Spectral Theory in Hilbert Space

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44 7. THE SPECTRAL THEOREM as a closed, densely defined operator. We need only show that this inverse is bounded, to see that its domain is all of H so that λ ∈ ρ(T ). But (T − λ)u2 = ∞ −∞ (t − λ)2 d〈Etu, u〉 ≥ ε2 ∞ −∞d〈Etu, u〉 = ε2u2 so the inverse of T − λ is bounded by 1/ε. Conversely, assume that Et is not constant near λ. Then there are arbitrarily short intervals ∆ containing λ such that E∆ = 0, i.e., there are non-zero vectors u such that E∆u = u. But then (T − λ)u ≤ |∆|u, where |∆| is the length of ∆. Hence we can find a sequence of unit vectors uj, j = 1, 2, . . . for which (T − λ)uj → 0 (a ‘singular sequence’). Consequently either T −λ is not injective, or else the inverse is unbounded so λ /∈ ρ(T ). Exercises for Chapter 7 Exercise 7.1. Suppose σ is increasing and ∞ −λ ∞ −∞ dσ(t) t−λ → ∞ −∞ the origin. In particular, ∞ −∞ −∞ dσ < ∞. Show that dσ as λ → ∞ along any non-real ray originating in dσ(t) t−λ → 0. Exercise 7.2. Suppose B(·, ·) is a sesqui-linear form on a complex linear space. Show the polarization identity B(u, v) = 1 3 i 4 k B(u + i k v, u + i k v) . k=0 Exercise 7.3. Show that if T is a closed operator on H and S is bounded and everywhere defined, then T S, but not necessarily ST , is closed. Exercise 7.4. Show that if T is selfadjoint and f is a continuous function defined on σ(T ), then f(T ) = ∞ −∞f(t) dEt defines a densely defined operator, which is bounded if f is and selfadjoint if f is realvalued. Also show that (f(T )) ∗ = f(T ), that (f(T )) ∗ has the same domain as f(T ) and commutes with it in a reasonable sense, and that fg(T ) = f(T )g(T ). This is the functional calculus for a selfadjoint operator, and also makes sense for arbitrary Borel functions. The integral is made sense of in the same way as in the statement of the spectral theorem. Exercise 7.5. Let T be selfadjoint and put H(t) = e −itT , t ∈ R, the exponential being defined as in the previous exercise. Show that H(t + s) = H(t)H(s) for real t and s (a group of operators), that H(t) is unitary and that if u0 ∈ D(T ), then u(t) = H(t)u0 solves the Schrödinger equation T u = iu ′ t with initial data u(0) = u0. Similarly, if T ≥ 0 and t ≥ 0, show that K(t) = e −tT is selfadjoint and bounded, that K(t + s) = K(t)K(s) for s ≥ 0 and t ≥ 0 (a semigroup of operators) and that if u0 ∈ H then u(t) = K(t)u0 solves the heat equation T u = u ′ t for t > 0 with initial data u(0) = u0.
CHAPTER 8 Compactness If a selfadjoint operator T has a complete orthonormal sequence of eigen-vectors e1, e2, . . . , then for any f ∈ H we have f = ˆ fjej where ˆfj = 〈f, ej〉 are the generalized Fourier coefficients; we have a generalized Fourier series. However, σp(T ) can still be very complicated; it may for example be dense in R (so that σ(T ) = R), and each eigenvalue can have infinite multiplicity. We have a considerably simpler situation, more similar to the case of the classical Fourier series, if the resolvent is compact. Definition 8.1. • A subset of a Hilbert space is called precompact (or relatively compact) if every sequence of points in the set has a strongly convergent subsequence. • An operator A : H1 → H2 is called compact if it maps bounded sets into precompact ones. Note that in an infinite dimensional space it is not enough for a set to be bounded (or even closed and bounded) for it to be precompact. For example, the closed unit sphere is closed and bounded, and it contains an orthonormal sequence. But no orthonormal sequence has a strongly convergent subsequence! The second point means that if {uj} ∞ 1 is a bounded sequence in H1, then {Auj} ∞ 1 has a subsequence which converges strongly in H2. Theorem 8.2. (1) The operator A is compact if and only if every weakly convergent sequence is mapped onto a strongly convergent sequence. Equivalently, if uj ⇀ 0 implies that Auj → 0. (2) If A : H1 → H2 is compact and B : H3 → H1 bounded, then AB is compact. (3) If A : H1 → H2 is compact and B : H2 → H3 bounded, then BA is compact. (4) If A : H1 → H2 is compact, then so is A ∗ : H2 → H1. Proof. If uj ⇀ u then uj − u ⇀ 0, and if A(uj − u) → 0 then Auj → Au. Thus the last statement of (1) is obvious. By Theorem 3.9 every bounded sequence has a weakly convergent subsequence, so if A maps weakly convergent sequences into strongly convergent ones, then A is compact. Conversely, suppose uj ⇀ u and A is compact. Since 45
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
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Page 57 and 58: CHAPTER 9 Extension theory We will
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Page 63: EXERCISES FOR CHAPTER 9 57 Exercise
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Page 89 and 90: CHAPTER 12 Inverse spectral theory
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Page 93 and 94: 2. UNIQUENESS THEOREMS 87 hand, the
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13. FIRST ORDER SYSTEMS 95 and (u2,
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13. FIRST ORDER SYSTEMS 97 a non-ze
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EXERCISES FOR CHAPTER 13 99 Exercis
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102 14. EIGENFUNCTION EXPANSIONS We
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104 14. EIGENFUNCTION EXPANSIONS Ex
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106 15. SINGULAR PROBLEMS We obtain
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108 15. SINGULAR PROBLEMS We will g
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110 15. SINGULAR PROBLEMS Proof. Le
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112 15. SINGULAR PROBLEMS in the pr
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114 15. SINGULAR PROBLEMS Since vK
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APPENDIX A Functional analysis In t
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A. FUNCTIONAL ANALYSIS 119 other wo
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122 B. STIELTJES INTEGRALS (4) C (5
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124 B. STIELTJES INTEGRALS Definiti
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126 B. STIELTJES INTEGRALS We obtai
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APPENDIX C Linear first order syste
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C. LINEAR FIRST ORDER SYSTEMS 131 s
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Spectral Theory in Hilbert Space

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