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Lecture Notes Spectral Theory Lecture Notes Spectral Theory 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 “ supkxk¤1kT 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 ¥ junp0qj “ 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 ¤ t ¤ 1{n; fnptq “ 2 #pt ´ 1{nq ; 1{n ¤ t ¤ 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 ¤ p ¤ 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 ¤ t ¤ 1{n; fnptq “ #0; 1{n ¤ t ¤ 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 ¤ c kxk for some c ¥ 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 ¤ 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 ¡ 0 such that kAxkY ¤ kxkA ¤ 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 ¤ 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.
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