Spectral Theory in Hilbert Space

More documents

Recommendations

Info

48 8. COMPACTNESS only if g ∈ L 2 (Ω, w) ⊗ L 2 (Ω, w), i.e., if and only if Ω×Ω |g(x, y)| 2 w(x)w(y) dx dy < ∞. Proof. Let {ej} ∞ 1 be a complete orthonormal sequence in the space L 2 (Ω, w). For fixed x ∈ Ω we may view Aej(x) as the j:th Fourier coefficient of g(x, ·) so Parseval’s formula gives |Aej(x)| 2 = Ω |g(x, y)|2 w(y) dy for a.a x ∈ Ω. By monotone convergence the product of this function by w is in L 1 (Ω) if and only if the Hilbert-Schmidt norm of A is finite. The theorem now follows by an application of Tonelli’s theorem (i.e., a positive, measurable function is integrable over Ω × Ω if and only if the iterated integral is finite). Example 8.8. Consider the operator T in L 2 (−π, π) with domain D(T ), consisting of those absolutely continuous functions u with deriva- tive in L 2 (−π, π) for which u(π) = u(−π), and given by T u = −i du dx (cf. Example 4.8). This operator is self-adjoint and its resolvent is given by Rλu(x) = π g(x, y, λ)u(y) dy where Green’s function g(x, y, λ) is −π given by ⎧ ⎪⎨ − g(x, y, λ) = ⎪⎩ e−iλπ 2 sin λπ eiλ(x−y) y < x, − eiλπ 2 sin λπ eiλ(x−y) y > x. The reader should verify this! Since |g(x, y, λ)| 2 dx dy < ∞ for noninteger λ the resolvent is a Hilbert-Schmidt operator, so it is compact. Now consider the operator of Example 4.6. Green’s function is now only defined for non-real λ and given by Im λ i |Im λ| (8.2) g(x, y, λ) = eiλ(x−y) if (x − y) Im λ > 0, 0 otherwise. The reader should verify this as well! In this case there is no value of λ for which g(·, ·, λ) ∈ L 2 (R 2 ) so the resolvent is not a Hilbert-Schmidt operator.
EXERCISES FOR CHAPTER 8 49 Exercises for Chapter 8 Exercise 8.1. Prove Theorem 8.2(2)–(4). Exercise 8.2. Show that if ∆1 and ∆2 are disjoint intervals and {Et}t∈R a resolution of the identity, then the ranges of E∆1 and E∆2 are orthogonal. Generalize to the case when ∆1 and ∆2 are arbitrary Borel sets in R. Exercise 8.3. Show the converse of Theorem 8.3, i.e., if the spectrum consists of isolated eigen-values of finite multiplicity, then the resolvent is compact. Hint: Let λ1, λ2, . . . be the eigenvalues ordered by increasing absolute value and repeated according to multiplicity and let the corresponding normalized eigen-vectors be e1, e2, . . . . Show that Rλu2 = |〈u,ej〉| 2 |λ−λj| 2 and use this to see that Rλuk → 0 if uk ⇀ 0. Exercise 8.4. Prove the last statement of Lemma 8.6. Exercise 8.5. Verify all claims made in Example 8.8. Exercise 8.6. Let T be a selfadjoint operator. Show that if the resolvent Rλ of T is a Hilbert-Schmidt operator and λj, j = 1, 2, . . . are the non-zero eigenvalues of T , then ∞ j=1 λ−2 j < ∞.
Page 1 and 2:
Spectral Theory in Hilbert Space Le
Page 3: Preface The aim of these notes is t
Page 6 and 7: iv CONTENTS Exercises for Chapter 1
Page 8 and 9: 2 0. INTRODUCTION cos( √ λ t), i
Page 10 and 11: 4 0. INTRODUCTION • Can the equat
Page 12 and 13: 6 1. LINEAR SPACES of u so one may
Page 14 and 15: 8 1. LINEAR SPACES Exercises for Ch
Page 16 and 17: 10 2. SPACES WITH SCALAR PRODUCT on
Page 18 and 19: 12 2. SPACES WITH SCALAR PRODUCT Pr
Page 20 and 21: 14 2. SPACES WITH SCALAR PRODUCT Ex
Page 22 and 23: 16 3. HILBERT SPACE Given a normed
Page 24 and 25: 18 3. HILBERT SPACE Two subspaces M
Page 26 and 27: 20 3. HILBERT SPACE ˆvn = limj→
Page 29 and 30: CHAPTER 4 Operators A bounded linea
Page 31 and 32: 4. OPERATORS 25 Proposition 4.2. A
Page 33 and 34: 4. OPERATORS 27 In the rest of this
Page 35 and 36: 4. OPERATORS 29 must both be 0. Hen
Page 37 and 38: CHAPTER 5 Resolvents We now conside
Page 39: EXERCISES FOR CHAPTER 5 33 Analytic
Page 42 and 43: 36 6. NEVANLINNA FUNCTIONS However,
Page 44 and 45: 38 6. NEVANLINNA FUNCTIONS Proof. B
Page 46 and 47: 40 7. THE SPECTRAL THEOREM function
Page 48 and 49: 42 7. THE SPECTRAL THEOREM as The r
Page 50 and 51: 44 7. THE SPECTRAL THEOREM as a clo
Page 52 and 53: 46 8. COMPACTNESS weakly convergent
Page 57 and 58: CHAPTER 9 Extension theory We will
Page 59 and 60: 2. SYMMETRIC RELATIONS 53 We will n
Page 61 and 62: 2. SYMMETRIC RELATIONS 55 then it i
Page 63: EXERCISES FOR CHAPTER 9 57 Exercise
Page 66 and 67: 60 10. BOUNDARY CONDITIONS the same
Page 68 and 69: 62 10. BOUNDARY CONDITIONS locally
Page 70 and 71: 64 10. BOUNDARY CONDITIONS of a sym
Page 72 and 73: 66 10. BOUNDARY CONDITIONS uniforml
Page 74 and 75: 68 10. BOUNDARY CONDITIONS Hint: If
Page 76 and 77: 70 11. STURM-LIOUVILLE EQUATIONS fu
Page 78 and 79: 72 11. STURM-LIOUVILLE EQUATIONS ob
Page 80 and 81: 74 11. STURM-LIOUVILLE EQUATIONS Ta
Page 82 and 83: 76 11. STURM-LIOUVILLE EQUATIONS Pr
Page 84 and 85: 78 11. STURM-LIOUVILLE EQUATIONS No
Page 86 and 87: 80 11. STURM-LIOUVILLE EQUATIONS Th
Page 89 and 90: CHAPTER 12 Inverse spectral theory
Page 91 and 92: 1. ASYMPTOTICS OF THE m-FUNCTION 85
Page 93 and 94: 2. UNIQUENESS THEOREMS 87 hand, the
Page 95 and 96: 2. UNIQUENESS THEOREMS 89 the bound
Page 97 and 98: CHAPTER 13 First order systems We s
Page 99 and 100: u(c) = 0 is given by (13.2) u(x) =
Page 101 and 102: 13. FIRST ORDER SYSTEMS 95 and (u2,
Page 103 and 104: 13. FIRST ORDER SYSTEMS 97 a non-ze
Page 105:
EXERCISES FOR CHAPTER 13 99 Exercis
Page 108 and 109:
102 14. EIGENFUNCTION EXPANSIONS We
Page 110 and 111:
104 14. EIGENFUNCTION EXPANSIONS Ex
Page 112 and 113:
106 15. SINGULAR PROBLEMS We obtain
Page 114 and 115:
108 15. SINGULAR PROBLEMS We will g
Page 116 and 117:
110 15. SINGULAR PROBLEMS Proof. Le
Page 118 and 119:
112 15. SINGULAR PROBLEMS in the pr
Page 120 and 121:
114 15. SINGULAR PROBLEMS Since vK
Page 123 and 124:
APPENDIX A Functional analysis In t
Page 125:
A. FUNCTIONAL ANALYSIS 119 other wo
Page 128 and 129:
122 B. STIELTJES INTEGRALS (4) C (5
Page 130 and 131:
124 B. STIELTJES INTEGRALS Definiti
Page 132 and 133:
126 B. STIELTJES INTEGRALS We obtai
Page 135 and 136:
APPENDIX C Linear first order syste
Page 137:
C. LINEAR FIRST ORDER SYSTEMS 131 s
show all

Spectral Theory in Hilbert Space

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?