Spectral Theory in Hilbert Space

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98 13. FIRST ORDER SYSTEMS Proof. Let [c, d] be a compact subinterval of I = (a, b) such that d c F ∗ (·, λ)W F (·, λ) is invertible and put v1 = v in (a, c] and v1 ≡ 0 in [d, b). Now let u1(x) = F (x, λ)(u(c) + J −1 x F ∗ (y, λ)W (y)v1(y) dy) . It is clear that u1 = u in (a, c] and if we choose v1 appropriately in [c, d] we can achieve that u1 ≡ 0 in [d, b). In fact, setting v(x) = −F (x, λ)( d c F ∗ (·, λ)W F (·, λ)) −1 Ju(c) in this interval will do. It follows that (u − u1, v − v1) ∈ T1 is 0 near a and equals (u, v) near b. We can therefore find a maximal non-degenerate space for Ba consisting of elements of T1 vanishing near b. Similarly, a maximal nondegenerate space for Bb consisting of elements of T1 vanishing near a. Thus Ba and Bb are independent, as claimed. Since the signature of the complete boundary form Bb − Ba is (n+, n−) the independence of Ba and Bb implies that n+ = r a − + r b + and n− = r a + + r b −, using the notation introduced above for the signatures of Ba and Bb. According to Corollary 9.15 T1 has selfadjoint restrictions precisely if n+ = n−. Reasoning like in the regular case it follows that there are selfadjoint restrictions defined by separated boundary conditions precisely if r a + = r a − and r b + = r b −, from which n+ = n− follows. In fact, from any two of these relations the third clearly follows. Consider finally the case when a is a regular endpoint but b possibly is singular. In this case Ba is given by Ba(u1, u2) = iu2(a) ∗ Ju1(a), with notation as above. Clearly r a + is the number of positive eigenvalues of iJ and r a − the number of negative eigenvalues. It follows that selfadjoint restrictions of T1 defined by separated boundary conditions exist if and only if the deficiency indices are equal and iJ has an equal number of positive and negative eigenvalues; in particular n must be even. In the Sturm-Liouville case all these conditions are fulfilled, as we already know. Exercises for Chapter 13 Exercise 13.1. Show that u and ũ are elements of the same equivalence class in L2 W if and only if W u = W ũ a.e. Exercise 13.2. Verify that T0 is the graph of an operator if (13.1) is equivalent to an equation of the type (10.1) (or more generally an equation of the type discussed in Exercise 13.3) and w > 0 a.e. in I. Also show that in this case Assumption 13.7 holds. Try to show this assuming only that w ≥ 0 but w > 0 on a subset of I of positive measure (this is considerably harder). c
EXERCISES FOR CHAPTER 13 99 Exercise 13.3. Show that the differential equation iu ′′′ = wv (here i = √ −1) can be written on the form (13.1). Also show that the equation m k=0 (pku (k) ) (k) = wv can be written on this form if the coefficients w and p0, p1, . . . , pm satisfy appropriate conditions (state these conditions!). Hint: Put U = in the first case. In the second case, let U be u hu ′ hu ′′ the matrix with 2m rows u0, . . . , u2m−1 where uj = u (j) and um+j = (−1) j m k=j+1 (pku (k) ) (k−j−1) for j = 0, . . . , m − 1. Exercise 13.4. Find all selfadjoint realizations of a regular Sturm- Liouville equation. More generally, assume J −1 = J ∗ = −J and show that the eigen-values of B are ±1, both with multiplicity n. Then describe all maximal isotropic spaces for B. Exercise 13.5. Suppose B is a Hermitian form of finite rank on a Hilbert space L, and that B is non-degenerate on a subspace M. Show that for any u ∈ L there is a unique v ∈ M, the B-projection on M, such that B(u − v, M) = 0. Also show that if, and only if, M is maximal non-degenerate, then B(u − v, L) = 0. Exercise 13.6. Suppose B1, B2, . . . is a sequence of Hermitian forms on L with finite rank, all of signature (r+, r−), and suppose Bj(u, v) → B(u, v) as j → ∞, for any u, v ∈ L. Show that B is a Hermitian form on L of finite rank (s+, s−), where s+ ≤ r+ and s− ≤ r−.
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Spectral Theory in Hilbert Space Le
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Preface The aim of these notes is t
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iv CONTENTS Exercises for Chapter 1
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2 0. INTRODUCTION cos( √ λ t), i
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4 0. INTRODUCTION • Can the equat
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6 1. LINEAR SPACES of u so one may
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8 1. LINEAR SPACES Exercises for Ch
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10 2. SPACES WITH SCALAR PRODUCT on
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12 2. SPACES WITH SCALAR PRODUCT Pr
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14 2. SPACES WITH SCALAR PRODUCT Ex
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16 3. HILBERT SPACE Given a normed
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18 3. HILBERT SPACE Two subspaces M
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20 3. HILBERT SPACE ˆvn = limj→
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CHAPTER 4 Operators A bounded linea
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4. OPERATORS 25 Proposition 4.2. A
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4. OPERATORS 27 In the rest of this
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4. OPERATORS 29 must both be 0. Hen
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CHAPTER 5 Resolvents We now conside
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EXERCISES FOR CHAPTER 5 33 Analytic
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36 6. NEVANLINNA FUNCTIONS However,
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38 6. NEVANLINNA FUNCTIONS Proof. B
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40 7. THE SPECTRAL THEOREM function
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42 7. THE SPECTRAL THEOREM as The r
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44 7. THE SPECTRAL THEOREM as a clo
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46 8. COMPACTNESS weakly convergent
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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 97 and 98: CHAPTER 13 First order systems We s
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Page 123 and 124: APPENDIX A Functional analysis In t
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Page 135 and 136: APPENDIX C Linear first order syste
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