P rp, p s. d ´ kp (b) If kp “ d, then k,p d q d W pR q ãÑ L pR q for all q P rp, 8q.

(c) If kp ą d, then there are either j P N0 and β P p0, 1q such that d d d k ´ p “ j ` β or k ´ p P N. In the latter case we set j :“ k ´ p ´ 1 P N0 and take any β P p0, 1q. Then k,p d j`β d W pR q ãÑ C0 pR q d for all x, y P R , |α| ď j, a constant c ą 0, and j`β d j d α α C0 pR q :“ tu P C pR q B upxq Ñ 0 as |x|2 Ñ 8 and B u is β–Hölder d ˇ continuous on R , for all 0 ď |α| ď ju. ˇ In parts (a) and (b), the embedding J is just the inclusion map, and in part (c) the function Jf is the continuous representative of f. We rephrase the above results in a slightly modified way using the ‘effec- d k,p tive regularity index’ k ´ p of W .

Corollary 3.18. Let k P N, j P N0, and p P r1, 8q. We have the following embeddings. d d (a) If q P rp, 8q and k ´ p ě j ´ q , then k,p d j,q d W pR q ãÑ W pR q. d (b) If q P rp, 8q and k ´ p “ j, then k,p d j,q d W pR q ãÑ W pR q. d (c) If β P p0, 1q and k ´ p “ j ` β, then k,p d j`β d W pR q ãÑ C0 pR q. Proof of Corollary 3.18. (a) If pk´jqp “ d, then Theorem 3.17(b) yields that k´j,p d q d W pR q ãÑ L pR q (3.13) k j,p d p d for all q P rp, 8q. If pk ´ jqp ą d, then W ´ pR q ãÑ L pR q by (3.11) k j,p d d and W ´ pR q ãÑ L8pR q by Theorem 3.17(c). Hence the interpolation inequality (3.14) below implies the embedding (3.13) for all q P rp, 8s in this case. Let pk ´ jqp ă d. By assumption, we have p ď q ď pdpd ´ pk ´ jqpq´1, so that (3.13) for these q is a consequence of Theorem 3.17(a). Applying the α k j,p d k,p d embedding (3.13) to B f P W ´ pR q for all |α| ď j and f P W pR q, we deduce α α kB fkq ď c kB fkk´j,p ď c kfkk,p. So (a) is true. Part (b) follows from (a), and (c) from Theorem 3.17(c). 3.2. Density and embedding theorems 47

1 8 d Example 3.19. Let d ě 2, α P p0, 1 ´ d q and ϕ P Cc pR q with supp ϕ Ď Bp0, 3{4q and ϕ “ 1 on Bp0, 1{2q. We deﬁne α ϕpxqp´ ln|x|2q , 0 ă |x|2 ď 3{4, fpxq :“ #0, |x|2 ą 3{4 or x “ 0. 1,d d d Then f P W pR qzL8pR q; i.e., Theorem 3.17(b) is sharp if d ě 2. p d Proof. Arguing as in Example 3.3c), one sees that f P L pR q for all d p ă 8, f R L8pR q and that, for all j P t1, . . . , du, we have α α´1 xj Bjfpxq “ pBjϕpxqqp´ ln|x|2q ´ αϕpxqp´ ln|x|2q 2 |x|2 for 0 ă |x|2 ă 3{4 and Bjfpxq “ 0 otherwise. Using polar coordinates, we further estimate 1{d 1{d 3{4 d αd d´1 |Bjf| dx ď c kBjϕk8 pln rq r dr d ˆżR ˙ ˜ż0 ¸ 1{d 3{4 pln rqpα´1qd ` c kϕk rd´1 dr 8 rd ˜ż0 ¸ 1{d 3{4 dr ď c ` c ă 8 rpln rqp1´αqd ˜ż0 ¸ for some constants c ą 0, since p1 ´ αqd ą 1. l j For the proof of Theorem 3.17 we setx ˆ “ px1, . . . , xj´1, xj`1, . . . , xdq P d 1 d R ´ for all x P R , j P t1, . . . , du and d ě 2. We start with a lemma. d´1 d´1 d´1 Lemma 3.20. Let d ě 2 and f1, . . . , fd P L pR q X CpR q. Set 1 d d 1 d fpxq “ f1pxˆ q ¨ ... ¨ fdpxˆ q for x P R . We then have f P L pR q and

kfkL1pRdq ď kf1kLd´1pRd´1q ¨ ... ¨ kfdkLd´1pRd´1q. Proof.6 If d “ 2, then Fubini’s theorem shows that

|fpxq| dx “ |f1px2q||f2px1q| dx1 dx2 “ kf1k1kf2k1, 2 żR żR żR as asserted. Assume that the assertion holds for some d P N with d ě 2. d d d d Take f1, . . . , fd`1 P L pR q X CpR q. Write y “ px1, . . . , xdq P R and d`1 j j d x “ py, xd`1q P R . For a.e. xd`1 P R, the mapsy ˆ ÞÑ |fjpyˆ , xd`1q| are d 1 integrable on R ´ for each j P t1, . . . , du due to Fubini’s theorem. Fix such a xd`1 P R and write d ˜ j fpy, xd`1q :“ fjpxˆ q. j“1 ź Using H¨older’sinequality, we obtain ˜ |fpy, xd`1q| dy “ |fpy, xd`1q| |fd`1pyq| dy d d żR żR 1{d1 ˜ d1 ď kfd`1kLdp dq |fpy, xd`1q| dy . R d ˆżR ˙ 6Not shown in the lectures. 3.2. Density and embedding theorems 48

j j d1 d`1 We set gjpyˆ q “ |fjpyˆ , xd`1q| for j P t1, . . . , du and x P R . Since 1 d´1 d´1 d pd´1q “ d, the maps gj belong to L pR q and the induction hypothesis yields ˜ d1 1 d |fpy, xd`1q| dy “ g1pyˆ q ¨ ... ¨ gdpyˆ q dy ď kg1kd´1 ¨ ... ¨ kgdkd´1 d d żR żR d 1 d´1 j d “ |fjpyˆ , xd`1q| dy . d´1 j“1 R ź ˆż ˙ Integrating over xd`1 P R, we thus arrive at d 1 d´1 d´1 d j d |f| dx ď kfd`1kd |fjpxˆ q| dy dxd`1. d`1 d´1 R R j“1 R ż ż ź ˆż ˙ Applying the d-fold Hölderinequality to the xd`1-integral, we conclude that 1 d 1 d d ¨d j d |f|dx ď kfd`1kd |fjpxˆ q| dy dxd`1 d`1 d´1 R j“1 ˜ R R ¸ ż ź ż ˆż ˙ “ kf1kd ¨ ... ¨ kfd`1kd. Recall from Analysis 3 that for f P LppUq X LqpUq and r P rp, qs with 1 ď p ă q ď 8, we have θ 1´θ kfkr ď kfkp kfkq ď θkfkp ` p1 ´ θqkfkq, (3.14) 1 θ 1´θ where θ P r0, 1s is given by r “ p ` q and we also used Young’s inequality from Analysis 1+2. Proof of Theorem 3.17.7 We only prove the case k “ 1, the rest can 1,p d p d be done by induction, see e.g. §5.6.3 in [Ev]. Since W pR q ãÑ L pR q, in view of (3.14) for assertion (a) it suffices to show 1,p d p˚ d W pR q ãÑ L pR q. 1 d ˚ d d 1) Let f P Cc pR q. Let first p “ 1 ă d, whence p “ d´1 . For x P R and j P t1, . . . , du, we then obtain

d d d´1 1 d kfk d ď g1pxˆ q ¨ ... ¨ gdpxˆ q dx ď kgjkLd´1pRd´1q d´1 d d L pR q R j“1 ż ź 7Not shown in the lectures. 3.2. Density and embedding theorems 49

d 1 d´1 j “ |Bjfpxq| dxj dˆx , d´1 j“1 R R ź ˆż ż ˙ d 1 d kfk d ď kBjfk 1 d ď k|∇f|1k1 ď kfk1,1. (3.15) L d´1 p dq L pR q R j“1 ź ˚ pd d´1 ˚ d´1 2) Next, let p P p1, dq and p “ d´p . Set t˚ :“ d p “ d´p p ą 1. An 1 d ˚ elementary calculation shows that pt˚ ´ 1qp “ t˚ d´1 “ p . Set t´1 t´1 g “ f|f| “ fpffq 2 for t ą 1. We compute

t´1 t ´ 1 t´1 ´1 B g “B f|f| ` f pffq 2 pB fqf ` fpB fq j j 2 j j t´1 t´3 “Bjf|f| ` pt ´ 1qf|f| Rep`fBjfq, ˘ t t´1 |g| “ |f| , |Bjg| ď t|Bjf||f| . Applying (3.15) to g, we estimate

d´1 d´1 t d d d d t d´1 d´1 kfk td “ |f| dx “ |g| dx d´1 d d ˆżR ˙ ˆżR ˙ d d 1 1 1 d d t´1 ď kBjgk1 ď t d |Bjf| |f| dx d j“1 j“1 ˆżR ˙ ź ź 1 d 1 p1d p dp pt´1qp1 ď t |Bjf| dx |f| dx d d j“1 R R ź´ż ¯ ˆż ˙ d 1 t´1 d d t´1 ď t k|∇f|pkp kfkpt´1qp1 “ t k|∇f|pkp kfkpt´1qp1 , j“1 ź where we used H¨older’sinequality. For t “ t˚, we use the properties of t stated above and obtain d ´ 1 d ´ 1 kfk ˚ ď p k|∇f|pkp ď p kfk1,p. (3.16) p d ´ p d ´ p 1,p d This estimate can be extended to all f P W pR q by density (see Theo- rem 3.15). Hence, the inclusion map is the required embedding in (a). 1 d 1 d 3) Let p “ d, f P Cc pR q, and t ą 1. Then p “ d´1 , and step 2) yields

1 1 1 1´ t t kfk d ď t t kfk k|∇f| k ď c kfk d ` k|∇f| k (3.17) t t 1 d p d pt´1q p d d´1 p ´ q d´1 d´1 ´ ¯ using Young’s inequality from Analysis 1+2. For t “ d, this estimate gives 2 d d f P L d´1 pR q and

kfk d2 ď c kfk1,d. d´1 d Here and below the constants c ą 0 do not depend on f. For q P pd, d d´1 q, inequality (3.14) further yields

kfkq ď c pkfkd ` kfk d2 q ď c kfk1,d. d´1 3.2. Density and embedding theorems 50

Now, we can apply (3.17) with t “ d ` 1 and obtain

kfk d2`d ď c pkfk d2 ` k|∇f|pkdq ď c kfk1,d. d´1 d´1

q d d`1 As above, we see that f P L pR q for d ď q ď d d´1 . We can now iterate this procedure with tn “ d ` n and obtain

kfkq ď cpqqkfk1,p for all q ă 8. As above, assertion (b) follows by approximation. 1 d r r d 4) Let p ą d, f P Cc pR q, Qprq “ r´ 2 , 2 s for some r ą 0, and x0 P Qprq. ´d d Set Mprq “ r Qprq f dx. We further put β :“ 1 ´ p P p0, 1q. Using |x ´ x | ď r for x P Qprq, the transformation y “ tpx ´ x q and H¨older’s 0 8 ş 0 inequality, we compute

d for constants C, c ą 0 only depending on d and p, using also that p1 ´d ą ´1 due to p ą d. A translation then gives

´d β fpx0 ` zq ´ r fpyq dy ď Cr k|∇f|pkp z`Qprq ˇ ż ˇ d ˇ ˇ for all z P R .ˇ Taking x “ z, x0 “ 0, r “ 1,ˇ and using H¨older’sinequality, we thus obtain

|fpxq| ď fpxq ´ f dy ` f dy x`Qp1q x`Qp1q ˇ ż ˇ ˇ ż ˇ ď Cˇ k|∇f| k ` kfk ďˇ c kˇfk ˇ (3.18) ˇ p p p ˇ ˇ 1,p ˇ d d for all x P R , where c only depends on d and p. Given x, y P R , we ﬁnd a cube Q of side length |x ´ y|8 “: r such that x, y P Q and Q is parallel to 3.2. Density and embedding theorems 51 the axes. Hence,

|fpxq ´ fpyq| ď fpxq ´ r´d f dy ` r´d f dy ´ fpyq Q Q ˇ ż ˇ ˇ ż ˇ ˇ βˇ ˇ ˇβ ď 2ˇ Ck|∇f|pkp |x ´ y|8ˇ ďˇ 2Ck|∇f|pkp |x ´ yˇ|2 . 1,p d 1 d 1,p d If f P W pR q, then there are fn P Cc pR q converging to f in W pR q. d By (3.18), fn is a Cauchy sequence in C0pR q. Hence, f has a representative ˜ d ˜ f P C0pR q such that fn Ñ f uniformly as n Ñ 8. So the above estimates imply that ˜ ˜ ˜ |fpxq ´ fpyq| kfk8 ` sup β ď c kfk1,p. x y ‰ |x ´ y|2 ˜ The map f ÞÑ f is the required embedding. Remark 3.21. Theorem 3.17 remains valied on any open set U instead of d k,p k,p R if we replace W by W0 . In fact, we obtain the desired estimate for 8 f P Cc pUq if we apply Theorem 3.17 to the 0–extension of f. The result k,p for f P W0 pUq then follows by density. ♦ 8 d Corollary 3.22. Let U Ď R be open and bounded. We then have Poincare’s inequality

p p |∇u|p dx ě δ |u| dx (3.19) żU żU 1,p for some δ ą 0 and all u P W0 pUq and p P r1, 8q. Proof. For p P r1, dq, the estimate (3.19) follows from (3.16) since ˚ Lp pUq ãÑ LppUq. Let p P rd, 8q. The case p “ d “ 1 is easy to check. For 1,p the other cases, ﬁx r P pp, 8q and u P W0 pUq. Then (3.14), (3.15) and 1,p r W0 pUq ãÑ L pUq imply

}u}p ď cε}u}1 ` ε}u}r ď cε}|∇u|1}1 ` cε}u}1,p

ď cε}|∇u|p}p ` cε}u}p ` cε}|∇u|p}p for all ε ą 0 and some constants cε, c ą 0 independent of u, where c does not depend on ε ą 0. Choosing a small ε, we derive (3.19). Remark 3.23. We can show (parts of) Theorem 3.17(a) and (c) for the case p “ 2 using the Fourier transform. We first recall some basic facts. For 1 d f P L pR q, we define Fourier transform by ˆ 1 í ξ¨x d fpξq “ pFfqpξq :“ d e fpxq dx, ξ P R , d p2πq 2 żR d where the dot denotes the scalar product in R . Plancherel’s theorem says 1 d 2 d that F can be extended from L pR q X L pR q to an isometric bijection 2 d 2 d F : L pR q Ñ L pR q. Moreover, F is a bijection on the Schwartz space Sd with inverse given by pF ´1gqpyq “ pFgqpýq. See e.g. Proposition 5.10 and Theorem 5.11 in [Sc1]. For Sobolev spaces we have the crucial identity k,2 d 2 d k 2 d W pR q “ tu P L pR q | |ξ|2 u P L pR qu (3.20)

8This result was not part of the lectures. p 3.2. Density and embedding theorems 52

k k for k P N, where |ξ|2 stands for the function ξ ÞÑ |ξ|2 (and analogously for k,2 d similar expressions). Moreover, the norm of W pR q is equivalent to the 2 k 2 2 k 2 one given by }u}2 ` } |ξ|2 u}2 “}u}2 ` } |ξ|2 u}2 and α α k,2 d FpB uq “ iξ u for u P W pR q, |α| ď k. (3.21) p p p See e.g. Theorem 5.21 in [Sc1]. We further use the Hausdorff–Young inequality p q1 d }Ff}q ď c }f}q1 for q P r2, 8s, f P L pR q, (3.22) see e.g. Satz V.2.10 in [We]. 8 d We come to the announced proofs. Let f P Cc pR q and k P N. Then 1 d f P Sd Ď L pR q. 1) Let k ą d{2. Using the Fourier inversion formula, (3.22) for q “ 8, Hölder’sinequality,p (3.20) and polar coordinates, we compute ´1 ´d{2 k ´1 k }f}8 “}F Ff}8 ď p2πq }p1 ` |ξ|2q p1 ` |ξ|2qf}1 ´d{2 k ´1 k ď p2πq }p1 ` |ξ| q }2 }p1 ` |ξ| qf}2 2 2 p 1 8 rd´1 {2 ď c }f} dr ď c }pf} k,2 p1 ` rkq2 k,2 ˆż0 ˙ k,2 d for constants c ą 0, since 2k ą d. Next, let g P W pR q. By Theorem 3.15, 8 d d k,2 d there are fn P Cc pR q Ď C0pR q which tend to g in W pR q and thus in 2 d L pR q. The above estimate implies that pfnq is a Cauchy sequence for the d sup-norm. The limit of pfnq in C0pR q is then a representative of its limit g in 2 d L pR q. In particular, g also satisfies the above estimate and we have shown k,2 d d Theorem 3.17(c) for p “ 2, j “ 0 and β “ 0; i.e., W pR q ãÑ C0pR q. 2) Let k ă d{2 and 2 ď q ă 2˚ “ 2dpd ´ 2kq´1. The latter is equivalent to 1 1 k ą ´ . q 2 d To apply Hölder’sinequality, we define the number s P p2, 8s by 1 1 1 1 1 1 k “ ´ “ ´ . Hence, ă . s q1 2 2 q s d As in part 1), we estimate k ´1 k k ´1 k }f}q ď c }p1 ` |ξ|2q p1 ` |ξ|2qf}q1 ď c }p1 ` |ξ|2q }s }p1 ` |ξ|2qf}2 1 8 rd´1 {s ď c }f} dpr ď c }f} , p k,2 p1 ` rkqs k,2 ˆż0 ˙ where we have used that sk ą d by the above relations between the expo- k,2 d q d nents. One can again conclude that W pR q ãÑ L pR q, which is Theo- ˚ 9 rem 3.17(a) for p “ 2 and q ă 2 . ♦ Problem: How to extend the Sobolev embedding to W k,ppUq? This problem can be solved by an extension operator; i.e., a map E P k,p k,p d BpW pUq,W pR qq satisfying Eu “ u on U. We state a rather advanced d theorem that provides such an operator for open sets U Ď R with a compact Lipschitz boundary. This means that each x PBU has a neighborhood Vx in

9One can show the limiting case q “ 2˚ by a reﬁnement of the above argument. 3.2. Density and embedding theorems 53

d d´1 R such that there is a Lipschitz function fx : R Ñ R such that (possibly after a rotation) we have U X Vx “ ty P Vx yd ă fxpy1, ¨ ¨ ¨ , yd´1qu. Due to compactness, we can then cover BU by finitely many such Vx, cf. §4.9 in ˇ [AF]. Clearly, compact boundaries of type Ckˇ as defined in Section 2.3 are also compact Lipschitz boundaries. We further need the upper and lower half spaces d 1 1 d R˘ “ tpx , yq x P R , y ż 0u. Theorem 3.24 (Stein extension theorem)ˇ . Let U have a compact Lipschitz d ˇ boundary or let U “ R`. Let k P N and p P r1, 8s. Then there exists k,p k,p d an operator Ek,p in BpW pUq,W pR qq with Ek,pu “ u on U. These operators are restrictions of each other. We thus write EU for all of them. An even more general result is proved in Theorem VI.5 of [St]. A sketch of the proof is given in §5.25 of [AF]. Remark 3.25. We sketch a proof of the extension theorem for W 1,ppUq with p P r1, 8s if BU is compact and of class C1.10 One needs a Ck–boundary to prove the corresponding result on W k,ppUq by the method below. 1) For any open U, p P r1, 8s and k P N0, we define the 0-extension p p d operator E0 : L pUq Ñ L pR q by E0fpxq “ fpxq for x P U and E0fpxq “ 0 d k,p d k,p for x P R zU and the restriction operator RU : W pR q Ñ W pUq by RU f “ f|U . Clearly, E0 is a linear isometry and RU is a linear contraction. 1,p d 1 d d Let f P W pR´q X C pR´q. We write elements in R˘ as py, tq. Define d fpy, tq, py, tq P R´, E´fpy, rq “ t d #4fpy, ´ 2 q ´ 3fpy, ´tq, py, tq P R`. 1 d Note that E´f belongs to C pR q. One can check that kE´fkW 1,ppRdq ď c kfk 1,p d for a constant c ą 0. W pR´q 1,p d 1 d 1,p d 2) We show that W pR´qXC pR´q is dense in W pR´q, so that E´ can 1,p d 1,p d be extended to an extension operator on W pR´q. In fact, let f P W pR´q 8 1,p d and ε ą 0. Theorem 3.15 yields a function g P C pH´q X W pR´q with 1 d´1 kf ´ gk1,p ď ε. Setting gnpy, tq “ gpy, t ´ n q for t ď 0, y P R and n P N, 1 d 1,p d we define maps gn in C pR´q X W pR´q. Observe that α α B gn “ R d SnE0B g R´ p d 1 for 0 ď |α| ď 1, where Sn P BpL pR qq is given by Snhpy, tq “ hpy, t ´ n q p d p d for h P L pR q. One can see that Snh Ñ h in L pR q as in Example 4.12 of 1,p d [Sc2]. Hence, gn converges to g in W pR´q implying the claim. 1 d 3) Since BU P C , there are bounded open sets U0,U1,...,Um Ď R such that U Ď U0 Y ¨ ¨ ¨ Y Um, U 0 Ď U and BU Ď U1 Y ¨ ¨ ¨ Y Um, as well as a 1 ´1 1 diffeomorphism Ψj : Uj Ñ Vj such that Ψj and pΨj q are bounded and d d´1 ΨjpUj X Uq Ď R´ and ΨjpUj XBUq Ď R ˆ t0u, for each j P t1, . . . , mu. 8 d Theorem 5.6 in [RR] provides us with functions 0 ď ϕj P Cc pR q with m supp ϕj Ď Uj for all j “ 0, 1, . . . , m and j“0 ϕjpxq “ 1 for all x P U.

10Not shown in the lectures. ř 3.2. Density and embedding theorems 54

´1 d Let j P t1, . . . , mu. Set Sjgpyq “ gpΨj pyqq for y P R´XVj and Sjgpyq “ 0 d 1,p 1,p d ˆ for y P R´zVj, where g P W pUj X Uq. For h P W pR q, set Sjhpxq “ ˆ d 8 d hpΨjpxqq for x P Uj and Sjhpxq “ 0 for x P R zUj. Take anyϕ ˜j P Cc pR q 1,p with suppϕ ˜j Ď Uj andϕ ˜j “ 1 on supp ϕj (see Lemma 3.4). Let f P W pUq. We now deﬁne m ˆ E1f “ E0ϕ0f ` ϕ˜jSjE´Sj RpUj XUqpϕjfq . j“1 ÿ ` ˘ Using part 2) and Propositions 3.8 and 3.9, we see that E1 belongs to 1,p 1,p d BpW pUq,W pR qq. Let x P U. If x P Uk for some k P t1, . . . , mu, d we have Ψkpxq P R´. If x R Uj, thenϕ ˜jpxq “ 0. Thus ´1 E1fpxq “ ϕ0pxqfpxq ` ϕ˜jpxqpϕjfqpΨj pΨjpxqqq j“1,...,m xÿPUj m “ ϕjpxqfpxq “ fpxq. j“0 ÿ If x P U0zpU1 Y ¨ ¨ ¨ Y Umq, we also have E1fpxq “ ϕ0fpxq “ fpxq. ♦ We can now easily extend and improve the above results. Theorem 3.26 (Sobolev–Morrey on U). Let U have a compact Lipschitz d boundary or let U “ R`. Theorem 3.17 and Corollary 3.18 then remain d j d true if we replace R by U and C0pR q by j j α α C0pUq “ tf P C pUq B f has a continuous extension to BU, B fpxq Ñ 0 ˇ as |x|2 Ñ 8 if U is unbounded, for all 0 ď |α| ď ju. ˇ Proof. Consider e.g. Theorem 3.17(a). We have the embedding k,p d p˚ d J : W pR q ãÑ L pR q given by the inclusion. Thanks to Theorem 3.24, the map k,p p˚ RU JEU : W pUq Ñ L pUq is continuous and injective. The other assertions are proved similarly. d Theorem 3.27. Let U have a compact Lipschitz boundary or let U “ R`, k P N and p P r1, 8q. Then W k,ppUq X C8pUq is dense in W k,ppUq, where C8pUq contains all f P C8pUq such that f and all its derivatives can continuously be extended to BU. k,p k,p d Proof. Let f P W pUq. Then EU f belongs to W pR q by The- 8 d orem 3.24. Theorem 3.15 yields functions gn in Cc pR q that converge to k,p d k,p 8 EU f in W pR q. Hence, RU gn is contained in W pUqXC pUq and tends k,p to f “ RU EU f in W pUq as n Ñ 8. Proposition 3.28. Let U have a compact C1–boundary. Then W 1,8pUq is isomorphic to 1´ Cb pUq “ tf P CbpUq f is Lipschitzu, ˇ ˇ 3.2. Density and embedding theorems 55 and the norm of W 1,8pUq is equivalent to

kfk 1´ “ kfk8 ` rfsLip, Cb where rfsLip is the Lipschitz constant of f.

Proof. The extension operator E1 from Remark 3.25 also is continu- 1´ 1´ d ous from Cb pUq to Cb pR q. Owing to Proposition 3.13 we have an iso- 1,8 d 1´ d morphism J from W pR q to Cb pR q given by choosing the continuous representative. Then RU JE1 is the asserted isomorphism. We continue with one of the main compactness results in analysis. d Theorem 3.29 (Rellich-Kondrachov). Let U Ď R be bounded and open with a Lipschitz boundary BU, k P N and 1 ď p ă 8. Then the following assertions hold. ˚ dp (a) If kp ď d and 1 ď q ă p “ d´kp P pp, 8s, then the inclusion map J : W k,ppUq ãÑ LqpUq is compact. (For instance, let q “ p.) d (b) If k ´ p ą j P N0, then the embedding J : W k,ppUq ãÑ CjpUq is compact, where Jf is the continuous representative. Recall that a compact embedding J : Y ãÑ X means that any bounded sequence pynq in Y has a subsequence such that pJynj qj converges in X. Proof of Theorem 3.29.11 We prove the result only for k “ 1 (and thus j “ 0), see e.g. Theorem 6.3 of [AF] for the other cases. Part (b) follows from the Arzela-Ascoli theorem since Theorem 3.26 gives constants β, c ą 0 such that |fpxq ´ fpyq| ď c |x ´ y|β and |fpxq| ď c for all x, y P U 1,p and f P W pUq with kfk1,p ď 1, where p ą d. 1,p In the case p ă d, take fn P W pUq with kfnk1,p ď 1 for all n P N. Fix d an open bounded set V Ď R containing U. Lemma 3.4 yields a function 8 ϕ P Cc pV q which is equal to 1 on U. Let EU be given by Theorem 3.24. Set 1,p d gn “ ϕEU fn P W pR q. These functions have support V and kgnk1,p ď ˚ c }ϕ}1,8kEU k “: M for all n P N. Take q P r1, p q and θ P p0, 1s with 1 θ 1´θ q “ 1 ` p˚ . Inequality (3.14) and Theorem 3.17 yield that θ 1´θ kfn ´ fmk q ď kgn ´ gmk q ď kgn ´ gmk 1 kgn ´ gmk ˚ L pUq L pV q L pV q Lp pV q θ 1´θ 1´θ ď c kgn ´ gmkL1pV qpkgnk1,p ` kgmk1,p q 1´θ θ ď 2cM kgn ´ gmkL1pV q for all n, m P N. So it suﬃces to construct a subsequence of gn which 1 converges in L pV q. For x P V , n P N and ε ą 0, we compute

|gnpxq ´ Gεgnpxq| “ kεpx ´ yqpgnpxq ´ gnpyqq dy d ˇżR ˇ ˇ ˇ ˇ ˇ 11Not shown in the lectures.ˇ ˇ 3.2. Density and embedding theorems 56

´d 1 ď ε kp ε px ´ yqq |gnpxq ´ gnpyq| dy żBpx,εq

|Gεgnpxq| ď kkεk8 kgnkL1pV q, and |∇Gεgnpxq| ď k∇kεk8 kgnkL1pV q for all x P V and n P N and each ﬁxed ε ą 0. The Arzela-Ascoli theorem now implies that the set Fε :“ tGεgn n P Nu is relatively compact in CpV q for each ε ą 0, and thus in L1pV q since CpV q ãÑ L1pV q. Let δ ą 0 be given δ ˇ and ﬁx ε “ 2C . Then there are indecesˇ n1, . . . , nl P N such that l δ l Fε Ď BL1 V pGεgn , q “: Bj. j“1 p q j 2 j“1 Hence, given n P N, thereď is an index nj such thatďGεgn P Bj. The estimates (3.23) and (3.6) then yield δ kgn ´ Gεgnj kL1pV q ď kgn ´ GεgnkL1pV q ` kGεpgn ´ gnj qkL1pV q ď Cε ` {2 “ δ.

We have shown that, for each δ ą 0, the set G :“ tgn n P Nu is covered by ﬁnitely many open balls B of radius δ{2; i.e., G is totally bounded in L1pV q. j ˇ Thus G contains a subsequence converging in L1pV q (seeˇ e.g. Corollary 1.39 ˚ in [Sc2]). In the case p “ d one simply replaces p by any r P pq, 8q. Remark 3.30. a) Theorem 3.29 is wrong for unbounded domains, in general. In fact, let k P N, p P r1, 8q and deﬁne the functions fn “ fp¨ ´ nq in k,p W pRq for any function 0 ‰ f P C8pRq with supp f Ď p´1{2, 1{2q. Then kfnkk,p and kfn ´ fmkq ą 0 do not depend on n ‰ m in N so that pfnq is k,p q bounded in W pRq and has no subsequence which converges in L with 1 ď q ă p˚. ˚ b) The embedding W 1,ppUq ãÑ Lp pUq is never compact, see Example 6.12 in [AF]. ♦ 3.2. Density and embedding theorems 57

We conclude this section with an interpolation estimate for first derivatives. Again there are plenty of variants. 2,p Proposition 3.31. Let 1 ď p ă 8. Let either f P W0 pUq or U have a compact Lipschitz boundary and f P W 2,ppUq. Then there are constants C, ε0 ą 0 such that d 1{p d 1{p C kB fkp ď ε kB fkp ` kfk , j p ij p ε p j“1 i,j“1 ˆ ÿ ˙ ˆ ÿ ˙ 2,p 2,p for all ε ą 0 if f P W0 pUq and for all 0 ă ε ď ε0 if f P W pUq. 12 2 d Proof. 1) Let f P Cc pUq and extend it to R by 0. Take j “ 1. Write d 1 d d 1 x “ pt, yq P R ˆ R ´ for x P R . Fix y P R ´ and set gptq “ fpt, yq for ε t P R. Let ε ą 0 and a, b P R with b ´ a “ ε. Take any r P pa, a ` 3 q and ε t P pb ´ 3 , bq. There there is a number s “ spr, tq P pa, bq such that gptq ´ gprq 3 |g1psq| “ ď p|gptq| ` |gprq|q. t ´ r ε ˇ ˇ For every s P pa, bq we thusˇ obtain ˇ ˇ ˇ s 3 b |g1psq| “ g1psq ` g2pτq dτ ď p|gprq| ` |gptq|q ` |g2pτq| dτ. s ε a ˇ ż ˇ ż Integrating firstˇ over r and then overˇ t, we conclude ˇ ˇ a` ε b ε 3 3 ε |g1psq| ď |gprq| dr ` |gptq| ` |g2pτq| dτ, 3 ε 3 ża ża 2 a` ε b 2 b ε 3 ε |g1psq| ď |gprq| dr ` |gptq| dt ` |g2pτq| dτ, 9 a b´ ε 9 a ż ż 3 ż b b 1 9 2 |g psq| ď 2 |gpτq| dτ ` |g pτq| dτ ε a a ż b ż b 1 9 1{p 1 1{p ď ε p1 |gpτq|p dτ ` ε p1 |g2pτq|p dτ ε2 ża ża ´ p ¯b ´ b ¯ 1{p p´1 p´1 9 ď ε p 2 p |gpτq|p dτ ` |g2pτq| dτ , ε2 ˆˆ ˙ ża ża ˙ 2 where we used Hölder’sinequality first for the integrals and then in R . We take now the p-th power and then integrate over s arriving at b 9p b b |g1psq|p ds ď εεp´12p´1 |gpτq|p dτ ` |g2pτq|p dτ . ε2p ża ˆ ża ża ˙ Now choose a “ ak “ kε and b “ bk “ pk ` 1qε for k P Z. Summing the d 1 integrals on rkε, pk ` 1qεq for k P Z and then integrating over y P R ´ , it follows that p 1 p p p´1 9 p 2 p |g pτq| dτ ď ε 2 2p |gpτq| dτ ` |g pτq| dτ , R ε R R ż ˆ ż p ż ˙ p p p 36 p |B1f| dx ď p2εq |B11f| dx ` |f| dx. (3.24) p2εqp żU żU żU 12Not shown in the lectures. 3.3. Traces 58

2,p 2) By approximation, (3.24) can be established for all f P W0 pUq. The same result holds for Bjf and Bjjf with j P t2, . . . , du. We now replace 2ε by ε, sum over j and take the p-th root to arrive at

d 1{p d 36p 1{p kB fkp ď εp kB fkp ` kfkp j p jj p εp p j“1 j“1 ˆ ÿ ˙ ˆ ÿ ˙ d 1{p 36 ď ε kB fkp ` kfk , (3.25) jj p ε p j“1 ˆ ÿ ˙ 2,p for all f P W0 pUq, as asserted. 3) If u P W 2,ppUq and U has a Lipschitz boundary, we use the extension 2,p 2,p d operator EU P BpW pUq,W pR qq from Theorem 3.24 to deduce from d (3.25) with U “ R that

d 1{p d 1{p p p kBjfk p ď kBjEU fk L pUq LppRdq j“1 j“1 ˆ ÿ ˙ ˆ ÿ ˙ d 1{p p 36 ď ε kBjjEU fk ` kEU fk p d LppRdq ε L pR q j“1 ˆ ÿ ˙ 36 ď ε kE fk 2,p d ` kE fk p U W pR q ε U L pUq c ď cε kfk 2,p ` kfk p W pUq ε L pUq d 1{p d 1{p c ď c ε kB fkp ` c ε kB fkp ` kfk 0 ij p 1 j p ε p i,j“1 j“1 ˆ ÿ ˙ ˆ ÿ ˙ where we assume that ε P p0, 1s and the constants c, c0, c1 do not depend on ε or f. Choosing ε “ mint 1 , 1u we arrive at 1 2c1 1 d 1{p d 1{p c kB fkp ď c ε kB fkp ` kfk 2 j p 0 ij p ε p j“1 i,j“1 ˆ ÿ ˙ ˆ ÿ ˙ if 0 ă ε ď ε1. This inequality implies the assertion for U with a Lipschitz ε c boundary, after replacing ε by {p2c0q and ε1 by ε0 “ mint 0{c1, 2c0u. Remark 3.32. Arguing as in part 1) of the above proof, one derives 2,p Proposition 3.31 on W pJq for every open interval J Ď R with uniform constants. ♦

3.3. Traces For p ą d and Lipschitz boundaries, we have W 1,ppUq ãÑ CpUq by Sobolev’s Theorem 3.26 so that the trace map f ÞÑ f|BU is well de- ﬁned from W 1,ppUq to CpBUq, say. Theorem 3.11 further implies that W 1,1pa, bq ãÑ Cpra, bsq for d “ 1. In other cases it is not clear at all how to give a meaning to the mapping f ÞÑ f|pBUq on W 1,ppUq since for reasonable open sets BU is a null set. The next result solves this problem. Moreover, 1,p 1,p it says that W0 pUq is the space of functions in W pUq with trace 0. 3.3. Traces 59

d Theorem 3.33 (Trace theorem). Let p P r1, 8q and U Ď R be open with 1 a compact boundary BU of class C . Then the trace map f ÞÑ f|BU from W 1,ppUqXCpUq to LppBU, σq has a bounded linear extension tr : W 1,ppUq Ñ p 1,p L pBU, σq whose kernel is W0 pUq, where σ is the surface measure on BU. Proof.13 1) Let u P C1pUq. By the definition of the surface integral, see Section 2.3 with a slightly different notation, there are finitely many 1 diffeomorphisms Ψj : Uj Ñ Vj and ϕj P Cc pUjq with 0 ď ϕj ď 1 such that p }u}LppBU,σq is dominated by r m r r ´1 ´1 p 1 c ϕj ˝ Ψj |u ˝ Ψj | dy j“1 Vj ÿ ż d where Uj and Vj are open subsets of R , the sets Uj cover BU, the functions 1 ϕj form a partition of unity subordinated to Uj, Vj :“ tpy , ydq P Vj yd “ 0u, V : r y1, y r V y 0 ,Ψ U U V r, and Ψ U U V . j` “ tp dq P j d ą u jp j XB q “ j jp j X ˇq “ j` We set v u Ψ´1 and ψ ϕ Ψ´1 C1 Vr and drop the indicesr ˇ j below. “ ˝ j ˇ “ j ˝ j P c p jq By means of Fubini’sr ˇ theorem and ther fundamental theoremr of calculus, we compute r

1 p 1 p ψ |vpy q| dy “ ´ Bdpψ |v| q dy żV żV` p p´2 “ ´ rpBdψq |v| ` pψ|v| RepvBdvqs dy żV` p p´1 p p´1 ď c r|v| ` |v| |Bdv|s dy ď c }v}p ` c }v}p }Bdv}p żV` p p p p ď c p}v} ` }Bdv} q ď c }v} 1,p ď c }u} 1,p . p p W pV`q W pUq Here we also used H¨older’sand Young’s inequality, Proposition 3.9 and the 1 p transformation rule. As a result, the map tr : pC pUq, } ¨ }1,pq Ñ L pBU, σq, tr u “ u|BU , is continuous. Theorem 3.27 allows to extend tr to an operator in LpW 1,ppUq,LppBU, σqq. If we start with a function u P W 1,ppUq X CpUq, 1 then we can construct approximations un P C pUq which converge to u in W 1,ppUq and in CpUq, see the proof of Theorem 5.3.3 in [Ev]. Hence, p tr un “ un|BU tends to u|BU uniformly on BU and to tr u in L pBU, σq, so that tr u “ u|BU . 1,p 2a) We next observe that the inclusion W0 pUq Ď Nptrq is a consequence 8 of the continuity of tr since tr vanishes on Cc pUq and this space is dense 1,p in W0 pUq by deﬁnition. To prove the converse, we start with the model 1,p case that v P W pV`q has a compact support in V` and tr v “ 0. Our 1 1,p density results yield vn P C pV`q converging to v in W pV`q, and hence p tr vn “ vn|V Ñ 0 in L pV q, as n Ñ 8. Observe that yd 1 1 1 |vnpy , ydq| ď |vnpy , 0q| ` |Bdvnpy , sq| ds, ż0 yd p 1 p 1 p 1 |vnpy , ydq| ď 2 |vnpy , 0q| ` 2 |Bdvnpy , sq| ds ˆ ż0 ˙ 13Not shown in the lectures. 3.3. Traces 60

1 1 for y P V and yd ě 0. Integrating over y und employing H¨older’sinequality, we obtain

yd 1 p 1 1 p 1 p´1 1 p 1 |vnpy , ydq| dy ď 2 |vnpy , 0q| dy ` 2yd |Bdvnpy , sq| ds dy . żV żV żV ż0 We can now let n Ñ 8 and arrive at yd 1 p 1 p´1 1 p 1 |vpy , ydq| dy ď 2yd |Bdvpy , sq| ds dy . (3.26) żV żV ż0 We next use a cutoﬀ argument to obtain a support in the interior of V`. 8 Choose a function χ P C pR`q such that χ “ 0 on r0, 1s and χ “ 1 on r2, 8q. Set χnpsq “ χpnsq for s ě 0 and n P N, and deﬁne wn “ χnv on V`. p Note that wn Ñ v in L pV`q as n Ñ 8, Bjwn “ χnBjv for j P t1, . . . , d ´ 1u 1 and Bdwn “ χnBdv ` nχ pn¨qv. Estimate (3.26) further implies

p |∇wn ´ ∇v|p dy żV` 2{n 2{n p p 1 p 1 p 1 ď c |1 ´ χn| |∇v|p dy ds ` cn |vpy , sq| dy ds ż0 żV ż0 żV 2{n 2{n s p 1 p p´1 1 p 1 ď c |∇v|p dy ds ` cn s |Bdvpy , τq| dy dτ ds ż0 żV ż0 ż0 żV 2{n 2{n p 1 1 p 1 ď c |∇v|p dy ds ` c |Bdvpy , τq| dy dτ ż0 żV ż0 żV 1,p for some constants c ą 0. Because of v P W pV`q, the above integrals tend 1,p to 0 as n Ñ 8, and so wn Ñ v in W pV`q as n Ñ 8. Since wn “ 0 for 8 yd P p0, 1{ns, we can mollify wn to obtain a function wn P Cc pV`q such that 1,p }wn ´ wn}1,p ď 1{n. This means that wn Ñ v in W pV`q as n Ñ 8. 1,p 2b) We come back to u P W pUq and consider thep sets Uj and Vj and the ´1 functionsp Ψj and ϕj from step 1). Let vpj “ pϕjuq ˝ Ψj . First, observe that ´1 r ´1 r the trace of vj to the set Vj is given by ptr ϕjq ˝ Ψj ptr uq ˝ Ψj if u P CpUq, in addition. By continuity one can extend this identity to all u P W 1,ppUq. j 1 Let tr u “ 0. Then we can apply part 2a) to vj and obtain wn P Cc pVj`q 1,p converging to vj in W pVj`q. The function m p j un “ wn ˝ pΨj|U X Ujq j“1 ÿ 1 p p r 1,p thus belongs to Cc pUq and converges to u in W pUq as n Ñ 8. Since 8 un has compact support, we can mollify un to a function un P Cc pUq with 1,p }un ´ un}1,p ď 1{n. This means that un Ñ u in W pUq as n Ñ 8, and 1,p hencep u P W0 pUq. p p Remark 3.34. As in Remark 3.25 we give a much shorter proof in the case p “ 2 of an even stronger continuity property of tr, but here only in d d´1 the half space case R` “ tpx, yq x P R , y ą 0u with d ě 2. 1 d 1 d d 1) Let u P Cc pR`q “ C pR`qXˇCcpR`q and let Fx be the Fourier transform d´1 ˇ 1 with respect to x P R . The map y ÞÑ pFxuqpξ, yq then belongs to Cc pR`q 3.3. Traces 61

d 1 for each ξ P R ´ . Since we deal with regular functions, we can calculate

2 |ξ|2 |pFxuqpξ, 0q| dξ Rd´1 ż 8 2 “ ´ |ξ|2 By |pFxuqpξ, yq| dy dξ Rd´1 0 ż 8 ż “ ´2 Re |ξ|2 Fxu BypFxuq dξ dy 0 Rd´1 ż ż 1 1 8 2 8 2 2 2 ď 2 |ξFxu|2 dξ dy |BypFxuq| dξ dy 0 Rd´1 0 Rd´1 „ż ż  „ż1 ż  1 8 2 8 2 2 2 “ 2 |Fxp∇xuq|2 dξ dy |FxpByuq| dξ dy 0 Rd´1 0 Rd´1 „ż ż 1  „ż ż 1  8 2 8 2 2 2 “ 2 |∇xu|2 dx dy |Byu| dx dy d´1 d´1 „ż0 żR  „ż0 żR  2 2 2 ď |∇xu|2 dz ` |Byu|2 dz ď }u} 1,2 d , d d W pR`q żR` żR` also using H¨older’sinequality, (3.21) and Plancherel’s theorem. For s ą 0 we deﬁne the Bessel potential space

s n 2 n s 2 n H pR q “ tv P L pR q |ξ|2 v P L pR qu endowed with the (Hilbertian) norm givenˇ by ˇ p

2 2s 2 }v}Hs “ p1 ` |ξ|2 q |vpξq| dξ. n żR k n k,2 n (Recall from (3.20) that H pR q “ W pRpq for k P N.) We have thus 1 d 1 d´1 shown that the trace map is continuous from pCc pR`q, } ¨ }1,2q to H 2 pR q. 1,2 d 1 d 1,2 d 2) Theorem 3.27 implies that W pR`q X C pR`q is dense in W pR`q. Using cut-off functions as in the first part of Theorem 3.15, one sees the 1 d 1,2 d 1 d density of Cc pR`q in W pR`q X C pR`q for } ¨ }1,2. Hence, the trace map 1,2 d 1 d´1 is continuous from W pR`q to H 2 pR q. 1,2 d 1 d´1 3) It is possible to show that tr : W pR`q Ñ H 2 pR q is surjective. One can show analogous boundedness and surjectivity results on domains U with a compact C1–boundary, and also for p P p1, 8q (employing somewhat different spaces of functions on BU). See e.g. Theorem 7.39 in [AF]. ♦ We can now prove the divergence theorem in Sobolev spaces, which is a crucial tool in many applications to partial differential operators

d d Theorem 3.35 (Gauß and Green formulas). Let U “ R or U Ď R have a compact boundary BU of class C1 with the outer unit normal ν, p P r1, 8s, 1 F P W 1,ppUqd, and ϕ P W 1,p pUq. We obtain

divpF qϕ dx “ ´ F ¨ ∇ϕ dx ` ϕ ν ¨ F dσ. (3.27) żU żU żBU 3.3. Traces 62

1 If u P W 2,ppUq and v P W 2,p pUq, we obtain

p∆uqv dx “ ´ ∇u ¨ ∇v dx ` pBνuqv dσ żU żU żBU (3.28) “ u∆v dx ` ppBνuqv ´ uBνvq dσ. żU żBU d If U “ R , these formulas hold without the boundary integrals. (We omit tr d in the boundary integrals and set Bνu “ j“1 νj tr Bju.) Proof. We ﬁrst observe that Green’sř formulas (3.28) are a straightforward consequence of (3.27) with F “ ∇u. 1) Let U be bounded. Gauß’ formula (3.27) is shown in analysis courses for F P C1pUqd and ϕ P C1pUq. 1 a) Let p P p1, 8q. For F P W 1,ppUqd and ϕ P W 1,p pUq, Theorem 3.27 1 d 1 provides functions Fn P C pUq and ϕn P C pUq that converge to F and ϕ 1 in W 1,ppUqd and W 1,p pUq as n Ñ 8, respectively, since p, p1 ă 8. The- p d orem 3.33 then yields that Fn|BU Ñ tr F in L pU, σq and ϕn|BU Ñ tr ϕ p1 1,q q in L pU, σq as n Ñ 8. Since Bj : W pUq Ñ L pUq is continuous for q P tp, p1u, we further obtain that the terms with derivatives converge in Lp, respectively in Lp1 . Formula (3.27) now follows by approximation. 1 b) Next, let p “ 1 and thus p “ 8. As above, div Fn and Fn converge 1 1 d to div F and F in L pUq and L pUq , respectively, as well as Fn|BU Ñ tr F in L1pBU, σqd. By Proposition 3.28, the function ϕ belongs to CpUq and d 8 d one can thus extend it a map ϕ P CcpR q. Set ϕn “ G 1 ϕ P Cc pR q. n Properties (3.7) and (3.6) and Lemma 3.4 imply that ϕn Ñ ϕ in CpUq, }∇ϕn}8 ď }∇ϕ}8 and ∇ϕn Ñ ∇ϕ pointwise a.e., as n Ñ 8, where we possibly pass to a subsequence also denoted by pϕnqn. We can now take the limit n Ñ 8 in the ﬁrst and the third integral of equation (3.27) for Fn and ϕn. For the second integral we also use the estimate

Fn ¨ ∇ϕn dx ´ F ¨ ∇ϕ dx ď }Fn ´ F }1 }∇ϕn}8 ` |F | |∇ϕn ´ ∇ϕ| dx U U U ˇż ż ˇ ż andˇ Lebesgue’s theorem for theˇ integral on the right-hand side. The case ˇ ˇ p “ 8 is treated analogously. 2) Let U be unbounded. There is a radius r ą 0 such that BU Ď Bp0, rq. For k P N with k ą r we deﬁne cut-oﬀ functions χk as the functions ϕn in part 1) of the proof of Theorem 3.15. For Fk “ χkF and ϕk “ χkϕ, the formula (3.27) follows from step 1) replacing U by U X Bp0, 2kq. Observe that Fk “ F and ϕk “ ϕ on BU. We compute

2 divpFkqϕk dx “ χk divpF qϕ dx ` χkϕF ¨ ∇χk dx, żU żU żU 2 Fk ¨ ∇ϕk dx “ χkF ¨ ∇ϕ dx ` χkϕF ¨ ∇χk dx. żU żU żU The products ϕ divpF q, ϕF and F ¨ ∇ϕ are integrable, 0 ď χk ď 1, χk Ñ 1 1 pointwise and }χk}8 ď c{k. Letting k Ñ 8, the theorem of dominated convergence implies (3.27) for F and ϕ. Note that this proof also works for d U “ R , where BU “H. 3.4. Diﬀerential operators in Lp spaces 63

3.4. Diﬀerential operators in Lp spaces We now use some of above results to study diﬀerential operators in Lp spaces and discuss their closedness, spectra and the compactness of their resolvents. d p d Example 3.36. Let U Ď R be open, 1 ď p ă 8, X “ L pUq, and α P N0. Then the operator A “Bα with DpAq “ tu P Dα,ppUq u, Bαu P Xu is closed. p k Let J Ď R be an open interval and X “ L pJq. Then also Ak “B with k,p ˇ DpAkq “ W pJq for k “ 1 or k “ 2 are closed. ˇ α Proof. Let un P DpAq with un Ñ u and Aun “B un Ñ v in X as n Ñ 8. By Lemma 3.5(b), the function u P X belongs to Dα,ppUq and Bαu “ v P X. Hence, u P DpAq and Au “ v; i.e., A is closed. 2 In particular, for J “ U the operators A1 and B on D2 “ tu P 2,p 2 p D pJq u, B u P L pJqu are closed. Since DpA2q Ď D2, it remains to show the converse inclusion. Let u P D2. Lemma 3.5(a) yields functions 8ˇ 2 2 p un P C ˇpJq such that un Ñ u and un ÑB u in LlocpJq, as n Ñ 8. From Proposition 3.31 and Remark 3.32 we obtain a constant c ą 0 such that for all compact intervals J 1 Ď J and g P C2pJq we have 1 2 }g }LppJ1q ď c p}g}LppJ1q ` }g }LppJ1qq. 1 p If we insert here un ´ um “ g, we see that un converges in LlocpJq to a function v and hence u belongs to D1,ppJq and Bu “ v by Lemma 3.5(b). We can now conclude that 2 }Bu}LppJ1q “ lim }Bun}LppJ1q ď c lim p}un}LppJ1q ` }B un}LppJ1qq nÑ8 nÑ8 2 2 “ cp}u}LppJ1q ` }B u}LppJ1qq ď cp}u}LppJq ` }B u}LppJqq.

As a result, Bu P X and u P DpA2q. l We note that in the one-dimensional case Bk is closed on W k,ppJq for all k P N (by similar proofs based on higher order versions of Proposition 3.31). This property relies on the fact that the domain W k,ppJq does not contain 1,p 2 derivatives in other directions. In contrast, B1 on W pp0, 1q q is not closed p 2 in L pp0, 1q q which can be shown by functions unpx, yq “ ϕnpyq where 1 p 1,p ϕn P C pr0, 1sq converges in L p0, 1q to a function ϕ R W p0, 1q. Next, in various settings we study the ﬁrst derivative in more detail. p Example 3.37. Let 1 ď p ă 8, X “ L pRq and A “B with DpAq “ 1,p W pRq. Then σpAq “ iR, σppAq “ H and 8 eλpt´sqfpsq ds, Re λ ą 0, pRpλ, Aqfqptq “ t ´ t eλpt´sqfpsq ds, Re λ ă 0, #ş ´8 for t P R and f P X, cf. Exampleş 1.21 for X “ C0pRq. 1 Proof. 1) Let Re λ ą 0 and ϕλ “ eRe λ R´ . By Rλfptq we denote the integral in the assertion. Since |Rλpfptqq| ď pϕλ ˚ |f|qptq for all t P R, Young’s inequality (see e.g. Theorem 2.14 in [Sc2]) yields that 1 kR fk ď kϕ k kfk “ kfk . λ p λ 1 p Re λ p 8 p d Let fn P Cc pRq converge to f in L pRq. We then compute dt Rλfn “ p λRλfn ´ fn, which tends to λRλf ´ f in L pRq by the above estimate. 3.4. Diﬀerential operators in Lp spaces 64

Since A is closed by Example 3.36, the function Rλf belongs to DpAq and ARλf “ λRλf ´ f; i.e., λI ´ A is surjective. 2) Let Au “ µu for some u P DpAq and µ P C. Theorem 3.11 says that u is continuous so that Remark 3.12c) implies the continuous differentiability 1 of u and so u “ µu. Consequently, u is equal to a multiple of eµ. Since eµ R X, we obtain u “ 0 and hence µI ´ A is injective. We have shown that σppAq “ H and λ P ρpAq with Rpλ, Aq “ Rλ if Re λ ą 0. In the same way, one sees that λ P ρpAq if Re λ ă 0, with the asserted resolvent. 1 ´1{p 3) Let λ P iR. Take ϕn P Cc pRq with ϕn “ n on rń, ns, ϕnptq “ 0 ´1{p for |t| ě n ` 1 and kϕnk8 ď 2n for every n P N. We then derive 1,p un “ ϕneλ P W pRq, Aun “ λun ` ϕneλ, n 1{p p ´1{p 1{p 1{p kunkp ě |ϕnptqeλptq| dt “ n p2nq “ 2 , ˆżń ˙ 1{p 1 1 p ´1{p 1{p kλun ´ Aunkp “ kϕneλkp “ |ϕnptqeλptq| dt ď 2n 2 , tnď|t|ďn`1u ´ż ¯ for every n P N. Consequently, λ P σappAq and so σpAq “ iR. l Example 3.38. Let 1 ď p ă 8. a) Let X “ Lpp0, 1q and A “B with DpAq “ W 1,pp0, 1q. Then σpAq “ σppAq “ C since eλ P DpAq and Aeλ “ λeλ for all λ P C. b) Let X “ Lpp0, 1q and A “B with DpAq “ tu P W 1,pp0, 1q up0q “ 0u (using the continuous representative of u P W 1,pp0, 1q from Theorem 3.11). t λpt´sq ˇ Then A is closed, σpAq “ H, Rpλ, Aqfptq “ ´ 0 e fpsq ds forˇ t P p0, 1q, f P X and λ P C, and A has compact resolvent. Proof. As in Example 3.37 one sees thatş the above integral defines a bounded inverse of λI ´ A for all λ P C. In particular, A is closed by Re- mark 1.11. (Alternatively, take un in DpAq such that un Ñ u and Aun Ñ v in X. Example 3.36 yields that u P W 1,pp0, 1q and Bu “ v. Using Theo- rem 3.11, we further infer that 0 “ unp0q Ñ up0q. Hence, u P DpAq and Au “Bu “ v.) Finally, Remark 2.14 and Theorem 3.29 imply that A has a compact resolvent. l c) Let X “ Lpp0, 8q and A “B with DpAq “ W 1,pp0, 8q. Then σpAq “ tλ P C Re λ ď 0u, σppAq “ tλ P C Re λ ă 0u and Rpλ, Aqfptq “ 8 eλpt´sqfpsq ds for t ą 0, f P X and Re λ ą 0. t ˇ ˇ Proof. Theˇ operator is closed by Exampleˇ 3.36. As in Example 3.37, ş one computes the resolvent for Re λ ą 0 and checks that iR does not contain eigenvalues. If Re λ ă 0, then eλ is an eigenfunction. Since σpAq is closed, it is equal to tλ P C Re λ ă 0u. l p 1,p d) Let X “ L p0ˇ, 8q and A “ ´B with DpAq “ W0 p0, 8q. Then A ˇ is closed, σpAq “ tλ P C Re λ ď 0u, σppAq “ H, and Rpλ, Aqfptq “ t e´λpt´sqf s ds for t 0, f X and Re λ 0. 0 p q ą ˇ P ą Proof. If Re λ ą 0, the formulaˇ for the resolvent is verified as in Exam- şple 3.37, so that A is closed. The point spectrum is empty since the only possible eigenfunctions eλ do not fulfill the boundary condition in DpAq. 1 Let Re λ ă 0 and f “ p0,1q. If u P DpAq satisfies λu ` Au “ f, then up0q “ 0 and Bu “ ´λu´f is continuous except at t “ 1. By Remark 3.12c), 3.4. Differential operators in Lp spaces 65 u is piecewise C1 and hence 1 uptq “ e´λt eλs ds, t ě 1. ż0 So u cannot belong to X and thus f is not in the range of λI ´ A; i.e., λ P σpAq. The result then follows from the closedness of the spectrum. l e) Let X “ Lpp0, 1q and A “B with DpAq “ tu P W 1,pp0, 1q up0q “ up1qu. Then A is closed, σpAq “ σppAq “ 2πiZ and A has a compact resolvent. ˇ (These facts can be proved as in Example 2.17 for X “ Cpr0,ˇ1sq, using now Theorem 3.29 for the compactness.) ♦ We now turn our attention to the second derivative and the Laplacian. p 2 Example 3.39. Let X “ L pRq, 1 ď p ă 8, and A “B with DpAq “ 2,p W pRq. Then σpAq “ R´. 1,p Proof. 1) Set A1 “B and DpA1q “ W pRq. Let µ P CzR´. There 2 exists a number λ P C with Re λ ą 0 such that µ “ λ . Observe that

pµI ´ Aqu “ pλI ´ A1qpλI ` A1qu (3.29) for all u P DpAq. Example 3.37 says that λI ˘ A1 are invertible. Hence, 1,p µI ´ A is injective. Next, for v P W pRq the function ´1 ´1 ´1 BpλI ` A1q v “ A1pλI ` A1q v “ ´λpλI ` A1q v ` v 1,p ´1 1,p belongs to W pRq. This means that pλI `A1q maps W pRq into DpAq. ´1 ´1 Given f P X, the map u :“ pλI ` A1q pλI ´ A1q f thus is an element of DpAq and µu ´ Au “ f in view of the factorization (3.29) . We have shown ´1 ´1 that µ P ρpAq and Rpµ, Aq “ pλI ` A1q pλI ´ A1q . 2 2) For µ ď 0, we have µ “ λ for some λ P iR and (3.29) is still true. The operator λI´A1 is not surjective since its range is not closed by Example 3.37 and Proposition 1.19. Equation (3.29) thus implies that µI ´ A is not surjective. As a result, σpAq “ R´. l Example 3.40. Let X “ Lpp0, 1q, 1 ď p ă 8, and A “B2 with DpAq “ 2,p 1,p 2 2 W p0, 1q X W0 p0, 1q. Then A is closed, σpAq “ σppAq “ t´π k k P Nu, and A has a compact resolvent. These facts can be proved as in Example 2.17 ˇ for X “ Cpr0, 1sq, using now Theorem 3.29 for the compactness. ˇ ♦ 2 d Example 3.41. Let X “ L pR q and A “ ∆ “B11 ` ... ` Bdd with 2,2 d DpAq “ W pR q. Then A is closed and σpAq “ R´. Proof. We use the Fourier transform F which is unitary on X, see e.g. Theorem 5.11 in [Sc1]. By (3.20) and (3.21), we have ´1 2 2 ∆u “ F p´|ξ|2uq for u P DpAq “ tu P X | |ξ|2 u P Xu. 2 ´1 d 1) Let λ P CzR´. Set mλpξq “ pλ ` |ξ|2q for ξ P R . Observe that }mλ}8 ď cλ where cλ “ 1p{| Im λ| if Re λ ď 0 and cλ “ 1{|λ| pif Re λ ą 0. Let ´1 f P X. Then mλf belongs to X so that Rλf :“ F pmλfq P X and

}Rλf}2 “}mλf}2 ď cλ }f}2 “ cλ }f}2 . p p 2 d 2 Moreover, since |ξ|2mλpξq is bounded for ξ P R , the function |ξ|2FRλf “ 2 p p |ξ|2mλf is an element of X; i.e., Rλf P DpAq. Similarly we see that ´1 2 ´1 pλI ´ ∆qRλf “ F pλ ` |ξ| qFF mλf “ f, p 2 p 3.4. Diﬀerential operators in Lp spaces 66

´1 ´1 2 RλpλI ´ ∆qu “ F mλFF pλ ` |ξ|2qu “ u, u P DpAq.

Thus, λ P ρpAq and Rpλ, Aq “ Rλ. In particular, A is closed and σpAq Ď R´. 2 ´1 d 2) Let λ ă 0. Define mλpξq “ pλ ` |ξ|2q forpξ P R with |ξ| ‰ ´λ and 2 d mλpξq “ 0 if |ξ| “ ´λ. Then mλ is unbounded. Suppose that mg P L pR q 2 d for all g P L pR q. Then the (closed) multiplication operator M : g ÞÑ mg in 2 d 2 d L pR q was defined on L pR q and thus bounded by the closed graph theo- d rem. Set Bn “ tξ P R |mλpξq| ě nu for n P N. Observe that the Lebesgue 1 measure ` of B is contained in 0, . Set g `´ {2 m m ´11 . n n ˇ p 8q n “ n λ | λ| Bn ´1{2 1 ˇ ´1{2 1 Since |gn| “ `n Bn and |Mgn| “ `n |mλ| Bn , we obtain }gn}2 “ 1 and }Mgn}2 ě n for all n P N. These relations contradict the continuity of M. So there exists a function h P X such that mλh R X. If there was an element u of DpAq with λu ´ ∆u “ h, we would obtain as above that 2 λu ` |ξ|2u “ h and hence mλh “ u P X which isp impossible. As a result, λ P σpAq and σpAq “ R´. l p p p d pExamplep 3.42. Let 1 ă p ă 8,pX “ L pR q, and A “ ∆ with DpAq “ 2,p d W pR q. Then A is closed. Proof. The Calderón–Zygmund estimate says that the graph norm of 8 d A is equivalent to k¨k2,p on Cc pR q, see e.g. Corollary 9.10 in [GT]. Let 2,p d 8 d u P W pR q. By Theorem 3.15, there are un P Cc pR q that converge to u 2,p d in W pR q as n Ñ 8, and hence un Ñ u and ∆un Ñ ∆u in X. We derive

}u}2,p “ lim }un}2,p ď lim c p}un}p ` }∆un}pq nÑ8 nÑ8 1 “ c p}u}p ` }∆u}pq ď c }u}2,p , so that } ¨ }A is equivalent to a complete one and thus A is closed. l d Example 3.43 (Dirichlet Laplacian). Let 1 ă p ă 8, U Ď R be bounded and open with BU P C2, X “ LppUq, and A “ ∆ with DpAq “ W 2,ppUq X 1,p W0 pUq. Then A is closed, invertible and has a compact resolvent. In particular, σpAq “ σppAq. Proof. The closedness of A follows from Theorem 9.14 in [GT]. Its bijectivity is shown in Theorem 9.15 of [GT]. Remark 2.14, Theorem 3.29 and Theorem 2.16 then imply the other assertions. l There are variants of Examples 3.42 and 3.43 for X “ L1pUq, X “ L8pUq and in other sup-norm spaces, see Theorem 5.8 in [Ta] as well as Sec- tions 3.1.2 and 3.1.5 in [Lu]. In theses cases, the descriptions of the domains are much more complicated and they are not just (subspaces of) Sobolev (or C2-) spaces. To indicate the diﬃculties, we note that there is a function u R W 2,8pBp0, 1qq such that ∆u P L8pBp0, 1qq, namely px2 ´ y2q lnpx2 ` y2q for px, yq ‰ p0, 0q, upx, yq “ #0 for px, yq “ p0, 0q. Then the second derivative 2 2 2 2 2 2 2 2 2 2 4x p6x ´ 2y qpx ` y q ´ 4x px ´ y q Bxxupx, yq “ 2 lnpx ` y q ` ` x2 ` y2 px2 ` y2q2 x2´y2 is unbounded around p0, 0q, but ∆upx, yq “ 8 x2`y2 is bounded. CHAPTER 4

Selfadjoint operators

In this chapter X and Y are Hilbert spaces with scalar product p¨|¨q, unless something else is said.

4.1. Basic properties Let A be a densely defined linear operator from X to Y . We define the Hilbert space adjoint A1 as in the Banach space case by DpA1q :“ ty P Y D z P X @ x P DpAq : pAx|yq “ px|zqu, 1 A y :“ z. ˇ (4.1) ˇ As in Remark 1.23 one sees that A1 :DpA1q Ñ X is a linear map, which is closed from Y to X. Let T P BpX,Y q. Then DpT 1q “ Y and T 1 is given by @ x P X, y P Y : x T 1y “ pT x|yq , as in (5.9) in [Sc2]. We recall from Proposition` ˇ ˘ 5.42 of [Sc2] that ˇ }T }“}T 1},T 2 “ T, pαT ` βSq1 “ αT 1 ` βS1, pUT q1 “ T 1U 1 for T,S P BpX,Y q, U P BpY,Zq, a Hilbert space Z, and α, β P C. Definition 4.1. A densely defined linear operator A on X is called selfadjoint if A “ A1 (in particular, DpAq “ DpA1q and A must be closed) skewadjoint if A “ Á1, and normal if AA1 “ A1A. We say that T P BpX,Y q is unitary if T is invertible with T ´1 “ T 1. Let Φ : X Ñ X˚ be the (antilinear) Riesz isomorphism given by pΦpxqqpyq “ py|xq for all x, y P X. For λ P C and X “ Y , we obtain 1 ´1 ˚ 1 ´1 ˚ λIX ´ T “ Φ pλIX˚ ´ T qΦ, λIX ´ A “ Φ pλIX˚ ´ A qΦ, with DpA1q “ Φ´1 DpA˚q, see p.109 of [Sc2]. Theorem 1.24 thus implies ˚ 1 ˚ 1 σpAq “ σpA q “ σpA q, σrpAq “ σppA q “ σppA q, (4.2) Rpλ, A1q “ Φ´1Rpλ, A˚qΦ “ Φ´1Rpλ, Aq˚Φ “ Rpλ, Aq1 for λ P ρpAq, where the bars mean complex conjugation of each element. Remark 4.2. a) Selfadjoint, skewadjoint or unitary operators (with X “ Y ) are normal. If A “ A1 and λ P C, then λI ´ A is normal. b) A densely defined linear operator A is skewadjoint if and only if iA is 1 selfadjoint. (Use that (4.1) yields piAq “ íA.) ♦ Theorem 1.16 says that the spectral radius rpT q is less or equal than }T } for each bounded operator T on a Banach space. Already for the matrix 0 1 T “ p 0 0 q one has the strict inequality rpT q “ 0 ă 1 “}T }. We next show kT k “ rpT q for normal operators which is the key to their deeper properties.

67 4.1. Basic properties 68

Proposition 4.3. Let X and Y be Hilbert spaces. If T P BpX,Y q, then kT 1T k “ kTT 1k “ kT k2. Let T P BpXq be normal. We then have kT k “ rpT q, and thus T “ 0 if σpT q “ t0u. Proof. For x P X we compute kT xk2 “ pT x|T xq “ T 1T x|xq ď kT 1T kkxk2, 2 2 1 1 2 kT k “ sup kT xk `ď kT T k ď kT kkT k “ kT k ; kxkď1 i.e., kT k2 “ kT 1T k. We infer that }T }2 “}T 1}2 “}T 2T 1}“}TT 1}. Next, let T be normal. From the first part we then deduce kT 2k2 “ kT 2pT 2q1k “ kTTT 1T 1k “}TT 1TT 1}“ kTT 1pTT 1q1k “ kTT 1k2 “ kT k4, 2 2 2n 2n so that kT k “ kT k . Iteratively it follows that kT k “ kT k for all n P N. Using Theorem 1.16, we conclude n ń rpT q “ lim kT mk1{m “ lim kT 2 k2 “ kT k. mÑ8 nÑ8 The following concepts turn out to be very useful to compute adjoints, for instance. Definition 4.4. Let A and B be linear operators from a Banach space X to a Banach space Y . We say that B extends A (and write A Ď B) if DpAq Ď DpBq and Ax “ Bx for all x P DpAq. Next, let X be a Hilbert space. A linear operator A on X is called symmetric if we have pAx|yq “ px|Ayq for all x, y P DpAq. Of course, a selfadjoint operator is symmetric. As we see in Example 4.8, the converse is not true in general. Remark 4.5. Let A and B be linear operators from a Banach space X to a Banach space Y . a) The operator B extends A if and only if its graph GpBq contains GpAq. Let A Ď B. Then A “ B is equivalent to DpBq Ď DpAq. b) Let A Ď B, A be surjective and B be injective. Then A and B are equal. Hence, if X “ Y and there is a λ P C such that λI ´ A is surjective and λI ´ B is injective, then A “ B. Proof. Let x P DpBq and set y “ Bx. The surjectivity of A yields a vector z P DpAq Ď DpBq such that y “ Az “ Bz. Since B is injective, we obtain x “ z P DpAq so that A “ B. l c) Let A be densely defined and symmetric on a Hilbert space X. Defini- tion (4.1) implies that A Ď A1. In particular, A is selfadjoint if and only if 1 DpA q Ď DpAq. ♦ In the next theorem we give spectral conditions implying that a symmetric operator is selfadjoint. We start with a simple, but crucial lemma. Lemma 4.6. Let A be symmetric, x P DpAq and α, β P R. Set λ “ α ` iβ. We have pAx|xq P R and kλx ´ Axk2 “ kαx ´ Axk2 ` |β|2 kxk2 ě |β|2 kxk2. 1 If A is also closed, then σappAqĎR and kRpλ, Aqk ď |Im λ| for all λPρpAqzR. 4.1. Basic properties 69

In particular, λ R σappAq if Im λ “ β ‰ 0. If λ P ρpAqzR and y P X, write x “ Rpλ, Aqy P DpAq. We then calculate 2 2 2 2 2 2 kyk “ kλx ´ Axk ě |Im λ| kxk “ |Im λ| kRpλ, Aqyk . Theorem 4.7. Let X be a Hilbert space and A be densely deﬁned, closed and symmetric. (a) Then σpAq is either a subset of R or σpAq “ C or σpAq “ tλ P C Im λ ě 0u or σpAq “ tλ P C Im λ ď 0u. (b) The following assertions are equivalent. ˇ ˇ (i)ˇ A “ A1. ˇ (ii) σpAq Ď R. (iii) iI ´ A1 and iI ` A1 are injective. (iv) piI ´ Aq DpAq and piI ` Aq DpAq are dense. (c) If ρpAq X R ‰H, then A is selfadjoint. (d) Let A be selfadjoint. Then we have 1 kRpλ, Aqk ď (4.3) |Im λ| for λ R R. Further, σpAq “ σappAq is non-empty and A has no selfadjoint extension B ‰ A. Proof. (a) Let λ P σpAq and µ P ρpAq. Suppose that Im λ ą 0 and Im µ ą 0. The line segment from λ to µ then must contain a point γ PBσpAq. This point belongs to σappAq by Proposition 1.19 and satisﬁes Im γ ą 0, which contradicts Lemma 4.6 since A is symmetric. Similarly we exclude that Im λ ă 0 and Im µ ă 0. Since the spectrum is closed, only the four cases in (a) remain. (b) Let A be selfadjoint. Lemma 4.6 yields σappAq Ď R. Due to (4.2) 1 we also have σrpAq “ σppA q “ σppAq. Hence, σrpAq “ σppAq Ď R. From Proposition 1.19 we thus deduce σpAq Ď R; i.e., (i) implies (ii). The im- 1 plication ‘(ii)ñ(iii)’ is obvious. Equation (4.2) also shows that ˘i P σppA q if and only if ¯i P σrpAq so that (iii) and (iv) are equivalent. Finally, let (iv) (and thus (iii)) be true. The range of the operator iI ˘ A is closed by Lemma 4.6 and Proposition 1.19. In view of (iv), the map iI ˘ A is then surjective. On the other hand, iI ˘ A1 is injective because of (iii), and hence A “ A1 thanks to Remark 4.5b). (c) If there is a point λ in ρpAq X R, then ρpAq contains a ball around λ by its openness. By part (a), the spectrum of A is thus contained in R, and so A is selfadjoint by (b). (d) Let A “ A1. If its spectrum was empty, then A is invertible with a selfadjoint inverse. Proposition 1.20 thus yields that σpA´1q is equal to t0u 4.1. Basic properties 70 so that A´1 “ 0 by Proposition 4.3, which is impossible. Hence, σpAq ‰ H. Since σpAq “ σpA1q Ď R by (b), the other parts of assertion (c) follow from Lemma 4.6 and Remark 4.5. 2 1,2 Example 4.8. a) Let X “ L pRq and A “ iB with DpAq “ W pRq. Then A is selfadjoint with σpAq “ R. Proof. For u, v P DpAq, integrating by parts we deduce

pAu|vq “ i pBuqv ds “ ´i uBv ds “ uiBv ds “ pu|Avq , żR żR żR see Theorem 3.35; i.e., A is symmetric. Example 3.37 and Proposition 1.20 2 further imply that σpAq “ iσp´iAq “ i R “ R. Hence, A is selfadjoint. l b) Let X “ L2p0, 8q and A “ iB on DpAq “ W 1,2p0, 8q. Then A is not symmetric. Proof. For u, v P DpAq with up0q “ vp0q “ 1, as above an integration by parts implies

8 8 pAu|vq “ i pBuqv ds “ í uBv ds ´ iup0qvp0q “ pu|Avq ´ i. l ż0 ż0 2 1,2 c) Let X “ L p0, 8q and A “ iB on DpAq “ W0 p0, 8q. Then A is symmetric, but not selfadjoint, and σpAq “ tλ P C Im λ ě 0u. Proof. Symmetry is shown as in a) using Theorem 3.35 and that now ˇ u, v P DpAq have trace 0 at s “ 0. From Example 3.38ˇ and Proposition 1.20 we deduce that σpAq “ íσpiAq “ ítλ P C Re λ ď 0u “ tλ P C Im λ ě 0u. Consequently, A is not selfadjoint. l ˇ ˇ 2 d ˇ 2,2 d ˇ d) Let X “ L pR q and A “ ∆ with DpAq “ W pR q. Then A is selfadjoint with σpAq “ R´. Proof. For u, v P DpAq, Theorem 3.35 yields

pAu|vq “ p∆uqv dx “ u∆v dx “ pu|Avq , d d żR żR so that A is symmetric. In Example 3.41 we have seen that σpAq “ R´, and hence A is self-adjoint. l d 2 e) Let U Ď R be open with C –boundary and A “ ∆ with DpAq “ 2,2 1,2 W pUq X W0 pUq. Then A is selfadjoint (and has compact resolvent by Example 3.43). In fact, the symmetry of A can be shown as in part d) because the traces of u, v P DpAq vanish. Then A is selfadjoint since it is invertible by Example 3.43. 2 2 2,2 f) Let X “ L p0, 1q, A0 “B with DpA0q “ W0 p0, 1q and A be as in e) with U “ p0, 1q. As in e) we see that A0 is symmetric. But A0 is not 1 1 2 selfadjoint, since A0 Ř A and A “ A . We further claim that A0 “B with 1 2,2 DpA0q “ W p0, 1q. Proof. For v P W 2,2p0, 1q and u P DpAq, we deduce from Theorem 3.35

1 1 2 2 1 2 pA0u|vq “ pB uqv ds “ uB v ds ` rvBu ´ uBvs0 “ pu|B vq; ż0 ż0 4.1. Basic properties 71

2 2,2 1 1 8 i.e., pB ,W p0, 1qq Ď A0. Conversely, take v P DpA0q. For u P Cc p0, 1q we 8 have u P Cc p0, 1q Ď DpAq and hence obtain 1 1 2 1 1 pB uqv ds “ pA0u|vq “ pu|A0vq “ uA0v ds. ż0 ż0 2,2 2 1 After complex conjugation, we see that v P D p0, 1q X X and B v “ A0v P X. The function v thus belongs to W 2,2p0, 1q by Example 3.36. l Theorem 4.9. Let A be densely deﬁned and selfadjoint on the Hilbert space X and let B be symmetric with DpAq Ď DpBq. Assume there are constants c ą 0 and δ P r0, 1{2q such that kBxk ď ckxk ` δkAxk for all x P DpAq. Then the operator A ` B with DpA ` Bq “ DpAq is selfadjoint.

Proof. By Theorem 4.7, the number it belongs to ρpAq for all t P Rzt0u. Take ε P p0, 1 ´ 2δq Ď p0, 1q and x P X. Using (4.3), we estimate kBpitI ´ Aq´1xk ď δkApitI ´ Aq´1xk ` c kpitI ´ Aq´1xk “ δ kitpitI ´ Aq´1x ´ xk ` c kpitI ´ Aq´1xk |t| c ď δ ` 1 kxk ` kxk ď p1 ´ εqkxk, |t| |t| c ´ ¯ whenever |t| ě 1´2δ´ε . Theorem 1.27 now implies that ˘it P ρpA ` Bq for such t. Moreover, A ` B is symmetric since ppA ` Bqx|yq “ pAx|yq ` pBx|yq “ px|Ayq ` px|Byq “ px|pA ` Bqyq for all x, y P DpAq. So, A ` B is selfadjoint due to Theorem 4.7. Actually, in the above theorem it suﬃces to assume that δ ă 1, see The- orem X.13 in [RS].

2 3 2,2 3 Example 4.10. Let X “ L pR q and A “ ∆ with DpAq “ W pR q. Set V upxq “ b upxq for x P 3, u P DpAq and some b P . Then A ` V with |x|2 R R domain DpAq is selfadjoint. 3 Proof. Since kp “ 4 ą d “ 3, we have DpAq ãÑ C0pR q by Theorem 3.17. Let 0 ă ε ď 1. Using polar coordinates and Example 3.41, we compute 2 2 2 2 |upxq| 2 |upxq| |V u| dx “ b 2 dx ` b 2 dx 3 |x| 3 |x| żR żBp0,εq 2 żR zBp0,εq 2 ε 2 2 2 r b 2 ď c kuk8 2 dr ` 2 |u| dx r ε 3 ż0 żR zBp0,εq b2 b2 ď cεkuk2 ` kuk2 ď cεkAuk2 ` cεkuk2 ` kuk2, 2,2 ε2 2 2 2 ε2 2 for constants c ą 0 independent of u P DpAq and ε. Moreover, V is symmetric on DpAq since b pV u|vq “ upxqvpxq dx “ pu|V vq 3 |x| żR 2 for all u, v P DpAq. For small ε ą 0, Theorem 4.9 implies that A ` V is selfadjoint. l 4.2. The spectral theorems 72

The spectra σpA ` V q and σppA ` V q and the eigenfunctions of A ` V are computed in §7.3.4 of [Tr], where b ą 0. The above operator A “ ∆ ` V is used in physics to describe the hydrogen atom, see also Example 4.19.

4.2. The spectral theorems Hermitian matrices are unitarily equivalent to diagonal matrices and thus very easy to treat. In this section we establish the infinite dimensional analogues of this basic result from linear algebra. The following results can be extended to normal operators, and the separability assumption made below can be removed. See e.g. Corollaries X.2.8 and X.5.4 in [DS2] or Theorems 13.24, 13.30 and 13.33 in [Ru2]. n Let T P BpZq for a Banach space Z and ppzq “ a0 ` a1z ` ... ` anz be a complex polynomial. We then define the operator polynomial n ppT q “ a0I ` a1T ` ... ` anT P BpZq. (4.4) This gives a map p ÞÑ ppT q from the space of polynomials to BpZq. For selfadjoint T on a Hilbert space one can extend this map to all f P CpσpT qq, as seen in the next theorem. We set p1pzq “ z. Theorem 4.11 (Continuous functional calculus). Let T P BpXq be selfadjoint on a Hilbert space X. There is exactly one map ΦT : CpσpT qq Ñ BpXq; f ÞÑ fpT q satisfying (C1) pαf ` βgqpT q “ αfpT q ` βgpT q, (C2) kfpT qk “ kfk8 (hence, ΦT is injective), (C3) 1pT q “ I and p1pT q “ T , (C4) pfgqpT q “ fpT qgpT q “ gpT qfpT q, (C5) fpT q1 “ fpT q for all f, g P CpσpT qq and α, β P C. In particular, we have ΦT ppq “ ppT q for each polynomial p, where ppT q is given by (4.4). Proof. We first show the properties (C1)-(C5) for polynomials pptq “ n n a0 ` a1t ` ... ` ant and qptq “ b0 ` b1t ` ... ` bnt with t P R and the map p ÞÑ ppT q defined by (4.4), where any aj, bj P C may be equal to 0. Clearly, 1 n j 1 (C1) and (C3) are true in this case, and ppT q “ j“0 ajpT q “ ppT q since T “ T 1. Moreover, ř 2n n n m j k ppqqpT q “ ajbk T “ ajT bjT “ ppT qqpT q m“0 0ďj,kďn j“0 k“0 ÿ ´ j`ÿk“m ¯ ÿ ÿ and so ppqqpT q “ pqpqpT q “ qpT qppT q; i.e., (C4) is shown for polynomials. Properties (C4) and (C5) imply that ppT q is normal. Hence, Proposition 4.3 and Lemma 4.12 below imply

kppT qk “ maxt|λ| λ P σpppT qqu “ maxt|λ| λ P ppσpT qqu “ kpk8. Let f P CpσpT qq. Sinceˇ σpT q Ď R is compact,ˇ Weierstraß’ approximation ˇ ˇ theorem yields real polynomials such that pn Ñ Re f and qn Ñ Im f in CpσpT qq as n Ñ 8, hence pn ` iqn Ñ f. We can thus extend the map p ÞÑ ppT q to a linear isometry ΦT : f ÞÑ fpT q from CpσpT qq to BpXq. By continuity, ΦT also satisﬁes (C4) and (C5) on CpσpT qq. 4.2. The spectral theorems 73

If there is another map Ψ : CpσpT qq Ñ BpXq satisfying (C1)-(C5), then Ψppq “ ppT q “ ΦT ppq for all polynomials by (C1), (C3) and (C4), so that Ψ “ ΦT by continuity and density. We observe that we have actually shown uniqueness in then calss of linear and continuous maps Ψ : CpσpT qq Ñ BpXq fulﬁlling (C3) and (C4). Lemma 4.12. Let T P BpZq for a Banach space Z and let p be a polynomial. Then σpppT qq “ ppσpT qq.

Proof. See Theorem 5.3 below or the exercises. Corollary 4.13. Let T P BpXq be selfadjoint and f P CpσpT qq. Then the following asssertions hold. (C6) If T x “ λx for some x P X and λ P C, then fpT qx “ fpλqx. (C7) The operator fpT q is normal. (C8) σpfpT qq “ fpσpT qq (spectral mapping theorem). (C9) fpT q is selfadjoint if and only if f is real-valued.

Proof. Take a sequence of polynomials pn converging to f uniformly. Let T x “ λx. Property (C6) holds for a polynomial since n n j j ppT qx “ ajT x “ ajλ x “ ppλqx. j“0 j“0 ÿ ÿ Using (C2), we obtain

fpT qx “ lim pnpT qx “ lim pnpλqx “ fpλqx. nÑ8 nÑ8 From (C5) and (C4) we infer fpT qfpT q1 “ fpT qfpT q “ fpT qfpT q “ fpT q1fpT q so that fpT q is normal. 1 We next show (C8). If µ R fpσpT qq, then g “ µ´f is an element of CpσpT qq. Thus (C3) and (C4) yield pµI ´ fpT qqgpT q “ gpT qpµI ´ fpT qq “ pgpµ1 ´ fqqpT q “ 1pT q “ I.

Hence, µ P ρpfpT qq. Let µ “ fpλq for some λ P σpT q. Then µn :“ pnpλq belongs to σppnpT qq for all n P N by Lemma 4.12. As above, the operators µnI ´pnpT q tend to µI ´fpT q in BpXq. Suppose that µI ´fpT q was invertible. Then also µnI ´pnpT q would be invertible for large n by Theorem 1.27. This is impossible, and so µ P σpfpT qq. For the last assertion, observe that fpT q “ fpT q1 if and only if pf´fqpT q “ 0 if and only if f ´ f “ 0, because ΦT is injective. 1 Corollary 4.14. Let n P N and T “ T P BpXq with σpT q Ď R` (in this case one writes T “ T 1 ě 0). Then there is a unique selfadjoint operator n W P BpXq with σpW q Ď R` such that W “ T .

1{n Proof. Consider wptq “ t for t P σpT q Ď R` and deﬁne W :“ wpT q. n n Then W “ w pT q “ p1pT q “ T . Properties (C8) and (C9) imply that W “ W 1 ě 0. For the proof of the uniqueness of W , we refer to Korollar VII.1.16 in [We] or the exercises. We next use basic properties of orthogonal projections and orthonormal bases which are discussed in, e.g., Chapter 3 of [Sc2]. 4.2. The spectral theorems 74

Theorem 4.15 (Compact case). Let X be a Hilbert space with dim X “ 8, T P BpXq be compact and selfadjoint, and A be densely defined, closed and selfadjoint on X having a compact resolvent. Then the following assertions hold. (a) i) There is an index set J P tH, N, t1,...,Nu N P Nu and eigenvalues λ ‰ 0, j P J, such that σpT q “ t0u Yt9 λ j P Ju Ď where λ Ñ 0 as j j ˇ R j j Ñ 8 if J “ N. ˇ ˇ ii) There is an orthonormal basis of NpTˇqK “ RpT q consisting of eigenvectors of T for the eigenvalues λj. iii) The eigenspace EjpT q :“ NpλjI ´ T q is finite-dimensional and the orthogonal projection Pj onto EjpT q commutes with T , for each j P J. iv) The sum T “ jPJ λjPj converges in BpXq. (b) i) We have σpAřq“σppAq“tµn n P Nu Ď R with |µn| Ñ 8 as n Ñ 8. ii) There is an orthonormal basis of X consisting of eigenvectors of A. ˇ iii) The eigenspaces EnpAq “ NpˇµnI ´ Aq are finite dimensional and the orthogonal projections Qn onto EnpAq satisfy Qn DpAq Ď DpAq and AQnx “ QnAx for all x P DpAq and n P N. 8 iv) The sum Ax “ n“1 µnQnx converges in X for all x P DpAq. Proof. 1) Theoremř 2.11, 2.16 and 4.7 show the assertions i) in (a) and (b), except for the claim that σpAq is infinite, and also that dim EjpT q and dim EnpAq are finite for all j and n. Let x, y P DpAq be eigenvectors of A for eigenvalues µn ‰ µk. Then

µn px|yq “ pAx|yq “ px|Ayq “ µk px|yq, so that px|yq “ 0. Similarly, one sees that EjpT q K EkpT q if j ‰ k. By the Gram-Schmidt procedure, each eigenspace EjpT q and EnpAq has an orthonormal basis of eigenvectors for λj ‰ 0 and µn, respectively. The union of these bases gives orthonormal sets BT and BA. 2) Let J “ N in a), the other cases are treated similarly. Observe that 1 1 1 1 BT Ď RpT q. Set j “ tλj u P CpσpT qq and ϕn “ 1 ` ... ` n for every 1 2 12 1 j, n P N. We then have jpT q “ j pT q “ jpT q. Moreover, (C4) and (C9) 1 imply that T 1jpT q “ 1jpT qT , 1jpT q “ 1jpT q and

pλjI ´ T q1jpT q “ ppλj1 ´ p1q1jqpT q “ ppλj ´ λjq1jqpT q “ 0.

If λjv “ T v for some v P X, we further deduce 1jpT qv “ 1jpλjqv “ v from (C6). As a result, 1jpT q is a selfadjoint projection onto EjpT q. For x P X and y P Np1jpT qq, we then obtain p1jpT qx|yq “ px|1jpT qyq “ 0 so that 1jpT q is orthogonal; i.e., 1jpT q “ Pj and ϕnpT q “ P1 ` ... ` Pn for all j, n P N. We have shown iii) in (a). Since TPj “ λjPj, the operator ϕnpT qT is a partial sum of the series in part iv). Employing also (C2), we thus derive iv) from

kT ´ ϕnpT qT k “ kpp1 ´ ϕnp1qpT qk “ kp1 ´ ϕnp1k8 “ sup |λj| ÝÑ 0, jěn`1 as n Ñ 8. It also follows that ϕnpT qy P lin BT converges to y as n Ñ 8 for all y P RpT q. Therefore, BT is an orthonormal basis of RpT q due to e.g. Theorem 3.15 in [Sc2]. Finally, (2.1) shows that RpT q “ K NpT 1q “ NpT qK because T “ T 1 and X is reﬂexive. 4.2. The spectral theorems 75

3) Fix t P ρpAq X R. Then Rpt, Aq1 “ Rpt, A1q “ Rpt, Aq by (4.2), and this operator is compact and has a trivial kernel. By 2), X “ NpRpt, AqqK pos- sesses an orthonormal basis of eigenvectors w of Rpt, Aq for the eigenvalues λ ‰ 0. Proposition 1.20 yields the eigenvector v “ λRpt, Aqw of A for the eigenvalue µ “ t ´ λ´1. Since dim X “ 8 and the eigenspaces of Rpt, Aq are ﬁnite dimensional, A has inﬁnitely many distinct eigenvalues; i.e., part i) in (b) is shown. Let x P X be orthogonal to all eigenvectors of A. For the above v, we obtain 0 “ px|vq “ px|λRpt, Aqwq “ λ pRpt, Aqx|wq. Since the eigenvectors w span X, we infer that Rpt, Aqx “ 0 and hence x “ 0. Consequently, BA is a basis of X and part ii) of (b) is true.

4) Let tvn,1, ¨ ¨ ¨ , vmn u be eigenvectors of A forming an orthonormal basis of EnpAq for n P N. From step 3) we then deduce

mn 8 Qnx “ px|vn,jq vn,j and x “ Qnx j“1 n“1 ÿ ÿ for all x P X. For x P DpAq it follows

mn mn mn QnAx “ pAx|vn,jq vn,j “ px|Avn,jq vn,j “ px|µnvn,jq vn,j j“1 j“1 j“1 ÿ ÿ ÿ mn mn “ px|vn,jq µnvn,j “ px|vn,jq Avn,j “ AQnX. j“1 j“1 ÿ ÿ We thus conclude n n 8 8 8 A Qkx “ QkAx ÝÑ QkAx “ AQkx “ µkQkx, k“1 k“1 k“1 k“1 k“1 ÿ ÿ ÿ ÿ ÿ as n Ñ 8, so that the closedness of A yields the last assertion. Remark 4.16. a) In the above proof we also obtain that

8 mn Ax “ µn px|vn,jq vn,j n“1 j“1 ÿ ÿ for all x P DpAq. An analogous result holds for T . b) Let T be selfadjoint and compact such that NpT q is separable. Let tzk k P J0u be an orthonormal basis for NpT q, where J0 Ď Z´ could be empty. Denote by λ the non-zero eigenvalues of T (repeated according ˇ l to theirˇ multiplicity) with corresponding orthonormal basis of eigenvectors 1 twl l P J1u. The union tbj j P J u of tzk k P J0u and twl l P J1u is an orthonormal basis of X. By e.g. Theorem 3.18 in [Sc2], the map ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ Φ: X `2 J 1 ;Φx x b , Ñ p q “ p | jq jPJ1 ´1 ` ˘ is unitary with Φ ppξjqjPJ1 q “ jPJ1 ξjbj. Moreover, the transformed operator ΦT Φ´1 acts on `2pJ 1q as the multiplication operator ř ´1 ΦT Φ pξjq “ ΦT ξjbj “ Φ λjξjbj “ pλjξjqjPJ1 , jPJ1 jPJ1 ÿ ÿ 4.2. The spectral theorems 76

´1 where λj :“ 0 if j P J0. Hence, ΦT Φ is represented as inﬁnite diagonal matrix with diagonal elements λj. Analogous results hold for A from Theorem 4.15, see e.g. Theorems 4.5.1-4.5.3 in [Tr]. ♦ If one wants to extend part a) of the above remark to the non-compact case one is led to the so-called spectral measures, see the exercises. We generalize below part b). Theorem 4.17 (Multiplicator representation, bounded case). Let T P BpXq be selfadjoint on a separable Hilbert space X. Then there is a σ–ﬁnite measure space pΩ, A, µq, a measurable function h :Ω Ñ σpT q and a unitary operator U : X Ñ L2pµq such that T x “ U ´1hUx for all x P X.

Proof (partly sketched). 1) Let v1 P Xzt0u. We deﬁne the linear subspaces Y1 :“ tfpT qv1 f P CpσpT qqu and X1 “ Y 1 of X. Since T fpT qv1 “ pp1fˇqpT qv1 P Y1 for every f P CpσpT qq, we obtain TY1 Ď Y1 and so TX1 Ď X1.ˇ We introduce the map

ϕ1 : CpσpT qq Ñ C; ϕ1pfq “ pfpT qv1|v1q , 2 2 which is linear and bounded because |ϕ1pfq| ď kfpT qkkv1k “ kfk8kv1k due to (C2). If f ě 0, then σpfpT qq Ď R` by (C8). So we can deduce from Corollary 4.14 that

1{2 1{2 1{2 2 pfpT qv1|v1q “ fpT q v1 fpT q v1 “ kfpT q v1k ě 0. The Riesz representation` theorem ofˇ CpσpT qq˘˚ now gives a positive measure ˇ µ1 on BpσpT qq such that

ϕ1pfq “ f dµ1, żσpT q 2 for all f P CpσpT qq Ď L pµ1q, see Theorem 2.14 of [Ru1]. For x “ fpT qv1 P 2 Y1, we deﬁne V1x :“ f P L pµ1q. We compute

In particular, if x “ fp`T qv1 “ gpT qv1 for some g P CpσpT qq, then 2 2 kf ´ gk2 “ kpfpT q ´ gpT qqv1kX “ 0, 2 2 and so f “ g in L pµ1q. As a result, V1 : Y1 Ñ L pµ1q is a linear isometric 2 map and can be extended to a linear isometry U1 : X1 Ñ L pµ1q. Observe that CpσpT qq Ď RpU1q. Riesz’ theorem also yields that µ1 is 1 2 regular. Theorem 3.14 in [Ru1] then shows that CpσpT qq is dense in L pµ1q,

1A positive measure µ is called regular, if if satisﬁes µpBq “ inftµpOq B Ď O, O is openu “ suptµpKq K Ď B,K is compactu for all B in the correspondingˇ Borel σ–algebra. ˇ ˇ ˇ 4.2. The spectral theorems 77 so that U1 has dense range. Hence, the isometry U1 is bijective and thus unitary by e.g. Proposition 5.52 in [Sc2]. Finally, we compute

U1T fpT qv1 “ V1pp1fqpT qv1 “ p1f “ p1U1fpT qv1, ´1 for f P CpσpT qq. By density, it follows T x “ U1 p1U1x for all x P X1. 2 2) We are done if there is an v1 P X such that X1 “ X. In general this is not true. Using Zorn’s Lemma (see Theorem I.2.7 of [DS1]), we instead ﬁnd an orthonormal system of spaces Xn as in step 1) which span X. To that aim, we introduce the collection E of all sets E having as elements at most countably many closed subspaces Xj Ă X of the type constructed in step 1) such that Xi K Xj for all Xi ‰ Xj in E. The system E is ordered via inclusion of sets. Let C be a chain in E; i.e., a subset of E such that

E Ď F or F Ď E for all E,F P C. We put C “ EPC E. Cleary, E Ď C for all E P C. Let Y,Z P C. Then Y and Z are closed subspaces of X as constructed in 1) and there are sets E,F P C suchŤ that Y P E and Z P F . We may assume that E Ď F and so X,Y P F . The subspaces Y and Z are thus orthogonal (if Y ‰ Z). As a result, C contains pairwise orthogonal subspaces of X. If x K y have norm 1, then }x ´ y}2 “}x}2 ` }y}2 “ 2. The separability of X then implies that at most countably many subspaces belong to C, so that C is an element of E and hence an upper bound of C. Zorn’s Lemma now gives a maximal element M “ tXj j P Ju in E, where J Ă N and Xj are pairwise orthogonal subspaces as constructed in step 1). ˇ Assume that there is a vector v P X being orthogonalˇ to all Xj. Let Z be the closed span of all vectors fpT qv with f P CpσpT qq. Let g P CpσpT qq and x “ gpT qvi P Xi for some i P J, where vi generates Xi as in step 1). We then obtain 1 px|fpT qvq “ pfpT q gpT qvi|vq “ ppfgqpT qvi|vq “ 0 since pfgqpT qvi P Xi. By density, it follows that Z is orthogonal to all Xj and thus M YtZu P E. The maximality on M now implies that Z belongs to M, so that v “ 0. Consequently, X is the closed linear span of the elements in the orthogonal subspaces Xj. (One writes X “ jPJ Xj.) We now deﬁne 2 Ω “ σpT q ˆ tju Ď R , A “ BpΩq, µÀpAq “ µjpAjq jPJ jPJ ď ÿ with Aj ˆ tju “ A X pσpT q ˆ tjuq and

hpλ ˆ tjuq “ λ, Ux “ Ujxj, jPJ ÿ where λ ˆ tju P Ω, Uj is given as in step 1), and x “ jPJ xj and xj P Xj. It can be seen that Ω, A, µ, h and U satisfy the assertions. ř We add an observation to the above proof. Let λ P σpT qzσppT q. Then Λ “ jPJ tλ ˆ tjuu Ă Ω is a µ–null set. In fact, otherwise the characteristic function f of any subset of Λ with measure in p0, 8q would be a non-zero elementŤ of L2pµq. Hence, x “ U ´1f ‰ 0 would be an eigenvector of T for the eigenvalue λ, since T x “ U ´1λf “ λx. We further note that in the proof

2Not shown in the lectures. 4.2. The spectral theorems 78 of Theorem 4.17 one could also take Ω “ jPJ σpTjqˆtju, where Tj “ T |Xj . It can be shown that σpT q “ jPJ σpTjq. The above representation of bounded selfadjointŤ operators as multiplica- Ť tion operators now leads to a multiplication representation and to a Bb– functional calculus for (possibly) unbounded selfadjoint operators A. Here BbpσpAqq is the Banach space of bounded Borel functions on σpAq endowed with the supremum norm. We use this space instead of L8pσpAqq to avoid ´1 certain technical problems. Set rλpzq “ pλ ´ zq for z P Cztλu. Theorem 4.18 (Unbounded A). Let A be a closed, densely defined and selfadjoint operator on a separable Hilbert space X. Then the following assertions hold. (a) There is a σ–finite measure space pΩ, A, µq, a measurable function h :Ω Ñ σpAq and a unitary operator U : X Ñ L2pµq such that DpAq “ tx P X hUx P L2pµqu and Ax “ U ´1hUx. (b) There is a contractiveˇ map ΨA : BbpσpAqq Ñ BpXq; ΨT pfq “ fpAq, satisfying (C1) and (C3)–(C5),ˇ where p1pT q “ T in (C3) is replaced by rλpAq “ Rpλ, Aq for λ P ρpAq. Moreover, if fn P BbpσpAqq are uniformly bounded and converge to f P BbpσpT qq pointwise, then fnpAqx Ñ fpAqx as n Ñ 8 for all x P X. Finally, for x P DpAq and f P BbpσpAqq the vector fpAqx belongs to DpAq and AfpAqx “ fpAqAx. 3 Proof. a) We additionally assume that σpAq ‰ R and fix t P ρpAqXR. Then Rpt, Aq P BpXq is selfadjoint and can be represented as Rpt, Aq “ U ´1mU on a space L2pΩ, µq as in Theorem 4.17. Recall that Proposi- tion 1.20 yields σpAq “ t ´ rσpRpt, Aqqzt0us´1. Set 1 1 hpλ ˆ tjuq “ t ´ “ t ´ P σpAq, mpλ ˆ tjuq λ for j P J and λ P σpRpt, Aqqzt0u. The sets t0 ˆ tjuu have µ–measure 0 in view of the obervation before the theorem, due to the injectivity of Rpt, Aq. We can thus extend h by 0 to a measurable function on Ω. Let x P DpAq. We put y “ tx ´ Ax P X. Since x “ Rpt, Aqy “ U ´1mUy, we obtain hUx “ hmUy “ ptm ´ 1qUy P L2pµq, U ´1hUx “ tU ´1mUy ´ y “ tx ´ y “ Ax. If x P X satisfies hUx P L2pµq, then we put y “ U ´1pt1 ´ hqUx P X and obtain mUy “ ptm ´ mhqUx “ Ux. Therefore, x “ U ´1mUy “ Rpt, Aqy belongs to DpAq, and part (a) is proved. b) We define ΨA : f ÞÑ fpAq by fpAqx “ U ´1pf ˝ hqUx 2 for f P BbpσpAqq and x P X. We further set Mf ϕ “ pf ˝ hqϕ for ϕ P L pµq. It is straightforward to check that fpAq P BpXq,ΨT is linear, 1pAq “ I and (C5) is true. Let λ P ρpAq. We have ´1 2 hUrλpAqx “ hprλ ˝ hqUx “ hpλ1 ´ hq Ux P L pµq

3 If σpAq “ R, we instead take t “ i and use below the version of Theorem 4.17 for the normal operator Rpi,Aq given in e.g. Satz VII.1.25 in [We]. 4.2. The spectral theorems 79 for all x P X. So part (a) yields that rλpAqX Ď DpAq and pλIÁqrλpAq “ I. Similarly, one sees that rλpAqpλx ´ Axq “ x for all x P DpAq, and so (C3) is shown. The contractivity follows from kfpAqk “ kMf k ď kfk8. For property (C4) we observe that pfgqpAqx “ U ´1pf ˝ hqpg ˝ hqUx “ U ´1pf ˝ hqUU ´1pg ˝ hqUx “ fpAqgpAqx, where f, g P BbpσpAqq and x P X. Let f, fn P BbpσpAqq be uniformly bounded by c such that fn Ñ f point- ´1 wise as n Ñ 8. For every x P X, we have fnpAqx ´ fpAqx “ U ppfn ´ fq ˝ hqUx. Since pfn ´ fq ˝ h Ñ 0 pointwise and |ppfn ´ fq ˝ hqUx| ď 2c |Ux|, Lebesgue’s convergence theorem shows that ppfn ´ fq ˝ hqUx tends to 0 in 2 L pµq and so fnpAqx Ñ fpAqx in X as n Ñ 8. Let x P DpAq. The above results yield that g :“ hpf ˝ hqUx “ pf ˝ hqUU ´1hUx “ pf ˝ hqUAx P L2pµq. On the other hand, g “ hUU ´1pf ˝ hqUx “ hUfpAqx. Part (a) thus implies that fpAqx P DpAq and ´1 ´1 ´1 AfpAqx “ U hUfpAqx “ U g “ U pf ˝ hqUAx “ fpAqAx. We conclude with one of the most important applications of the above theorem. Example 4.19 (Schrödinger’sequation). Let X be a Hilbert space and H be a closed, densely defined and selfadjoint operator on X. For a given 1 u0 P DpHq we claim that there is exactly one function u P C pR,Xq X CpR, rDpHqsq solving d dt uptq “ íHuptq, t P R, up0q “ u0. (4.5)

(H is called Hamiltonian.) The solution is given by uptq “ T ptqu0 for unitary operators T ptq on X satisfying T p0q “ I, T ptqT psq “ T psqT ptq and T ptq´1 “ T p´tq for t, s P R. Moreover, t ÞÑ T ptqx P X continuous for t P R and all x P X. An example for this setting is X “ L2p 3q and H “ ´p∆ ` b q R |x|2 2,2 3 with DpHq “ W pR q, see Example 4.10. ítξ Proof. For t P R, we consider the bounded function ftpξq “ e , ξ P R. Theorem 4.18 allows us to define T ptq “ ftpHq P BpXq. Since f0 “ 1 and ftfs “ ft`s, we obtain T p0q “ I and T ptqT psq “ T psqT ptq for t, s P R. With s “ ´t it follows that T ptq has the inverse T p´tq. Moreover, T ptq is unitary 1 since T ptq “ ftpHq “ f´tpHq “ T p´tq. Because of kftk8 “ 1 and the continuity of t ÞÑ ftpξq for fixed ξ P R, the function R Q t ÞÑ T ptqz P X is continuous for each x P X. Let u0 P DpHq. We set y “ τu0 ´ Hu0 for some τ P ρpHq, so that u0 “ Rpτ, Hqy “ rτ pHqy. We then obtain 1 1 pT ptqu0 ´ T psqu0q “ pftpHq ´ fspHqqrτ pHqy t ´ s t ´ s 1 “ pft ´ fsqrτ pHqy “: gt,spHqy t ´ s ˆ ˙ íξ for all t ‰ s. Observe that gt,spξq Ñ τ´ξ fspξq “: mpξqfspξq as t Ñ s for ξ all ξ P σpHq and kgt,sk8 ď kmk8 “ supξPσpHq| τ´ξ | ă 8 for all t ‰ s. So 4.2. The spectral theorems 80

Theorem 4.18 shows that there exists d dt T ptqu0 “ mpHqT ptqpτu0 ´ Hu0q “ T ptqmpHqy. We further compute ´1 ´1 mpT qy “ Upm ˝ hqU y “ Uppíp1rτ q ˝ hqU y ´1 ´1 “ ÚihU Uprτ ˝ hqU y “ íHRpτ, Hqy “ íHu0 by means of Theorem 4.18. Hence, we arrive at d dt T ptqu0 “ íT ptqHu0 “ íHT ptqu0, using Theorem 4.18 once more. Due to these equations, u “ T p¨qu0 belongs 1 to C pR`,Xq X CpR`, rDpHqsq and solves (4.5). 1 Let v P C pR`,Xq X CpR`, rDpHqsq be another solution of (4.5). For 0 ď s ď t and h ‰ 0 we compute 1 1 h pT pt ´ s ´ hqvps ` hq ´ T pt ´ sqvpsqq ´ T pt ´ sqpv psq ` iHvpsqq 1 1 1 “ T pt ´ s ´ hq h pvps ` hq ´ vpsqq ´ v psq ` pT pt ´ s ´ hq ´ T pt´sqqv psq 1 ´ ´h pT pt ´ s`´ hq ´ T pt ´ sqqvpsq ´ T p˘t ´ sqiHvpsq ÝÑ 0 d as h Ñ 0. For any y P X we thus obtain ds pT pt ´ sqvpsq|yq “ 0, since v1 “ íHv. Consequently, pT ptqx|yq “ pT ptqvp0q|yq “ pT p0qvptq|yq “ pvptq|yq , which gives uptq “ vptq for all t ě 0. Thus the ‘strongly continuous unitary group’ pT ptqqtPR solves (4.5) uniquely. l CHAPTER 5

Holomorphic functional calculi

We come back to the case of Banach spaces X and Y . We want to introduce functional calculi for non-selfadjoint operators on X, using now complex curve integrals.

5.1. The bounded case

Let U Ď C be open, g : U Ñ Y be holomorphic (i.e., complex differentiable) and Γ Ď U be a piecewise C1 curve with parametrization γ : ra, bs Ñ U. This means that there are a “ a0 ă a1 ă ¨ ¨ ¨ ă am “ b such 1 that γ P C prak´1, aks, Cq for all k P t1, . . . , mu, where γ does not need to be continuous. We write Γ “ Γ1 Y ¨ ¨ ¨ Y Γn with Γk “ γkprak´1, aksq and deﬁne the curve integral

b m ak g dz “ gpγptqqγ1ptq dt :“ gpγptqqγ1ptq dt Γ a k“1 ak´1 ż ż ÿ ż as a Banach space-valued Riemann integral (having the same deﬁnition, results and proofs as for Y “ R in Analysis 1). Using Riemann sums, one easily checks that T Γ g dz “ Γ T g dz for all T P BpY,Zq and Banach spaces Z. The index of a closed curve (i.e., γpaq “ γpbq) is given by ş ş 1 dw npΓ, zq “ 2πi w ´ z żΓ for all z P CzΓ. The index is the number of times that Γ winds around z, counted with orientation. If Γ is closed and npΓ, zq “ 0 for all z R U, then Cauchy’s integral theorem and formula

g dz “ 0, (5.1) żΓ 1 gpwq npΓ, zqgpzq “ dw, (5.2) 2πi w ´ z żΓ are valid for all z P UzΓ. This fact is shown for Y “ C in Theorems IV.5.4 and IV.5.7 of [Co1]. For a general Banach space Y , the formulas (5.1) and (5.2) are thus satisﬁed by functions z ÞÑ xgpzq, y˚y for every y˚ P Y ˚. Hence, ˚ ˚ ˚ x Γ g dz, y y “ 0 for all y P Y so that a corollary of the Hahn–Banach theorem yields (5.1) in Y . Similarly one deduces (5.2). ş For compact, non-empty subsets K Ď C, we introduce the space HpKq “tf :Dpfq Ñ C K Ď Dpfq Ď C, Dpfq is open, f is holomorphicu. Let K Ď U Ď C, K be compact,ˇ and U be open. By Proposition VIII.1.1 in [Co1] and its proof thereˇ exists an admissible curve Γ for K and U (or, in

81 5.1. The bounded case 82

UzK) which means that Γ Ď UzK is piecewise C1, npΓ, zq “ 1 for all z P K and npΓ, zq “ 0 for all z P CzU. Let T P BpXq, f P HpσpT qq, and Γ be admissible for σpT q and Dpfq. We then deﬁne 1 fpT q :“ fpλqRpλ, T q dλ P BpXq. (5.3) 2πi żΓ This integral exists in the Banach space BpXq since λ ÞÑ fpλqRpλ, T q is 1 holomorphic on ρpT q X Dpfq Ě Γ. Writing Rpλ, T q as “ λ´T ” one sees the similarity of (5.3) and (5.2), but here Rpλ, T q does not exist on σpT q, whereas 1 in (5.2) the function w ÞÑ w´z is deﬁned on Cztzu. If Γ1 is another admissible curve for σpT q and Dpfq, then we set Γ2 “ Γ Y p´Γ1q, where “´” denotes the inversion of the orientation. We then have 1 ´ 1 “ 0, z P σpT q, npΓ2, zq “ npΓ, zq ´ npΓ1, zq “ #0 ´ 0 “ 0, z P Cz Dpfq. So we can apply (5.1) on U “ DpfqzσpT q obtaining

0 “ fpλqRpλ, T q dλ “ fpλqRpλ, T q dλ ´ fpλqRpλ, T q dλ. 2 1 żΓ żΓ żΓ Consequently, (5.3) does not depend on the choice of the admissible curve. 1 We recall that rλpzq “ λ´z and p1pzq “ z for λ, z P C with λ ‰ z. Theorem 5.1. Let T P BpXq for a Banach space X. Then the map

ΦT : HpσpT qq Ñ BpXq, f ÞÑ fpT q, deﬁned by (5.3) is linear and satisﬁes

(H1) kfpT qk ď c supλPΓ|fpλq| for a constant c “ cpΓ,T q ą 0, (H2) 1pT q “ I, p1pT q “ T , rλpT q “ Rpλ, T q, (H3) fpT qgpT q “ gpT qfpT q “ pfgqpT q, (H4) fpT q˚ “ fpT ˚q, (H5) if fn Ñ f uniformly on compact subsets of Dpfq, then fnpT q Ñ fpT q in BpXq as n Ñ 8, for all λ P ρpT q and f, g, fn P HpσpT qq with Dpfnq “ Dpfq for every n P N. ΦT is the only linear map from HpσpT qq to BpXq satisfying (H1)–(H3). For a polynomial p, the operators ppT q in (5.3) and in (4.4) coincide. If X 1 is a Hilbert space and T “ T , the above ΦT is the restriction of the map ΦT from Theorem 4.11. Proof. It is clear that f ÞÑ fpT q is linear. Property (H1) follows from 1 kfpT qk ď `pΓq supkRpλ, T qk sup|fpλq| “: cpΓ,T q sup|fpλq|. 2π λPΓ λPΓ λPΓ

Replacing here fpT q by fpT q ´ fnpT q “ pf ´ fnqpT q we also deduce (H5). To check (H4), we recall that σpT q “ σpT ˚q and Rpλ, T q˚ “ Rpλ, T ˚q from Theorem 1.24. Hence, 1 fpT q˚ “ fpλqRpλ, T q˚ dλ “ fpT ˚q. 2πi żΓ We next show (H3). We choose a bounded open set U Ď C such that σpT q Ď U Ď U Ď Dpfq X Dpgq and admissible curves Γf in UzσpT q and 5.1. The bounded case 83

Γg in pDpfq X DpgqqzU. We then have npΓf , µq “ 0 for all µ P Γg Ď CzU and npΓg, λq “ 1 for all λ P Γf Ď U. Using the resolvent equation, Fubini’s theorem in BpXq (see e.g. Theorem X.6.16 in [AE]) and (5.2) in C, we compute 1 1 fpT qgpT q “ fpλqRpλ, T q gpµqRpµ, T q dµ dλ 2πi 2πi żΓf żΓg 1 2 1 “ fpλqgpµq pRpλ, T q ´ Rpµ, T qq dµ dλ 2πi µ ´ λ ˆ ˙ żΓf żΓg 1 1 gpµq “ fpλqRpλ, T q dµ dλ 2πi 2πi µ ´ λ żΓf żΓg 1 1 fpλq ` gpµqRpµ, T q dλ dµ 2πi 2πi λ ´ µ żΓg żΓf 1 “ fpλqgpλqRpλ, T q dλ “ pfgqpT q. 2πi żΓf This identity also yields pfgqpT q “ pgfqpT q “ gpT qfpT q. To check (H2), we take f “ 1 with Dpfq “ C. We choose Γ0 “ BBp0, 2kT kq. Theorem 1.16 then leads to 1 1 8 1pT q “ Rpλ, T q dλ “ T nλń´1 dλ 2πi 2πi Γ0 Γ0 n“0 ż ż ÿ 8 1 “ T n λń´1 dλ “ I, 2πi n“0 Γ0 ÿ ż since the series converges in BpXq uniformly on Γ and λ´m dλ is equal to 0 Γ0 2πi if m “ 1 and equal to 0 for m P zt1u. The property p pT q “ T is shown Z ş 1 similarly. For λ P ρpT q, consider fλpzq “ λ ´ z. The previous results imply fλpT q “ λI ´ T and fλpT qrλpT q “ rλpT qfλpT q “ prλfλqpT q “ 1pT q “ I so that rλpT q “ Rpλ, T q. Let Ψ : HpσpT qq Ñ BpXq be any linear map satisfying the assertions (H1)–(H3). The linearity, (H2) and (H3) imply that Ψppq “ ppT q “ ΦT ppq for every polynomial p. If f P HpσpT qq and Γ Ď DpfqzσpT q is admissible, then there are polynomials pn converging uniformly to f on the compact set Γ. Hence, pnpT q Ñ Ψpfq by (H1) and thus Ψ “ ΦT . The final assertion is shown similarly. d Example 5.2. Let E “ CpKq for a compact set K Ď R and let m P CpKq. We define Mϕ “ mϕ for ϕ P E. Proposition 1.14 shows that 1 M P BpEq, σpMq “ mpKq, and Rpλ, Mqϕ “ λ´m ϕ for all λ P ρpMq. Let f P HpmpKqq, Γ be an admissible curve in DpfqzmpKq, ϕ P E, and x P K. Using that the map ψ ÞÑ ψpxq is continuous and linear from E to C and Cauchy’s formula (5.2) for µ “ mpxq P mpKq, we compute 1 1 pfpMqϕqpxq “ fpλqRpλ, Mqϕ dλpxq “ fpλqpRpλ, Mqϕqpxq dλ 2πi 2πi żΓ żΓ 1 fpλq “ dλ ϕpxq “ fpmpxqqϕpxq. 2πi λ ´ mpxq żΓ 5.1. The bounded case 84

As a result, fpMqϕ “ pf ˝ mqϕ is also a multiplication operator. ♦ Theorem 5.3. Let T P BpXq and f P HpσpT qq. We then have the spectral mapping theorem σpfpT qq “ fpσpT qq. Proof. Let µ R fpσpT qq. After possibly shrinking Dpfq, the map g “ 1 µ1´f belongs to HpσpT qq. The properties of the calculus yield gpT qpµI ´ fpT qq “ pµI ´ fpT qqgpT q “ pgpµ1 ´ fqqpT q “ I, so that µ P ρpfpT qq. fpzq´µ Conversely, let µ “ fpλq for some λ P σpT q. We set gpzq “ z´λ for z P Dpfqztλu and gpλq “ f 1pλq. Since g is bounded, it is holomorphic on Dpfq by Riemann’s theorem on removable singularities. Moreover, gpzqpz ´ λq “ fpzq ´ µ for all z P Dpfq and so pλI ´T qgpT q “ gpT qpλI ´T q “ pgpλ1´p1qqpT q “ pµ1´fqpT q “ µI ´fpT q. Because the operator pλI ´ T q is not surjective or not injective, µI ´ fpT q has the respective properties. Hence, µ is contained in σpfpT qq. The next result can in particular be used to solve linear ordinary differential equations governed by matrices A. tz Example 5.4. Let A P BpXq. For t P R we set ftpzq “ e , z P C, and tA define e “ ftpAq P BpXq. As in Example 4.19, one sees that ept`sqA “ etAesA “ esAetA, e0A “ I, retAs´1 “ e´tA tA 1 for all t, s P R. Moreover, t ÞÑ e belongs to C pR, BpXqq with d tA tA tA dt e “ Ae “ e A. tA 1 Hence, the map uptq “ e u0 is the unique solution in C pR,Xq of d dt uptq “ Auptq, t P R, up0q “ u0, where u0 P X is given. Theorem 5.3 further yields r etA “ max |µ| µ P σpetAq “ etσpAq “ max et Re λ λ P σpAq “ etspAq for` the˘spectral boundˇ spAq :“ maxtRe(λ λ P σp Aqu. Therefore,ˇ ( if spAq ă 0 ˇ ˇ (i.e., σpAq Ď tλ Re λ ă 0u), then we deduce from Theorem 1.16 that ˇ ˇ ˇ1 ą rpeAq “ lim kpeAqnk1{n “ lim kenAk1{n. ˇ nÑ8 nÑ8 NA So we can fix an index N P N with ke k “: q ă 1. Writing any given t ě 0 as t “ kN ` τ for some k P N0 and 0 ă τ ď N we estimate k ln q ketAk “ k eNA eτAk ď qk keτAk ď max keτAk exp Nk ď Me´wt 0ďτďN N ` ln q˘ τA |ln´q| ¯ where w :“ ´ N ą 0 and M :“ max0ďτďN ke k e . So spectral infor- mation on the given operator A implies the exponential decay kuptqk ď Me´wtkxk, t ě 0, of the solutions u. ♦ 5.1. The bounded case 85

Let S P BpXq and P “ P 2 P BpXq be a projection with SP “ PS. Set X1 “ RpP q and X2 “ NpP q. Then, X “ X1 ‘ X2 by e.g. Lemma 2.16 of [Sc2]. Moreover, if y “ P x P RpP q, then Sy “ SP x “ P Sx P RpP q. If x P NpP q, then P Sx “ SP x “ 0 and so Sx P NpP q. As a result, S leaves invariant X1 and X2 and the restrictions S|Xj P BpXjq are well deﬁned.

Theorem 5.5 (Spectral projection). Let T P BpXq and σpT q “ σ1Y9 σ2 for two disjoint closed sets σj ‰H in C. Then there is a projection P P BpXq with fpT qP “ P fpT q for all f P HpσpT qq such that σpTjq “ σj for j P t1, 2u, where Tj “ T|Xj P BpXjq, X1 “ RpP q and X2 “ NpP q. Moreover,

X “ X1 ‘ X2 and Rpλ, Tjq “ Rpλ, T q|Xj for λ P ρpT q “ ρpT1q X ρpT2q. We further have 1 P “ Rpλ, T q dλ, (5.4) 2πi żΓ1 where Γ1 is an admissible curve for σ1 and any open set U1 Ě σ1 such that U1 X σ2 “H.

Proof. There are open sets Uj with U 1 X U 2 “H and σj Ď Uj for j P t1, 2u. Define h P HpσpT qq by h “ 1 on U1 and h “ 0 on U2. We set P “ hpT q P BpXq. We then deduce P 2 “ h2pT q “ hpT q “ P and fpT qP “ P fpT q for all f P HpσpT qq from (H3) and (H2). The above remarks show that X “ X1 ‘ X2 and that the operators Tj “ T|Xj P BpXjq are well defined. The formula (5.4) follows by choosing Γ “ Γ1YΓ2, where Γj are admissible curves for σj and Uj for j P t1, 2u. Let λ R σ1. We may shrink U1 so that λ R U1 since P does not depend on the choice of Γ and thus not on the 1 choice of U1. We define gpzq “ λ´z for z P U1 and gpzq “ 0 for z P U2. Then g P HpσpT qq and

gpT qpλI ´ T q “ pλI ´ T qgpT q “ ppλ1 ´ p1qgqpT q “ hpT q “ P.

Setting R “ gpT q|X1 P BpX1q, we thus obtain

RpλIX1 ´ T1q “ pλIX1 ´ T1qR “ IX1 .

This means that λ P ρpT1q, and so σpT1q Ď σ1. Similarly, one sees that σpT2q Ď σ2. In particular, σpT1q and σpT2q are disjoint. Let λ P ρpT1q X ρpT2q. Given x P X, we have unique x1 P X1 and x2 P X2 such that x “ x1 ` x2. If λx ´ T x “ 0, then 0 “ λx1 ´ T1x1 ` λx2 ´ T2x2 P X1 ‘ X2 so that xj belongs to NpλI ´ Tjq “ t0u for j P t1, 2u; i.e., x “ 0. Given y P X, we deﬁne xj “ Rpλ, Tjqyj P Xj for j P t1, 2u. Setting x “ x1 ` x2, we derive

λx ´ T x “ λx1 ´ T1x1 ` λx2 ´ T2x2 “ y1 ` y2 “ y.

We have proved that λ P ρpT q, Rpλ, T qy “ Rpλ, T1qy1 ` Rpλ, T2qy2,

Rpλ, T q|Xj “ Rpλ, Tjq, and

σpT q “ σ1Y9 σ2 Ď σpT1qY9 σpT2q Ď σ1Y9 σ2, which implies that σpTjq “ σj for j P t1, 2u. We use the above concept to reﬁne the results in Example 5.4 about the long-term behavior of etA. 5.2. Sectorial operators 86

Example 5.6 (Exponential dichotomy). In the setting of Example 5.4, assume that σpAq X iR “H. Hence, σpAq “ σ1Y9 σ2 where σ1 Ď tλ P C Re λ ă 0u and σ2 Ď tλ P C Re λ ą 0u. Let P be the spectral projection of A for σ1 and define A1 and A2 as the restrictions of A to X1 “ RpP q and ˇ ˇ tA tAj X2ˇ “ NpP q, respectively, as inˇ Theorem 5.5. Then pe q|Xj “ e : Xj Ñ Xj and there are constants δ, N ą 0 such that }etA1 } ď Ne´δt and }e´tA2 } ď Ne´δt, t ě 0. In other words, X can be decomposed into etA-invariant subspaces on which etA decays exponentially in forward and in backward time, respectively. Proof. Let Γ1 be given as in Theorem 5.5. For x P X1 we compute 1 etAx “ etAP x “ pf hqpAqx “ etλRpλ, Aqx dλ t 2πi żΓ1 1 “ etλRpλ, A qx dλ “ etA1 x, 2πi 1 żΓ1 tλ where ftpλq “ e for t P R and h is given by the proof of Theorem 5.5. tA tA In the same way one derives e x “ e 2 x for all x P X2 and t ě 0. Since tA σpA1q “ σ1, we have spA1q ă 0 and so Example 5.4 shows that ke x1k ď ´ωt Me kx1k for all t ě 0 and x1 P X1 and some constants M, ω ą 0. Moreover, σpA2q “ σ2 and so spÁ2q ă 0. Note that the curve Γ˜ “ t´λ λ P Γu is admissible for σpÁq “ ´σpAq. Substituting µ “ ´λ, we conclude that ˇ ˇ 1 1 e´tA “ e´tλpλI ´ Aq´1 dλ “ etµpµI ´ pÁqq´1 dµ “ etpÁq 2πi 2πi ˜ żΓ żΓ for all t P R. For x2 P X2 we thus obtain

´tA2 ´tA tp´Aq tp´A2q e x2 “ e x2 “ e x2 “ e x2 ´tA 1 ´ω1t 1 1 so that ke 2 x2k ď M e kx2k for all t ě 0 and some M , ω ą 0. l

5.2. Sectorial operators We extend the above results to certain unbounded operators A, restricting ourselves to the exponential etA.1 For φ P p0, πs we deﬁne the open sector

Σφ “ tλ P Czt0u |arg λ| ă φu.

We also set Σπ{2 “: C` and C´ “ ´C`.ˇ Note that Σπ “ CzR´. ˇ Definition 5.7. A closed operator A is called sectorial of angle φ P p0, πs if there is a constant K ą 0 such that Σφ Ď ρpAq and K kRpλ, Aqk ď , λ P Σ . |λ| φ In the literature several small variations of the above deﬁnition are used. Note that a sectorial operator of angle φ is also sectorial of angle φ1 P p0, φq. In applications often arise operators A such that A´ωI is sectorial for some ω P R, cf. Remark 5.13.

1See [KW] for a detailed study of a corresponding functional calculus. 5.2. Sectorial operators 87

Example 5.8. Let X be a Hilbert space and A be closed, densely deﬁned and selfadjoint on X. We further suppose that σpAq Ď R´. Then A is π sectorial of any angle φ P p 2 , πq. π 1 Proof. Let φ P p 2 , πq and λ P Σφ. Since Rpλ, Aq “ Rpλ, Aq, the operator Rpλ, Aq is normal. Propositions 4.3 and 1.20 then yield 1 , Re λ 0, ´1 ´1 |λ| ě kRpλ, Aqk “ rpRpλ, Aqq “ dpλ, σpAqq ď dpλ, R´q “ 1 # |Im λ| , Re λ ă 0. ˘iθ π If Re λ ă 0, we can write λ “ |λ|e for some θ P p 2 , φq. We then have |Im λ| |λ| “ | sin θ| ě sin φ ą 0, and thus 1 sin φ Kφ kRpλ, Aqk ď “: , λ P Σ . l |λ| |λ| φ

Note that Kφ Ñ 8 as φ Ñ π in the above example. Example 5.9. Let X “ Cpr0, 1sq and Au “ u2 with DpAq “ tu P C2pr0, 1sq up0q “ up1q “ 0u. Then A is sectorial of any angle φ ă π. Proof. We ﬁrst recall from Example 3.40 that σpAq “ σppAq “ 2 2 ˇ 2 t´π k kˇP Nu. Let λ P Σπ and λ “ µ for some µ P C with Re µ ą 0. Let f P X. There exists a unique function u P DpAq and λu ´ Au “ f; i.e., ˇ u Pˇ C2pr0, 1sq, u2 “ µ2u ´ f on r0, 1s, up0q “ up1q “ 0. The solution of this ordinary boundary value problem is given by 1 1 uptq “ aeµt ` be´µt ` e´µ|t´s|fpsq ds, 0 ď t ď 1, 2µ ż0 1 1 up0q “ a ` b ` e´µsfpsq ds “ 0, 2µ ż0 e´µ 1 up1q “ aeµ ` be´µ ` eµsfpsq ds “ 0, 2µ ż0 where the complex numbers a “ apf, µq and b “ bpf, µq still have to be determined. The two boundary conditions above are equivalent to

´µ 1 1 ´µs apf, µq 1 e ´1 ´ 2µ 0 e fpsq ds “ µ ´µ bpf, µq e´µ ´ eµ ´e 1 ´ e 1 eµsfpsq ds ˆ ˙ ˆ ˙ ˜ 2µş 0 ¸ 1 e´µ 1peµs ´ eş´µsqfpsq ds “ 0 . 2µpe´µ ´ eµq 1peµe´µs ´ e´µeµsqfpsq ds ˜ 0 ş ¸ We thus obtain λ P ρpAq and ş 1 1 Rpµ2,Aqfptq “ apf, µqeµt ` bpf, µqe´µt ` e´µ|t´s|fpsq ds 2µ ż0 2 for all µ “ λ P CzR´, Re µ ą 0, f P X and t P r0, 1s. π iθ Fix φ P p 2 , πq. Take λ P Σφ and thus µ P Σ φ . Hence µ “ |µ| e with 2 φ φ 0 ď |θ| ă 2 and Re µ “ |µ| cos θ ě |µ| cos 2 . So we can estimate kfk t }Rpλ, Aqf} ď |apf, µq| eRe µ ` |bpf, µq| ` 8 sup e´ Re µ|r| dr 8 2|µ| tPr0,1s żt´1 5.2. Sectorial operators 88

kfk 1 8 eRe µs e´ Re µs ds ď µ ´µ ` 2|µ| |e ´ e | 0 ´ż 1 ` ˘ kfk ` peRe µe´ Re µs ` e´ Re µeRe µsq ds ` 8 |µ| Re µ ż0 kfk ¯ “ 8 peRe µ ´ 1 ` 1 ´ e´ Re µq 2 Re µ|µ||eµ ´ e´µ| ` kfk ` eRe µp1 ´ e´ Re µq ` e´ Re µpeRe µ ´ 1q ` 8 |µ| Re µ 1 Re µ ´ Re µ Re µ ˘ ´ Re µ cospφ{2q pe ´ e q ` pe ´ e q ď kfk ` 1 |µ|2 8 2peRe µ ´ e´ Re µq 2 ´ ¯ cospφ{2q “ kfk . l |λ| 8

Note that DpAq “ tu P X up0q “ up1q “ 0u is not equal to X for the above ’Dirichlet-Laplacian’, cf. Example 1.19 in [Sc2]. ˇ 2 Similarly one can show thatˇ the ‘Neumann-Laplacian’ A1u “ u with 2 1 1 DpA1q “ tu P C pr0, 1sq u p0q “ u p1q “ 0u is sectorial for every angle φ ă π withˇ on X “ Cpr0, 1sq. Moreover, its ˇ 2 2 spectrum is given by σpA1q “ σppA1q “ t´π k k P N0u with eigenfunctions u ptq “ cospkπtq. (See exercises.) Here DpA1q is dense in X. k ˇ p ˇ Example 5.10. Let X “ L pRq, 1 ď p ă 8, and Au “ u1 for DpAq “ 1,p π W pRq. Then A is sectorial of any angle φ P p0, 2 q. 1 Proof. Example 3.37 says that σpAq “ iR and kRpλ, Aqk ď Re λ for π Re λ ą 0. If φ P p0, 2 q and λ P Σφ, we have |Re λ| ě |λ| cos φ and hence 1{ cos φ kRpλ, Aqk ď . |λ| π Because of its spectrum, A is not sectorial of angle φ “ 2 . l d 2 Let 1 ă p ă 8, U Ď R be open and bounded with BU of class C , and p d X “ L pR q. The operators 2,p 1,p A0 “ ∆, DpA0q “ W pUq X W0 pUq, d 2,p A1 “ ∆, DpA1q “ tu P W pUq Bνu “ νj tr Bju “ 0 on BUu, j“1 ˇ ÿ are sectorial on X with angle φ ą π{2.ˇ Here tr : W 1,ppUq Ñ LppBUq is the trace operator and ν is the outer unit normal. There are variants for the spaces X “ L1pUq and X “ CpUq as well as for more general differential operators and boundary conditions. See e.g. Chapter 3 in [Lu] and also [Ta]. To obtain these results, one has to rely on methods from partial differential equations. One can however define the operators Aj in a more abstract way (in the ‘weak sense’) and show their sectoriality by purely functional analytic methods, cf. [Ou]. In this case, however, the domains are not known explicitly. 5.2. Sectorial operators 89

π Let A be sectorial of angle φ P p 2 , πq with constant K. Take any r ą 0 π and θ P p 2 , φq. We define íθ Γ1 “ tλ “ γ1psq “ p´sqe ´ 8 ă s ď ´ru, iα Γ2 “ tλ “ γ2pαq “ re ´ θˇ ď α ď θu, ˇ iθ Γ3 “ tλ “ γ3psq “ se ˇr ď s ă 8u, ˇ Γ “ Γpr, θq “ Γ1 Y Γ2 ˇY Γ3. ˇ For t ą 0, we introduce the operator 1 1 etA “ etλRpλ, Aq dλ “ lim etλRpλ, Aq dλ, (5.5) 2πi RÑ8 2πi żΓ żΓR where ΓR “ Γ X Bp0,Rq for R ą r. We first have to show that the limit in (5.5) exists in BpXq. Lemma 5.11. Under the above assumptions, the integral in (5.5) converges absolutely in BpXq and gives an operator etA P BpXq which does not depend π tA on the choice of r ą 0 and θ P p 2 , φq. Moreover, ke k ď M for all t ą 0 and a constant M “ MpK, θq ą 0.

K Proof. Since }Rpλ, Aq} ď |λ| on Γ, we can estimate R exppts Re eíθq ketλRpλ, Aqk dλ ď K |eíθ| ds |seíθ| ˇżΓR ˇ żr ˇ ˇ θ iα ˇ ˇ expptr Re e q ˇ ˇ ` K |ireiα| dα |reiα| ż´θ R exppts Re eiθq ` K |eiθ| ds |seiθ| żr 8 ets cos θ θ ď K 2 ds ` etr cos α dα s ˆ żr ż´θ ˙ 8 e´σ dσ ď K 2 p´t cos θq ` 2θetr σ ´t cos θ ˜ żrt|cos θ| ¸ “: Kcpr, t, θq, for all R, t ą 0, where we substituted σ “ ´st cos θ. Thus the limit in (5.5) exists absolutely in BpXq by the majorant criterium, and ketAk ď Kcpr, t, θq. If we take r “ 1{t, then cp1{t, t, θq “: cpθq does not depend on t ą 0. So it remains to check that the integral in (5.5) is independent of r ą 0 π 1 1 1 1 1 and θ P p 2 , φq. To this aim, we define Γ “ Γpr , θ q for some r ą 0 and θ P π 1 1 1 p 2 , φq, where we may assume that θ ě θ. We further set ΓR “ Γ X Bp0,Rq 1 ` ´ and choose R ą r, r . Let CR and CR be the circle arcs from the endpoint 1 1 of ΓR to that of ΓR in tIm λ ą 0u and tIm λ ă 0u, respectively. (If θ “ θ , ˘ ` 1 ´ then CR contain just one point.) Then SR “ ΓR Y CR Y p´ΓRq Y pĆR q is a closed curve in the starshaped domain Σφ. So (5.1) shows that

etλRpλ, Aq dλ “ 0. żSR 5.2. Sectorial operators 90

We further estimate θ1 tλ tR Re eiα K iα 1 tR cos θ e Rpλ, Aq dλ ď e iα |iRe | dα ď Kpθ ´ θqe Ñ 0, C` θ |Re | › ż R › ż › › ´ as› R Ñ 8, and analogously› for CR . So we conclude that etλRpλ, Aq dλ “ lim etλRpλ, Aq dλ “ lim etλRpλ, Aq dλ Γ RÑ8 Γ RÑ8 Γ1 ż ż R ż R tλ “ e Rpλ, Aq dλ. 1 żΓ We next establish some of the fundamental properties of the operators tA tA e . In view of these results one calls pe qtě0 the ‘holomorphic semigroup generated by A’. Actually, the theorem admits a converse. We refer to Section 2.1 of [Lu] for this and other facts. π tA Theorem 5.12. Let A be sectorial of angle φ ą 2 . Deﬁne e as in (5.5) for t ą 0, and set e0A “ I. Then the following assertions hold. (a) etAesA “ esAetA “ ept`sqA for all t, s ě 0. (b) The map t ÞÑ etA belongs to C1pp0, 8q, BpXqq. Moreover, etAX Ď d tA tA tA C DpAq, dt e “ Ae and kAe k ď t for a constant C ą 0 and all t ą 0. We also have AetAx “ etAAx for all x P DpAq and t ě 0. (c) Let x P X. Then etAx converges as t Ñ 0 in X if and only if x P DpAq. In this case, etAx tends to x as t Ñ 0. 1 π 1 Proof. (a) Let t, s ą 0. Take 0 ă r ă r and 2 ă θ ă θ ă φ. Set Γ “ Γpr, θq and Γ1 “ Γpr1, θ1q. Using the resolvent equation and Fubini’s theorem, we compute

tA sA 1 tλ sµ e e “ 2 e e Rpλ, AqRpµ, Aquddµ dλ p2πiq 1 żΓ żΓ 1 1 esµ “ etλRpλ, Aq dµ dλ 2πi 2πi 1 µ ´ λ żΓ żΓ 1 1 etλ ` esµRpµ, Aq dλ dµ. 2πi 1 2πi λ ´ µ żΓ żΓ 1 1 iα 1 Fix λ P Γ and take R ą maxtr, r , |λ|u. We deﬁne CR “ tz “ Re θ ď α ď 2π ´θ1u and S1 “ Γ1 YC1 . Since npS1 , λq “ 1, Cauchy’s formula (5.2) R R R R ˇ yields ˇ 1 esµ dµ “ esλ. 2πi S1 µ ´ λ ż R As in Lemma 5.11, we further compute esµ esµ dµ ÝÑ dµ and Γ1 µ ´ λ Γ1 µ ´ λ ż R ż sµ s Re µ e e 1 2πR dµ ď 2πR sup ď esR cos θ ÝÑ 0 ˇ C1 µ ´ λ ˇ µPC1 |µ ´ λ| R ´ |λ| ˇż R ˇ R ˇ ˇ as R Ñˇ 8. Consequently,ˇ ˇ ˇ 1 esµ esλ “ dµ. 2πi 1 µ ´ λ żΓ 5.2. Sectorial operators 91

iα Closing ΓR with the circle arc CR “ tz “ Re θ ď α ď 2π ´ θu for suﬃciently large R ą r, one veriﬁes in the same way that ˇ eλt ˇ 0 “ dλ. λ ´ µ żΓ We thus conclude that 1 etAesA “ eλtesλRpλ, Aq dλ “ ept`sqA “ esAetA. 2πi żΓ (b) Let x P X, t ą 0, ε ą 0, and R ą r. Observe that the Riemann sums for etλRpλ, Aq dλ converge in rDpAqs since λ ÞÑ Rpλ, Aq is continuous in ΓR BpX, rDpAqsq. We thus obtain ş A eλtRpλ, Aq dλ “ etλARpλ, Aq dλ żΓR żΓR “ eλtλRpλ, Aq dλ ´ etλ dλ I. (5.6) żΓR żΓR iα Take again CR “ tµ “ Re θ ď α ď 2π ´ θu. Using (5.1), one shows as in part (a) that ˇ ˇ etλ dλ “ ´ etλ dλ ď 2πR sup etR cos α ď 2πR eεR cos θ ÝÑ 0 ˇżΓR ˇ ˇ żCR ˇ θďαď2π´θ ˇ ˇ ˇ ˇ asˇ R Ñ 8,ˇ uniformlyˇ for tˇě ε. Moreover, as in the proof of Lemma 5.11 (withˇ r “ 1{ˇt) weˇ estimate ˇ 8 s θ kλetλRpλ, Aqk dλ ď K 2 ets cos θ ds ` recos αdα Γ 1 s ´θ ˇż R ˇ ż t ż ˇ ˇ ´ ¯ ˇ ˇ 2K 2eKθ C1 ˇ ˇ ď ` “: . t|cos θ| t t Therefore, the right hand side of (5.6) converges to

λeλtRpλ, Aq dλ żΓ as R Ñ 8. Since A is closed, it follows that etAX Ď DpAq and 1 C1 AetA “ λeλtRpλ, Aq dλ, kAetAk ď 2πi 2πt żΓ for all t ą 0. In a similar way one sees that 8 2K λetλRpλ, Aq dλ ď 2K ets cos θ ds ď eRε cos θ ÝÑ 0 ε|cos θ| ˇ ΓzΓR ˇ R ˇż ˇ ż ˇ ˇ as RˇÑ 8, uniformly for tˇ ě ε. As a result, ˇ ˇ d λeλtRpλ, Aq dλ “ eλtRpλ, Aq dλ dλ żΓR żΓR converges in BpXq uniformly for t ě ε, and so t ÞÑ etA P BpXq is continuously d tA tA diﬀerentiable for t ą 0 with dt e “ Ae . For x P DpAq, we further obtain 1 AetAx “ lim eλtRpλ, AqAx dλ “ etAAx. RÑ8 2πi żΓR 5.2. Sectorial operators 92

(c) Let x P DpAq, R ą r, and t ą 0. As in part (a), Cauchy’s formula (5.2) yields 1 eλt 1 eλt dλ “ lim dλ “ 1 2πi λ RÑ8 2πi λ ´ 0 żΓ żΓR Observing that λRpλ, Aqx ´ x “ Rpλ, AqAx, we conclude that 1 1 1 eλt etAx ´ x “ eλt Rpλ, Aq ´ x dλ “ Rpλ, AqAx dλ. 2πi λ 2πi λ żΓ żΓ ´ c¯ Since the integrand is bounded by |λ|2 on Γ for all t P p0, 1s, Lebesgue’s convergence theorem implies the existence of the limit 1 1 lim etAx ´ x “ Rpλ, AqAx dλ “: z. tÑ0 2πi λ żΓ iα Let KR “ tRe ´ θ ď α ď θu. Cauchy’s theorem (5.1) shows that 1 ˇ Rpλ, AqAx dλ “ 0. ˇ λ żΓRYp´KRq Since also 1 2πRK Rpλ, AqAx dλ ď kAxk ÝÑ 0 λ R2 ›ż´KR › › › tA as R Ñ 8, we› arrive at z “ 0. Because› of the uniform boundedness of e , it follows that› etAx Ñ x as t Ñ 0 for› all x P DpAq. Conversely, if etAx Ñ y as t Ñ 0, then y P DpAq by part (b). Moreover, Rp1,AqetAx “ etARp1,Aqx tends to Rp1,Aqx as t Ñ 0, since Rp1,Aqx P DpAq. We thus obtain Rp1,Aqy “ Rp1,Aqx, and so x “ y P DpAq. π Remark 5.13. Let A ´ ωI “ Aω be sectorial of angle greater than 2 for some ω P R. We then compute 1 1 eωtetAω “ etpλ`ωqRpλ ` ω, Aq dλ “ eµtRpµ, Aq dµ “: etA 2πi 2πi żΓ żω`Γ for all t ą 0. It is easy to see that etA “ eωtetAω has the analogous properties as in the case ω “ 0. ♦ We can now easily solve the evolution equation (5.7) governed by a sectorial operator A with angle φ ą π{2. One calls such problems ‘of parabolic type’ since diffusion problems are typical applications, see Example 5.15. In contrast to the Schrödingerequation in Example 4.19 we can allow for initial values in DpAq. π Corollary 5.14. Let A be sectorial of angle φ ą 2 and let u0 P DpAq. tA 1 Then uptq “ e u0, t ě 0, is the unique solution in C pp0, 8q,Xq X Cpp0, 8q, rDpAqsq X CpR`,Xq of the initial value problem 1 u ptq “ Auptq, t ą 0, up0q “ u0. (5.7) Proof. Existence follows from Theorem 5.12 and Remark 5.13. Let v be another solution of (5.7). Let 0 ă ε ď s ď t ´ ε ă t. Theorem 5.12 then implies that d pt´sqA pt´sqA pt´sqA 1 ds e vpsq “ é Avpsq ` e v psq “ 0. 5.2. Sectorial operators 93

As in Example 4.19, this fact yields ept´εqAvpεq “ eεAvpt´εq. Letting ε Ñ 0, tA τA one obtains that e u0 “ vptq since τ ÞÑ e x is continuous for τ ě 0 and each x P DpAq. We only give one of the possible examples. Example 5.15. Let X “ Cpr0, 1sq and Aϕ “ ϕ2 with DpAq “ tϕ P 2 C pr0, 1sq ϕp0q “ ϕp1q “ 0u. Let u0 P C0p0, 1q “ DpAq. Then the function uptq “ etAu , t ě 0, belongs to ˇ 0 ˇ 2 1 CpR`,Cpr0, 1sqq X Cp0, 8q,C pr0, 1sqq X C pp0, 8q,Cpr0, 1sqq and uniquely solves the partial diﬀerential equation

Btupt, xq “ Bxxupt, xq, t ą 0, x P r0, 1s, upt, 0q “ upt, 1q “ 0, t ą 0, (5.8) up0, xq “ u0pxq, x P r0, 1s. ♦ The formula (5.5) and Theorem 5.12 allow to establish a theory for parabolic problems which is similar to the case of ordinary diﬀerential equations, see e.g. [He] and [Lu]. We only add a basic result on the longterm behavior that extends Example 5.4. The next theorem can be applied to (5.8) thanks to Example 5.9. Theorem 5.16. Let A be sectorial of angle φ ą π{2 and spAq “ suptRe λ λ P σpAqu ă ´δ ă 0. Then there is a constant N ě 1 such that }etA} ď Ne´δt for all t ě 0.2 ˇ ˇ Proof. The assumptions imply that A´δ “ A ` δI is sectorial of some angle ψ P pπ{2, φq. Take Γ “ Γpr, θq with r ą 0 and θ P pπ{2, ψq. Re- mark 5.13 then yields that etA “ e´δtetA´δ , where however etA is deﬁned by the curve integral (5.5) on the shifted path Γ1 :“ ´δ `Γ. Lemma 5.11 shows that etA´δ is uniformly bounded for t ě 0. It thus remains to verify that

etµRpµ, Aq dµ “ etλRpλ, Aq dλ, t ą 0. (5.9) ż´δ`Γ żΓ ˘ To this end, let R ą r and SR be the horizontal line segments connecting 1 the end points on ΓR and ΓR in tIm λ ą 0u and tIm λ ă 0u, respectively. ` 1 ´ Let CR “ ΓR Y SR Y p´Γ q Y p´SR q. This path is contained in ρpAq and npCR, zq “ 0 for all z P σpAq. Cauchy’s theorem (5.1) now implies

etλRpλ, Aq dλ “ 0. żCR ˘ Observe that the segments SR have ﬁxed length δ and that Re λ ď R cos θ ă ˘ ˘ 0 and |λ| ě R for all λ P SR . Because of θ ă φ, the sets SR belong to Σφ for all suﬃciently large R. We can thus estimate δK etλRpλ, Aq dλ ď eRt cos θ. S˘ R › ż R › › › Since the right-hand› side vanishes as R›Ñ 8, we have shown (5.9).

2Not shown in the lectures. Bibliography

[AF] R.A. Adams and J.F. Fournier, Sobolev Spaces. 2nd edition. Academic Press, Am- sterdam, 2007. [AE] H. Amann and J. Escher, Analysis III. Birkhäuser,Basel, 2008. [Co1] J.B. Conway, Functions of One Complex Variable. 2nd edition. Springer–Verlag, Heidelberg, 1978. [Co2] J.B. Conway, A Course in Functional Analysis. 2nd edition. Springer–Verlag, New York, 1990. [DS1] N. Dunford and J.T. Schwartz, Linear Operators, Part 1, General Theory. Wiley- Interscience, New York, 1958. [DS2] N. Dunford and J.T. Schwartz, Linear Operators, Part 2, Spectral Theory, Self Adjoint Operators in Hilbert Space. Wiley-Interscience, New York, 1963. [Ev] L.C. Evans, Partial Differential Equations. 2nd edition. Amercian Mathematical Society, Providence (RI), 2010. [GT] D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order. 2nd edition. Springer–Verlag, Berlin, 1998. [He] D. Henry, Geometric Theory of Semilinear Parabolic Equations. Springer–Verlag, New York, 1981. [Ke] O.D. Kellogg, Foundations of Potential Theory. Springer–Verlag, Berlin, 1967. [KW] P.C. Kunstmann and L. Weis, Maxial Lp–Regularity for Parabolic Problems, Fourier Multiplier Theorems and H8–Calculus, in: M. Iannelli, R. Nagel and S. Piazzera (Eds.): ‘Functional Analytic Methods for Evolution Equations’, Springer-Verlag, Berlin 2004, pp. 65–311. [La] S. Lang, Algebra. Revised 3rd Edition. Springer-Verlag, 2002. [Lu] A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems. Birkhäuser,Basel, 1995. [Ou] E,M Ouhabaz, Analysis of Heat Equations on Domains, Princeton University Press, 2005. [PW] M.H. Protter and H.F. Weinberger, Maximum Principles in Differential Equations. Corrected reprint of the 2nd printing, Springer-Verlag, New York, 1984. [RS] M. Reed and B. Simon, Fourier Analysis, Self-Adjointness. Academic Press, New York, 1975. [RR] M. Renardy and R.C. Rogers, An Introduction to Partial Differential Equations. Springer-Verlag, New York, 2004. [Ru1] W. Rudin, Real and Complex Analysis. McGraw-Hill, Singapore, 1987. [Ru2] W. Rudin, Functional Analysis. McGraw-Hill, Boston, 1991. [Sa] B. Sandstede, Stability of travelling waves. In B. Fiedler (Ed.): “Handbook of Dy- namical Systems, Vol. 2,” North-Holland, Amsterdam, 2002, pp. 983 - 1055. [Sc1] R. Schnaubelt, Functional Analysis. Class Notes, Karlsruhe, 2015. [Sc2] R. Schnaubelt, Functional Analysis. Class Notes, Karlsruhe, 2018. [Sc3] R. Schnaubelt, Nonlinear Evolution Equations. Class Notes, Karlsruhe, 2019. [St] E.M. Stein, Singular Integrals and Differentiability Properties of Functions. Prince- ton University Press, 1970. [Ta] H. Tanabe, Functional Analytic Methods for Partial Differential Equations. Dekker, New York, 1997. [Tr] H. Triebel, Higher Analysis. Huthig Pub Ltd, 1997. [We] D. Werner, Funktionalanalysis. 5te erweiterte Auflage. Springer-Verlag, Berlin, 2005.