Generalized Eigenfunction Expansions

Home , Eigenfunction, Rigged Hilbert space

Universitat¨ Karlsruhe (TH) FakultätfürMathematik Institut fürAnalysis Arbeitsgruppe Funktionalanalysis

Diploma Thesis of Emin Karayel

supervised by Prof. Dr. Lutz Weis

August 2009 Erklärung Hiermit erkläreich, dass ich diese Diplomarbeit selbständigverfasst habe und keine anderen als die angegebenen Quellen und Hilfsmittel verwendet habe.

Emin Karayel, August 2009

Anschrift Emin Karayel Werner-Siemens-Str. 21 75173 Pforzheim Matr.-Nr.: 1061189 E-Mail: [email protected]

2 Contents

1 Introduction 5

2 Eigenfunction expansions on Hilbert spaces 9 2.1 Spectral Measures ...... 9 2.2 Phillips Theorem ...... 11 2.3 Rigged Hilbert Spaces ...... 12 2.4 Basic Eigenfunction Expansion ...... 13 2.5 Extending the Operator ...... 15 2.6 Orthogonalizing Generalized Eigenvectors ...... 16 2.7 Summary ...... 21

3 Hilbert-Schmidt-Riggings 23 3.1 Unconditional Hilbert-Schmidt Riggings ...... 23 3.1.1 Hilbert-Schmidt-Integral Operators ...... 24 3.2 Weighted Spaces and Sobolev Spaces ...... 25 3.3 Conditional Hilbert-Schmidt Riggings ...... 26 3.4 Eigenfunction Expansion for the Laplacian ...... 27 3.4.1 Optimality ...... 27 3.5 Nuclear Riggings ...... 28

4 Operator Relations 31 4.1 Vector valued multiplication operators ...... 31 4.2 Preliminaries ...... 33 4.3 Interpolation spaces ...... 37 4.4 Eigenfunction expansion ...... 38 4.5 Semigroups ...... 41 4.6 Expansion for Homogeneous Elliptic Selfadjoint Diﬀerential Oper- ators ...... 42 4.7 Conclusions ...... 44

5 Appendix 47 5.1 Acknowledgements ...... 47 5.2 Notation ...... 47 5.3 Conventions ...... 49

1 Introduction

Eigenvectors and eigenvalues are indispensable tools to investigate (at least compact) normal linear operators. They provide a diagonalization of the operator i.e. X Ax = λkhx, ekiek (1.1) k from there it is possible to form the inverse, build a functional calculus, analyze the stability of the operator in numeric situations, determine whether the operator is positive or negative definite etc. They play a role even in non-linear situations like the classification of quadratic forms. In quantum mechanics they correspond to the stationary states of the system. For normal operators with a continuous spectrum there is a diagonalization using a projection valued spectral measure Eλ but the eigenvectors do not exist in the space the operator is defined in. Hence it is necessary to consider larger spaces and to replace the sum in (1.1) with an integral (since the spectrum of the operator is not discrete). Thus the generalization of the above formula would look like: Z Ax = λhx, ek(λ)iek(λ)dλ (1.2) σ(A) − where the vectors ek(λ) are defined in a generalized space X (implying that the vectors x must be in a smaller space X+ in the above formula). An idea which may already have been anticipated by Dirac in his bra and ket notation and for which a rigorous mathematical foundation has been developed in the 1950’s by Y. Berezansky, L. Garding, I. Gelfand, G. Kats, A. Kostyuchenko and K. Maurin.1 The framework in which this is possible is the Rigged Hilbert space (also called Gelfand triplet):2

X+ ⊂ H ⊂ X− where X− is the space of continuous antilinear functionals on X+. Several conditions on the rigging have been established that lead to an eigenfunction expansion (1.2), the most basic one beeing that the embedding X+ into H is Hilbert-Schmidt, provided that X+ itself is a Hilbert space. If X+ is only a Banach space, the nu- clearity of the embedding is also a suﬃcient condition. In [BSU96] and [PSW89] also conditional riggings3 are considered, which require for a given operator A only

1[de 05], [BSU96], [Mau68] 2Usually denoted by Φ ⊂ H ⊂ Φ×. 3In [BSU96] they are called non-nuclear riggings.

5 1 Introduction that f(A): X+ → H is Hilbert-Schmidt, provided that f is bounded away from zero. There is an explicit eigenfunction expansion for the Laplace operator A := −∆ n n on the euclidean space L2(R ) which implies that for any x ∈ L2(R ) there are locally integrable functionsx ˆ(λ) s.t.

Z ∞ f(A)x = f(λ)ˆx(λ) dλ 0 where the generalized eigenfunctionsx ˆ(λ) are in the weighted spaces

n n s n L2,s(R ) := {f ∈ L1,loc(R ) | ((1 + k · k ) f(·)) ∈ L2(R )}

−1 for s > 2 . If the function f itself is in L2(R) then the above is a Bochner integral. At first sight there is no reason to expect that this result4 does not fit into the general framework above. However we discovered that - this is not the case, in fact the only map n f(A): L ( n) → L ( n) for s < 2,s R 2 R 2 n−1 which is Hilbert-Schmidt is the zero map. To close this gap (of 2 ) we have investigated a different approach, based on an operator relation of the following form BA − AB = h(A) leading to an eigenfunction expansion on riggings:5

+ × − X ⊂ [D(B); H]1/2,1 ⊂ H ⊂ [D(B); H]1/2,1 ⊂ X where X+ and (its antidual) X− are reflexive Banach spaces. In the case of homogeneous self-adjoint elliptic differential operators (e.g. the Laplace operator) this leads to an eigenfunction expansion on the rigging L2,s ⊂ H ⊂ L2,−s for all 1 6 s > 2 . The common denominator between the latter result and the classic results for (conditional) Hilbert-Schmidt and nuclear riggings is the existence of a σ− finite measure s.t. the following estimate holds

2 2 kχE(A)xkH ≤ µ(E)kxk+

For example in the case of Hilbert-Schmidt-Riggings the measure can be deﬁned by µ(E) = hχE(A)xk, xki

4It can be derived from the Plancharel formula in [Str89][p. 64]. 5Theorem 4.4.4 6Theorem 4.6.3

6 + where xk is an orthonormal system of X . On the other in the case of the operator relation Z µ(E) = |h−1(λ)||2 + h0(λ)|dλ. E The latter means that (outside of the zeros of h) the measure µ and hence the generalized eigenfunctions themselves are absolutely continuous with respect to the Lebesgue measure on σ(A).

2 Eigenfunction expansions on Hilbert spaces

The first two sections introduce some preliminaries, namely spectral measures for normal operators and the Phillips theorem. The main object of our investigation will be rigged Hilbert spaces X+ ⊂ H ⊂ X−, i.e. Hilbert spaces which have a dense linear subset of test vectors X+ and a space X− of generalized functions (defined as antil-linear functionals on X+). The triple of spaces (X+,H,X−) is sometimes also called Gelfand triple. In general the spaces X+ and X− are topological vector spaces. However we will only consider the case where X+ and X− are reflexive Banach spaces. We will introduce and investigate them in section 2.3. Provided there is a spectral measure µ for a self adjoint operator A s.t. the estimate

2 2 + kχE(A)vkH ≤ kvk+µ(E) for all v ∈ X (2.1) holds we will see that there is a complete system of generalized eigenvectors. If the operator A is closable on X− then the generalized eigenvectors are in fact real eigenvectors of the closure A−. The last section contains a summary with the results to which we will refer in the next chapters.

2.1 Spectral Measures

The spectral theorem for normal operators gives a measure on the spectrum of the operator, which we call spectral measure. But it is not necessarily unique (w.r.t to the operator). As mentioned before we will find spectral measures with the special condition (2.1). Thus it is reasonable to have a precise definition of them, which we will do using the spectral theorem (cf. [Wer05, p. 350, VII.3.1]): Theorem 2.1.1 (Spectral theoerem) For A self-adjoint operator on a separable complex Hilbert space H there is a Borel set Ω ⊂ C, a σ-finite measure ν on the Borel-σ-Algebra of Ω, a continuous function m :Ω → R and a unitary operator U : L2(Ω; ν) → H s.t. the operator M on L2(Ω; ν) defined by

(Mx)(ω) := m(ω)x(ω) with D(M) := {x ∈ L2(Ω; µ) | Mx ∈ L2(Ω; ν)} is a representation of A i.e. D(A) = UD(M) and AUx = UMx for all x ∈ D(M).

9 2 Eigenfunction expansions on Hilbert spaces

Deﬁnition 2.1.2 A σ-ﬁnite measure µ on the Borel-σ-Algebra of σ(A) is called spectral measure for the operator A if and only if

µ(E) = 0 ⇔ ν(m−1(E)) = 0 for all Borel sets E ⊂ σ(A). There is allways such a measure. We can define one for example by using a countable disjoint decomposition Ek of Ω s.t. ν(Ek) < ∞. The measure µ defined by −1 X ν(m (E) ∩ Ek) µ(E) := k 2 ν(Ek) k is an (actually finite) spectral measure for A. There is an algebra homomorphism from the bounded µ-measurable functions to the bounded operators commuting with A, which we call the functional calculus of A: Proposition 2.1.3 For a self-adjoint operator A on a separable Hilbert space, and a spectral measure µ, there is a continuous map

ΦA : L∞(σ(A); µ) → B(H)

−1 s.t. for all µ ∈ ρ(A) it is true that Φ(rλ) = (λI − A) where

−1 rλ(t) := (λ − t) .

Using the notation f(A) for ΦA(f); for all f, g, fn ∈ L∞(σ(A); µ) it is true that

• χσ(A)(A) = I,

• (fg)(A) = f(A)g(A),

∗ • f(A) = (f(A)) and

• kf(A)k = kfk∞.

• If fn → f µ-a.e. and kfnk∞ < C then fn(A)x → f(A)x for all x ∈ H.

• For a B ∈ B(H) s.t. there is just one λ ∈ ρ(A) with Brλ(A) = rλ(A)B also Bf(A) = f(A)B for all f ∈ L∞(σ(A); µ).

An implication of the theorem is that the spectral projections χE(A) are orthogonal projection operators on H, with the property that χE(A) = 0 if and only if µ(E) = 0 for a spectral measure µ. Actually this is a second method to define the notion of a spectral measure. In fact this is done in [PSW89]. We did not do this here since we used the spectral measure to define χE(A) in the first place. (However for a Borel set E the function χE is in L∞(σ(A); µ) w.r.t. to any Borel-measure µ on σ(A).) In the next section we will continue our preparation with the Radon-Nikodym property for Banach spaces.

10 2.2 Phillips Theorem

2.2 Phillips Theorem

The eigenfunction expansions build upon (conditional) Hilbert-Schmidt-riggings are directly or indirectly based on the Radon-Nikodym theorem for scalar-valued measures. But we will later consider eigenfunction expansions based on operator relations for which we will need a vector valued representation theorem (namely Phillips theorem). Luckily we will be able to investigate both approaches using this method. Proposition 2.2.1 Let (Ω; Σ; µ) be a measure space, X a Banach space. For 1 1 p, q ∈ (1, ∞) s.t. p + q = 1 there is an isometry 0 0 π : Lq(Ω; µ; X ) → Lp(Ω; µ; X) with Z (f, π(g)) = (f(λ), g(λ))dµ Ω 0 for all f ∈ Lp(Ω; µ; X) and g ∈ Lq(Ω; µ; X ). On the left hand side of the equation 0 the notation (·, ·) denotes the Banach space duality between Lp(X) and Lq(X ) while on the right hand side it is the inner product between X and X0.

Proof. 1 Using the generalized Hölderinequality we get that Z |(f, π(g))| ≤ |(f(λ), g(λ))| dµ ≤ kfkpkgkq Ω which implies that the map π is bounded with norm 1. We just need to show that kπ(g)kp0 ≥ kgkq. If there is a counter example g ∈ Lq(X) with kgkq = 1 s.t. kπ(g)k < 1 − ε (2.2) ε Pn there is also a step functiong ˜ ∈ Lq(X) s.t. kg˜ − gkq < 2 , whereg ˜ = k=1 χEk yk 0 with disjoint measurable Ek ⊂ Ω and yk ∈ X . We can find xk ∈ X s.t. kxkk = q−1 q ˜ Pn kykk and (xk, yk) = kykk thus we have for f := k=1 χEk xk that n ˜ X q (f, π(˜g)) = µ(Ek)(xk, yk) = kg˜kq k=1 where2 Z n n ˜ p X p X q q kfk dµ = µ(Ek)kxkk = µ(Ek)kykk = kg˜kq Ω k=1 k=1 ε which means kπ(˜g)kp0 ≥ 1 and thus kπ(g)kp0 ≥ 1 − 2 (contradicting (2.2)). Now, we can state Phillips theorem [Phi43] simply as: Theorem 2.2.2 (Phillips) The isometry from the previous proposition is surjective if X is a reflexive Banach space.

1Adapted from the scalar valued version. [Wer05, p.51] 2The function f˜ is called peaking element forg ˜.

11 2 Eigenfunction expansions on Hilbert spaces

2.3 Rigged Hilbert Spaces

Deﬁnition 2.3.1 Let X+, X− be reﬂexive Banach spaces, H a Hilbert space, s.t. X+ ⊂ H ⊂ X−, i.e. X+ is a continuously embedded dense linear subset of H and H itself is a continuously embedded dense linear subset of X−. Then we call the triple (X+,H,X−) a rigged Hilbert space if

+ kvk− := sup |hv, wiH | | for all w ∈ X and kwk+ = 1 .

Remark 2.3.2 The inner product h·, ·i of H can be extended by continuity to the pairs X+ × X− (similarly to the pairs X− × X+). There is an anti-linear3 surjective isometry i : X− → (X+)0 s.t.

(v, i(w)) = hv, wi. (2.3)

Proof. We can define a map ˜i : H → (X+)0 by (2.3) which is anti linear and we ˜ have by definition that ki(w)k(X+)0 = kwk− for all w ∈ H. Hence the map can be extended by continuity to i defined on all of X−. The norm equation implies the closedness of the image of i in X−. But the image of ˜i is already dense in X−, which means that i must be surjective. The latter also implies that the inner product extends by continuity to the pairs X+ × X−.

In the special case that X+ = H = X− the above remark is just the Riesz representation theorem, i.e. Remark 2.3.3 Let H be a Hilbert space, then there is an antilinear surjective isometry j : H → H0 s.t. (x, j(y)) = hx, yi (2.4) From a diﬀerent point of view, the space X− is just the space of continuous anti linear functionals on X+ and the embedding of H to X− is just the adjoint of the embedding of X+ into H.4 With the above remarks, we can use the theorems from Banach space theory e.g. Hahn-Banach or Phillips theorem and transfer them to our case using the isometry i. For example: Remark 2.3.4 For a continuous linear functional f on X+ there exists a w ∈ X− s.t. f(v) = hv, wi for all v ∈ X+ and similarly if f is a continuous linear functional on X− there is a v ∈ X+ s.t.

f(w) = hw, vi for all w ∈ X−.

3i.e. i(av + bw) = ai(v) + bi(w) 4This rises the question why we did not just deﬁne X− as the set of continuous antilinear functionals. The problem is that in most cases X+ and X− will be Hilbert spaces in which case we intuitively identify the dual of a Hilbert space with itself. If we - as proposed here - identify H with a subspace of X−, we end up with an inconsistent set of identiﬁcations. The same approach - though only for Hilbert spaces - has been chosen in [BSU96].

12 2.4 Basic Eigenfunction Expansion

Proof. In the ﬁrst case w is simply i−1(f). In the second case the map v0 → f(i−1(v0)) is a continuous functional on (X+)0. Since X+ is reﬂexive there is a v ∈ X+ s.t. f(i−1(w)) = (v, w) for all w ∈ (X+)0 which means f(w) = (v, i(w)) = hv, wi.

Let us see how a simple rigged Hilbert space looks like:

Example 2.3.5 For s ∈ R we denote the Hilbert spaces l2,s consisting of the ∞ sequences (xn)1 ⊂ C s.t.

X 2s 2 (1 + k) |xk| ≤ ∞. k∈N

The triples l2,s ⊂ l2,0 ⊂ l2,−s are rigged Hilbert spaces. And for s > 1/2 they are actually Hilbert-schmidt-riggings (cf. section 3).

2.4 Basic Eigenfunction Expansion

Let A be a self-adjoint operator on a rigged Hilbert space H and let us assume we have established that there is a σ-ﬁnite spectral measure µ on σ(A) with the estimate 2 2 + kχE(A)vkH ≤ kvk+µ(E) for all v ∈ X (2.5) We can easily see that there is a unique linear continuous map

+ Φ: L2(σ(A); µ; X ) → H s.t. for all Borel sets E ⊂ σ(A) with µ(E) < ∞ and for all v ∈ X+

Φ(χEv) = χE(A)v. The σ-ﬁniteness of the spectral measure µ is essential here5, otherwise the step + functions (with ﬁnite support) would not be dense in L2(σ(A); µ; X ). Using Phillips’ theorem 2.2.2 we know that there is a canonical isomorphism π + 0 + 0 mapping L2(σ(A); µ; X ) onto L2(σ(A); µ;(X ) ) with Z (f, g) = (f(λ), (πg)(λ)) dµ. σ(A) To get our eigenfunction expansion we compose the adjoint of Φ with that isomorphism, use the isomorphism j from the Riesz representation theroem on H and the isomorphism i between (X+)0 and X−, i.e. (Fx)(λ) := i−1 (π ◦ Φ0 ◦ j)(x) (λ)

5This is mentioned since finding a non-σ-finite measure fulfilling the above estimate is trivial.

13 2 Eigenfunction expansions on Hilbert spaces which we can visualize using a commutative diagram

F − H / L2(σ(A); µ; X ) O i1 pointwise

+ 0 j L2 σ(A); µ;(X ) O π H0 / L (σ(A); µ; X+)0 Φ0 2

− thus F maps H into L2(σ(A); µ; X ) and we can derive for a Borel set E ⊂ σ(A) with µ(E) < ∞ that for all x ∈ H and v ∈ X+

Z Z hv, (Fx)(λ))idµ = v, π Φ0 (j(x)) (λ) dµ E E 0 = χEv, Φ (j(x))

= (Φ(χEv), j(x))

= hχE(A)v, xi.

This means for all x ∈ H that Z (Fx)dµ = χE(A)x E which implies the equation Z f(λ)(Fx)(λ)dµ = f(A)x σ(A) for all x ∈ H and all step functions f on σ(A). Since the left hand side is continuous with respect to the L2(σ(A); µ) of f (according to the H¨older inequality) and the right hand side is continuous with respect to L∞(σ(A); µ) (according to 2.1.3) and since the step functions are dense in the space L∞(σ(A); µ) ∩ L2(σ(A); µ) we have prooven that: Proposition 2.4.1 Let X+ ⊂ H ⊂ X− be a rigged Hilbert space, A a self-adjoint 2 2 operator on H. If there is a spectral measure µ s.t. kχE(A)vkH ≤ µ(E)kvk+ for + − all v ∈ X . Then there is a continuous map F : H → L2(σ(A); µ; X ) s.t. Z f(λ)(Fx)(λ)dµ = f(A)x σ(A) for all x ∈ H and f ∈ L∞(σ(A); µ) ∩ L2(σ(A); µ).

14 2.5 Extending the Operator

2.5 Extending the Operator

From the above equation we can derive: Proposition 2.5.1 Under the conditions of proposition 2.4.1, if v ∈ X+ ∩ D(A), and Av ∈ X+ then h(Fy)(λ), Avi = λh(Fx)(λ), vi for all x ∈ H and almost all λ ∈ σ(A).

Proof. For all measurable E ⊂ σ(A) it is true that Z (h(Fx)(λ), Avi − λh(Fx)(λ), vi) dµ = hχE(A)x, Avi − hχE(A)Ax, vi = 0. E

However, we cannot show that A(Fx)(λ) = λ(Fx)(λ) since the operator A cannot be applied to elements of X−. We could think about extending the operator to X−, but this is more than just a formal problem. The following is an example where a bounded operator is not closable on X−:

Example 2.5.2 The operator A deﬁned on l2,0 by

∞ ! X x(k) Ax = , x(0), x(1), ··· 1 + k k=0

q π2 is bounded with norm 6 + 1. However it does not permit a closed extension to l2,−1 E.g. the sequence xn := χ[n...2n] converges to 0 in l2,−1, but Axn is converging to a non-zero vector. We cannot expect that a self-adjoint operator A automatically extends to a closed operator A− on X−. Or, for that matter, that its restriction to X+ is densely deﬁned. Curiuosly these conditions are the same, i.e.: Proposition 2.5.3 Let X+ ⊂ H ⊂ X− be a rigged Hilbert space, A a self-adjoint operator on H. The operator A is closable on X− if and only if the restriction of A to X+ has dense domain in X+, i.e. if

D(A+) := {v ∈ X+ ∩ D(A) | Av ∈ X+} is dense in X+.

− − − Proof. Let ΓA− denote the closure of the graph of A on X × X . For a w ∈ X + we will show that (0, w) ∈ ΓA− if and only if hv, wi = 0 for all v ∈ D(A ). − − If-Part: Let xn ⊂ D(A) be a sequnce s.t. xn → 0 in X and s.t. Axn → w in X . For a v ∈ D(A+) we have

hv, wi = lim hv, Axni = lim hAv, xni = 0. n→∞ n→∞

15 2 Eigenfunction expansions on Hilbert spaces

Only-If-Part: If (0, w) ∈/ ΓA− there is (according to Hahn-Banach) a functional − − 0 φ ∈ (X × X ) s.t. φ((0, w)) = 1 and φ(ΓA− ) = 0. We can represent φ by a sum − 0 − 0 of two functionals φ1 ∈ (X ) , φ2 ∈ (X ) i.e. φ((w1, w2)) = φ1(w1) + φ2(w2) and + according to remark 2.3.4 there are v1, v2 ∈ X s.t. φ(w1, w2) = hw1, v1i+hw2, v2i. Since φ((0, w) = 1 we have

h0, v1i + hw, v2i = 1 and that φ vanishes on the graph of A means that for all x ∈ D(A)

hx, v1i + hAx, v2i = 0.

∗ ∗ + The latter implies v2 ∈ D(A ) with A v2 = −v1, where v1, v2 ∈ X . And since A + is self adjoint, we have v2 ∈ D(A ). Actually, there is another way to see this. The adjoint (A+)0 defined on (X+)0 of A+ is aequivalent to A− w.r.t. to the isometry between (X+)0 and X− (cf. remark 2.3.2). This also indicates that σ(A+) = σ(A−). In fact the above proof is just an adaptation of the fact that a closed densely defined operator on a reflexive Banach space, has a closed densely defined adjoint. (cf. [Haa06, A.4]). We would like to note another condition that A is closable on X−. (This one however is not necessary.) Remark 2.5.4 If there is a µ ∈ ρ(A) s.t. R(µ, A)X+ ⊂ X+ then A is closable on X− moreover µ ∈ ρ(A+) and µ ∈ ρ(A−).

Proof. By the closed graph theorem R(µ, A) is bounded on X+ which implies that R(µ, A) is bounded on X−. Thus A is closable on X−.

Using proposition 2.5.1 we can conclude: Proposition 2.5.5 Assuming the conditions of propositon 2.4.1, and that the operator A is closable on X−. For all x ∈ H and almost all λ ∈ σ(A) it is true that (Fx)(λ) ∈ D(A−) and A−(Fx)(λ) = λ(Fx)(λ) where A− denotes the closure of A on X−.

2.6 Orthogonalizing Generalized Eigenvectors

We remember from linear algebra that for a symmetric matrix there exists a complete orthonormal system of eigenvectors, and we can write X X hx, yi = hx, ek,λihek,λ, yi. (2.6) λ∈σ(A) k

If σ(A) is discrete the above is still possible. But in the case of a continous spectrum the eigenvectors are deﬁned on a generalized space X−, and they are

16 2.6 Orthogonalizing Generalized Eigenvectors deﬁned only almost everywhere on the spectrum. The former implies that we have to apply them on vectors v, w on X+, and the latter implies that we have to replace the outter sum with an integral. So we expect the generaliziation of 2.6 to look like: Z X hv, wi = hv, ek(λ)ihek(λ), wi dµ. σ(A) k The next proposition will show how to ﬁnd each of the measurable functions − ek : σ(A) → X , the following one will show how to get a complete system of them. And the latter ones will deal with the question whether we can put the sum inside (once in the weak sense as above, and once in the strong sense). Proposition 2.6.1

6 G := {f(A)x ∈ H|f ∈ L∞(σ(A); µ)}.

Let P be the orthogonal projection onto G. If there is a v ∈ X+ s.t. x = P v then − there exists a bounded µ-measurable function e(λ): σ(A) → X s.t. ke(λ)k− ≤ kFk and for all w ∈ X+ it is true that

(F(P w))(λ) = hw, e(λ)ie(λ).

Proof. We will start by showing the following intermediate results

1. For all y ∈ G there is fy with (Fy)(λ) = fy(λ)(Fx)(λ).

2. The term hv, (Fx)(λ)i is real and non-negative µ− a.e. Indeed we have that 2 k(Fx)(λ)k− ≤ hv, (Fx)(λ)i. 3. For a.e. λ ∈ σ(A) is true that hv, (F(P w))(λ)i = h(Fx))(λ), wi.

∞ Step 1 By deﬁnition there is a sequence (fn)n=1 ⊂ L∞(σ(A); µ) s.t. fn(A)x → y. Since F is continuous and linear this means that Z 2 kfn(λ)(Fx)(λ) − (Fy)(λ)k− dµ → 0 σ(A) hence there exists a subsequence nk s.t. for almost all λ ∈ σ(A)

kfnk (λ)(Fx)(λ) − (Fy)(λ)k− → 0

which implies that fnk is converging almost everywhere on the support of Fx. Let fy be the zero function on the set where Fx vanishes and the limit of fnk on the support of Fx. Then we have

fy(λ)(Fx)(λ) = (Fy)(λ) a.e.

6i.e. x is a star-cyclic vector in the subspace G ([PCK03])

17 2 Eigenfunction expansions on Hilbert spaces

Step 2 Follows from Z Z 2 2 k(Fx)(λ)k− dµ ≤ kFk hx, χE(A)xi = hv, χE(A)P vi = hv, (Fx)(λ)idµ. E E Step 3 Follows from Z Z hv, (F(P w))(λ)i = hv, χE(A)P wi = hχE(A)P v, wi = h(Fx)(λ), wi E E where we used that P v = x and that P and χE(A) commute (the latter is true ∗ since G is χE(A) and χE(A) invariant). Proof of the proposition Using the second step we can deﬁne the measurable − function e in L∞(σ(A); µ; H ) by

e(λ) := (hv, (Fx)(λ)i)−1/2 (Fx)(λ).

+ For a w ∈ X , there is a measurable fw s.t. (Fw)(λ) := fw(λ)(Fx)(λ). According to step 3 we have hv, fw(λ)(Fx)(λ)i = h(Fx)(λ), wi and hence

hw, (Fx)(λ)i hw, e(λ)ie(λ) = (Fx)(λ) h(Fx)(λ), vi hF(P w)(λ), vi = (Fx)(λ) h(Fx)(λ), vi hf (λ)(Fx)(λ), vi = w (Fx)(λ) h(Fx)(λ), vi

= fw(λ)(Fx)(λ) = (F(P w))(λ).

Now we can conclude our result: Proposition 2.6.2 Under the conditions of proposition (2.4.1) and assuming that + X is separable, there is a denumerable set of measurable functions ek : σ(A) → X− s.t. for all w ∈ X+ it is true that

∞ X Z f(A)w = f(λ)hw, ek(λ)iek(λ) dµ. (2.7) k=0 σ(A)

+ Proof. Let vk be a dense subset of X . We can deﬁne inductively

k−1 X xk := vk − Pjvk j=0

Gk := {f(A)xk|f ∈ L∞(σ(A); µ)}

18 2.6 Orthogonalizing Generalized Eigenvectors

∞ where Pj is the orthogonal projection onto Gj and (Gk)k=1 is an orthogonal decomposition of H into subspaces, having the star-cyclic vectors xk (moreover 7 xk = Pkvk). Thus we can use the previous position and ﬁnd ek(λ) s.t.

hw, ek(λ)iek(λ) = (F(Pkw))(λ) meaning that Z f(λ)hw, ek(λ)iek(λ)dµ = f(A)Pkw. σ(A)

This implies: Proposition 2.6.3 Under the conditions of the previous proposition we have for all v, w ∈ X+

Z X hf(A)v, wi = f(λ)hv, ek(λ)ihek(λ), wi dµ σ(A) k where the sum converges absolutely µ− a.e.

Proof. In the last proposition we had

∞ X Z hχE(A)v, vi = hv, ek(λ)ihek(λ), vidµ k=0 E ∞ Z X 2 = |hv, ek(λ)i| dµ, k=0 E thus hv, ek(λ)i is square integrable in the product measure space (N; ν)×(σ(A); µ) (ν being the counting measure N). Hence using H¨olderwe have that

hv, ek(λ)ihek(λ), wi is integrable in the product space. Meaning that we can use Fubini:

Z X X Z f(λ)hv, ek(λ)ihek(λ), wi dµ = f(λ)hv, ek(λ)ihek(λ), wi dµ σ(A) k k σ(A) = hf(A)v, wi

We needed to go to the weak version of the equation (2.7) to put the sum inside. In general the vector valued sum does not converge absolutely. But we have unconditional convergence, which we will deﬁne ﬁrst:

7The same method can be used to proof the actual spectral theorem. [PCK03, p. 16]

19 2 Eigenfunction expansions on Hilbert spaces

Deﬁnition 2.6.4 [Hei97] A series xn in a Banach space X is unconditionally con- vergent, if all permutations xπ(n) of it are converging, i.e. if for all permutations π of N the sequences of partial sums n X Sn,π := xπ(k) k=0 are converging. Remark 2.6.5 In that case the limit is independent of the permutation.

Proof. Given a permutation π let xπ be the limit of the sequence Sn,π. Let us observe ﬁrst that for any ﬁnite set F and ε > 0 there is a super set S of F s.t. P kxπ − k∈S xkk < ε. (There is an n0 s.t. kSn,π − xπk < ε for all n > n0, we can choose n > n0 s.t. F ⊂ π({1, . . . , n}) and set S := π({1, . . . , n}.)

Assuming there are permutations π1 and π2 s.t. the sequences Sn,π1 resp. Sn,π2 1 are converging to different points y1 resp. y2. For an ε = 3 ky1 −y2k we can find an increasing sequence of finite subsets of N (i.e. F1 ⊂ F2 ⊂ F3 ... ) s.t. the distance of the sum of xk over Fi is within ε to y1 when i is even and with in ε to y2 when i is odd. Traversing the elements of Fi in order we can create a permutation π3 which is definitively not Cauchy.

Proposition 2.6.6 Under the conditions of proposition 2.6.3 we have for all v ∈ X+ that Z X f(A)v = f(λ)hv, ek(λ)iek(λ) dµ σ(A) k where the sum is converging unconditionally µ− a.e.

+ R P 2 Proof. For a w ∈ X it is true that E k |hw, ek(λ)i| dµ = hχE(A)w, wi ≤ 2 µ(E)kwk+, i.e. X 2 2 |hw, ek(λ)i| ≤ kwk+ (2.8) k + for µ a.e. λ ∈ σ(A). Let us choose a denumerable dense subset wj of X . There is a µ−null set N s.t. for all λ ∈ σ(A) \ N the equation above holds for all wj and also for v. Let π be a permutation of N (i.e. a bijective map from N to N) we want to show that the sequence

n X Sn := hv, eπ(k)(λ)ieπ(k)(λ) k=0 is Cauchy for all λ ∈ σ(A) \ N. Since the sum in equation (2.8) is converging absolutely, its permutation w.r.t. π is also converging . Thus for an ε > 0 there is an n0 > 0 s.t. ∞ X 2 |hv, eπ(k)(λ)i| ≤ ε k=n0

20 2.7 Summary

Let n > m > n0 then there is a wj with kwjk < 1 + ε s.t. kSn − Smk− ≤ |hSn − Sm, wji| + ε. The latter however is

m X hv, ek(λ)ihek(λ), wji k=n which is (using H¨older)smaller than ε(1 + ε).

2.7 Summary

Let us deﬁne what we mean by an eigenfunction expansion: Deﬁnition 2.7.1 Let X+ ⊂ H ⊂ X− be a rigged Hilbert space where X+ is seperable, A a self-adjoint operator on H. We say that A admits an eigenfunction expansion for the rigging if there is a spectral measure µ, a continuous map F : − H → L2(σ(A); µ; X ) and a denumerable set of bounded measurable functions − ek : σ(A) → X s.t. for all f ∈ L∞(σ(A); µ) ∩ L2(σ(A); µ) it is true that Z f(A)x = f(λ)(Fx)(λ)dµ for all x ∈ H σ(A) and for all v, w ∈ X+ it is true that

Z X f(A)v = f(λ) f(λ)hv, ek(λ)iek(λ) dµ (2.9) σ(A) k Z X hf(A)v, wi = f(λ) f(λ)hv, ek(λ)ihek(λ), wi dµ (2.10) σ(A) k where the sum in (2.9) is converging unconditionally µ-a.e. and the sum in (2.10) is converging absolutely µ-a.e. Remark 2.7.2 If A is closable on X−, the vectors (Fx)(λ) for an x ∈ H as well − as ek(λ) are in D(A ) for µ− a.e. λ ∈ σ(A) and of course

A−(Fx)(λ) = λ(Fx)(λ) − A ek(λ) = λek(λ)

The following is an intermediate result showing us how to deﬁne the measure µ: Proposition 2.7.3 For a rigged Hilbert space X+ ⊂ H ⊂ X− and a self-adjoint operator A, there is a possibly not σ-ﬁnite measure µ s.t. the estimate

2 2 kχE(A)xkH ≤ µ(E)kxk+ (2.11) holds for all Borel sets E ⊂ σ(A). If there is a second Borel-measure ν s.t. the esimtate holds, then µ(E) ≤ ν(E).

21 2 Eigenfunction expansions on Hilbert spaces

2 + 2 Proof. Withµ ˜(E) := sup kχE(A)xk | x ∈ X , kxk+ ≤ 1 we can deﬁne µ as the variation ofµ ˜(E) i.e. ( ) X µ(E) := sup µ˜(Ek) | where Ek is a countable partitioning of E . k∈N

That µ(∅) = 0 is obvious. We want to show σ− additivity, therefore let Ek be a countable partitioning of E. We have by deﬁnition that X µ(E) ≥ µ(Ek). k∈N

For the less-than-part: Let us ﬁrst observe that if a partitioning Ek is a reﬁnement of a second partitioning Fk then the sum ofµ ˜(Ek) is at least as big as the sum of µ˜(Fk). This follows from the equation

2 X 2 kχE(A)xk = kχEk (A)xk . k∈N

For an ε > 0 there is a countable partitioning Fk s.t. X µ(E) ≤ µ˜(Fk) + ε. k∈N

We can ﬁnd a countable partitioning Gk which is a reﬁnement of both Ek and Fk for which we have X X X µ(E) ≤ µ˜(Fk) + ε ≤ µ˜(Gk) + ε ≤ µ(Ek) + ε. k k k

Assuming there is a second measure ν s.t. the estimate holds (i.e.µ ˜(Ek) ≤ ν(Ek)). We can be sure that also µ(E) ≤ ν(E) since ν has to be σ-additive.

For the measure constructed above we have that µ(E) = 0 if and only if χE(A) = 0 (we are recalling that we expect X+ to be dense in H). Hence if µ is σ−ﬁnite it becomes automatically a spectral measure, which implies: Theorem 2.7.4 Let X+ ⊂ H ⊂ X− be rigged Hilbert space, A a self-adjoint operator then A has an eigenfunction expansion (cf. 2.7.1) on that rigging if there is a σ− ﬁnite measure ν on σ(A) s.t.

2 2 kχE(A)xkH ≤ ν(E)kxk+. (2.12)

Proof. Using the last proposition we can ﬁnd a spectral measure µ which is smaller than ν (and hence is also σ-ﬁnite) on which we can apply the propositions 2.4.1, 2.6.3 and 2.6.6.

22 3 Hilbert-Schmidt-Riggings

There are several methods to establish the estimate (2.12). The basic one beeing the Hilbert-Schmidt rigging, which can be improved via conditional Hibert- Schmidt-Riggings. The same is also possible for (conditional) nuclear riggings. Applying the Hilbert-Schmidt expansion theorem on the Laplace Operator on n n L2(R ) leads to an eigenfunction expansion on the weighted rigging L2,s(R ) ⊂ n n n L2(R ) ⊂ L2,−s(R ) for s > 2 . We will also verify that this not possible for any n s < 2 .

3.1 Unconditional Hilbert-Schmidt Riggings

Deﬁnition 3.1.1 Let again X+ ⊂ H ⊂ X− be a rigged Hilbert space (according to deﬁnition 2.3.1). If X+ is itself a separable Hilbert space and the embedding i : X+ → H is a Hilbert-Schmidt operator (cf. [Wer05, VI.6, p.292]), the triple X+ ⊂ H ⊂ X− is called Hilbert-Schmidt rigging ([Mau68]). + Under these circumstances we have for an orthonormal system xk of X that

X 2 kxkkH < ∞, k particularly if we had a positive self adjoint operator A, and a Borel measurable E ⊂ σ(A) we would have X 2 kχE(A)xkk < ∞. k Indeed the following is a ﬁnite measure on σ(A) X µ(E) := hχE(A)xk, xki.

We already saw that the sum exists (and is positive and real). That µ(∅) = 0 is obvious, and we already saw that µ(σ(A)) < ∞. To show that µ is σ-additive, let Ej be a series of disjoint Borel-measurable sets and let E be their union. The spectral theorem tells us that for an xk we have X χEj (A)xk = χE(A)xk j thus X hχEj (A)xk, xki = hχE(A)xk, xki j

23 3 Hilbert-Schmidt-Riggings which is converging absolutely (since all terms are positive). Moreover X µ(E) = hχE(A)xk, xki k X X = hχEj (A)xk, xki k j X X X = hχEj (A)xk, xki = µ(Ej). j k j

More to the point, the measure fullﬁlls our desired estimate:

2 2 kχE(A)xk ≤ µ(E)kxk+. Using the Hilbert space structure on X+ we have

2 X 2 kxk+ = |hx, xki+| X x = hx, xki+xk

2 P 2 Since kxk+ = |hx, xki+| we have (using H¨older)

X 1/2 kχE(A)xkH ≤ |hx, xki+| kχE(A)xkkH ≤ kxk(µ(E)) k Thus we have that Theorem 3.1.2 On a Hilbert-Schmidt rigging X+ ⊂ H ⊂ X−, there is an eigenfunction expansion for all self-adjoint operators on H.

3.1.1 Hilbert-Schmidt-Integral Operators How do we know that an embedding is Hilbert Schmidt? Of course we can always try to ﬁnd an orthonormal system of the spaces and try to sum. But we can also use the following (cf. [Wer05, p.296, VI.6.3]).

Proposition 3.1.3 Let (Ω; µ) be a σ-ﬁnite measure space, k ∈ L2((Ω; µ) × (Ω; µ)) then the operator Z (T f)(x) := k(x, y)f(y)dµ(y) Ω is Hilbert-Schmidt, with kT kHS := kkkL2 .

Proof. Let fn be an orthonormal basis of L2(Ω; µ) then we have using Parseval’s identity for a.e. x ∈ Ω Z 2 X 2 X 2 |(T fn)(x)| = k(x, y)fn(y)dµ(y) = kk(x, ·)k n n thus

24 3.2 Weighted Spaces and Sobolev Spaces

Z Z X 2 X 2 X 2 kT fnk = kT fn(x)k dµ = kT fn(x)k dµ n nZ n 2 = kk(x, ·)k dµ = kkkL2 .

We are going to use this proposition in the following form Proposition 3.1.4 Let X+ ⊂ H ⊂ X− be a rigged Hilbert space, T a unitary surjective operator from H to X+. The rigging is Hilbert Schmidt if and only if T as an operator on H is Hilbert Schmidt.

Proof. Let ek be an orthonormal basis of H; if T is Hilbert-Schmidt on H then

X 2 kT ekkH ≤ ∞. k

+ But the sequence T ek is an orthonormal basis of X .

3.2 Weighted Spaces and Sobolev Spaces

2 r n Let wr(u) := (1 + kuk ) for u ∈ R then we can deﬁne the scale of fractional Sobolev spaces (using the Fourier transform) by

s n n H := {x ∈ L2(R ) | (wsxˆ) ∈ L2(R )} for s ≥ 0. Similarly, we can deﬁne the weighted fractional Sobolev spaces by

s,r n n H := {x ∈ L2(R ) | wswdrx ∈ L2(R )} for s, r ≥ 0, Z hx, yis,r := wdrx(u) wdry(u) ws(u) du. Rn −s,−r The negative spaces H can be deﬁned as the closure of L2 w.r.t. the norm

kxk−s,−r := sup{|hx, yi| | y ∈ Hs,r, kyks,r = 1}

s,r n n −s,−r n Then the triple H (R ) ⊂ L2(R ) ⊂ H (R ) is a rigged Hilbert space. n n Moreover it is a Hilbert-Schmidt-rigging if s > 2 and r > 2 . To see this let us consider the operator Ts,rx := w−rxw[−s. n s,r n It is a unitary surjective map from L2(R ) to H (R ) and it is Hilbert-Schmidt n as an operator on L2(R ) since we can write it as Z − (Ts,rx)(u) = w−r(u)e iu · vw−s(v)x(v) dv Rn i.e. it has an L2-kernel (cf. proposition 3.1.3). And according to proposition 3.1.4 s,r n n the embedding H (R ) into L2(R ) is Hilbert-Schmidt.

25 3 Hilbert-Schmidt-Riggings

n Hence, for any self-adjoint operator A on L2(R ) there is an eigenfunction ex- s,r n n −s,−r n pansion (cf. 2.7.1) w.r.t. the rigging H (R ) ⊂ L2(R ) ⊂ H (R ). In the case of a multiplication operator (with a polynomial for example) or the Laplace operator we can also be sure that A is closable on H−s,−r (the Schwarz functions are A-invariant and dense in Hs,r). Thus in that case the operator A is closable on H−s,−r and we also have the conclusions of remark 2.7.2.

3.3 Conditional Hilbert-Schmidt Riggings

We used the fact that the embedding X+ → H is Hilbert-Schmidt to derive the estimate 2 2 kχE(A)xk ≤ µ(E)kxk+. (3.1) However there is a weaker condition [PSW89]. Let therefore γ be a non-zero bounded continuous function on R. (On compact subsets of R this implies that |γ| is bounded away from zero.) If the operator γ(A) is a Hilbert-Schmidt operator from X+ to H then we can still derive the estimate (3.1). Let again

X µ(E) := hχE(A)xk, xki k

+ for an orthonormal system xk of X . For a compact subset E of σ(A) we have an ε > 0 s.t. |γ1| ≤ ε, which implies

X 2 µ(E) := kχE(A)xkk k X −1 2 <= k[χEγ ](A)γ(A)xkk k −1 2 X 2 <= kgamma χEk∞ kγ(A)xkk k 2 2 <= ε kγ(A)kHS thus µ is σ−ﬁnite. Btw. that µ is σ additive can be shown exactly as in section 3, hence Proposition 3.3.1 Let X+ ⊂ H ⊂ X− be a rigged Hilbert space and let A be a self-adjoint operator, if X+ is a separable Hilbert space and there is a continuous bounded non-zero function γ s.t. its restriction to X+ (i.e. γ(A): X+ → H) is a + Hilbert-Schmidt-operator or equivalently if for an orthonormal system xk of X the sum X 2 kγ(A)xkkH k is ﬁnite, there is an eigenfunction expansion for A on X+ ⊂ H ⊂ X−.

26 3.4 Eigenfunction Expansion for the Laplacian

3.4 Eigenfunction Expansion for the Laplacian

2 (r/2) n Let wr(u) := (1 + kuk ) for u ∈ R then we can deﬁne the scale of weighted L2 spaces by (cf. [PSW89, p. 404])

n n L2,r(R ) := {(wrx) ∈ L2(R )}, Z 2 hx, yir := x(u) y(u) |wr(u)| du. Rn n The self-adjoint operator A under investigation is the Laplace operator on L2(R ) i.e. n X d2 A := − 2 . (duk) k=1 2 −r/2 n The operator (Mrf)(u) := f(u)(1 + |u| ) is a unitary map from L2(R ) to n −4A n n L2,r(R ). We are intending to show that the map e : L2,r(R ) → L2(R ) for all r > n/2 is Hilbert-Schmidt. Since Mr is unitary we only need to show that −4A n n G := e ◦ Ms is Hilbert-Schmidt from L2(R ) to L2(R ). The operator looks like: Z 2 (Gf)(u) = (16π)−n/2 f(v)(1 + |v|2)(−r/2)e−|u−v| dv

But this is an integral operator with kernel

2 k(u, v) := (16π)−n/2(1 + |v|2)(−r/2)e−|u−v| and we have

Z Z Z 2 |k(u, v)|2 d(x, y) = (16π)n (1 + |v|2)−re−2|u−v| dx dy Rn×Rn Rn Rn Z Z 2 = (16π)n (1 + |v|2)−rdy e−2|u| du Rn Rn n −4A which is ﬁnite if r > 2 . Hence G is Hilbert-Schmidt on L2 and so is e : L2,r → L2, which implies that (proposition 3.3.1) the operator A has an eigenfunction expansion 2.7.1 w.r.t. to the rigging L2,r ⊂ L2 ⊂ L2,−r. As mentioned before the Schwarz class is dense in L2,r and A-invariant, i.e. A’s restriction to L2,r is densely deﬁned, which means that A is closable on L2,−r (cf. 2.5.3)). Thus the generalized eigenfunctions are true eigenfunctions for the closure of the operator (cf. 2.7.2).

3.4.1 Optimality It is not possible to improve on that using conditional Hilbert-Schmidt-riggings.

Proposition 3.4.1 Let again A := −∆ and let f ∈ L∞(σ(A); µ) then for r ≤ n/2 the map n n f(A): L2,r(R ) → L2(R ) is Hilbert-Schmidt if and only if f(A) = 0.

27 3 Hilbert-Schmidt-Riggings

Proof. The operator f(A) is translation invariant - meaning that it commutes with the translation operator Ty defined by (Tyf)(x) = f(x − y). If f(A) is non-zero, n we will find an x ∈ L2(R ) with compact support s.t. f(A)x 6= 0. We can find n an axial parallel cube with side length a > 0 and center p ∈ R containing the support of x, i.e. supp(x) ⊂ [p1 − a/2; p1 + a/2] × · · · × [pn − a/2; pn + a/2]. Then n we can define for all s ∈ Z xs := Tas−px i.e. xs is a translation of x such that its support is in the cube [as1 − a/2; as1 + n a/2)]×· · ·×[asn −a/2; asn +1/2]. This means hxs, xsi = 0 if s, s ∈ Z with s 6= s. This also means kx k ≤ kxk(1 + kask2 + (a/2)2/n)r/2. Then the set y := xs s r s kxskr for s ∈ Z is an (incomplete) orthonormal set. Using linearity and translation invariance of f(A) we have

2 2 kf(A)xsk 2 2/n −r kf(A)ysk = 2 ≥ C(1 + kask + (a/2) ) kxskr where C = kf(A)xk2kxk−2 > 0, hence

X 2 X 2 2/n −r kf(A)ysk ≥ C (1 + kask + (a/2) ) r∈Zn r∈Zn Z ≥ C (1 + kask2)−r ds Rn n n which is not ﬁnite, so f(A): L2,r(R ) → L2(R ) is not Hilbert-Schmidt.

3.5 Nuclear Riggings

Deﬁnition 3.5.1 Let X,Y be Banach spaces, a linear map ϕ : X → Y is called 0 0 0 nuclear if there are sequences yn ∈ Y, xn ∈ X , αn ∈ C s.t. kynkY = kxnkX0 = 1, αn is an absolutely conveging series and

X 0 ϕ(x) = αnynxn(x). n As in the case of Hilbert-Schmidt-Riggings we can also treat conditional nuclear riggings i.e. Proposition 3.5.2 Let X+ ⊂ H ⊂ X− be a rigged Hilbert space and let A be a self-adjoint operator, and γ be a continuous non-zero bounded function. If the map γ(A): X+ → H is nuclear then there is an eigenfunction expansion for A on X+ ⊂ H ⊂ X−.

0 + 0 Proof. By assumption there are αn ∈ C, xn ∈ (X ) and hn ∈ H with C = P 0 n |αn| < ∞ and kxnk(X+)0 = khnk = 1 s.t.

X 0 γ(A)x = αnhnxn(x) n

28 3.5 Nuclear Riggings for all x ∈ X+. Let us deﬁne the measure ν on σ(A) by X ν(E) = |αn|hχE(A)hn, hni n for all Borel sets E ⊂ σ(A). The measure is ﬁnite and we can verify

X 0 kχE(A)γ(A)xkH = kχE(A) αnhnxn(x)kH n X ≤ |αn|kxk+kχE(A)hnk n X 1/2 1/2 ≤ kxk+ |αn| |αn| kχE(A)hnk n !1/2 !1/2 X X 2 ≤ kxk+ |αn| |αn|kχE(A)hnk n n 1/2 1/2 ≤ kxk+C (ν(E)) .

Using the fact that γ−1 is bounded on compact intervals we can deﬁne the (σ− ﬁnite) measure

X −1 µ(E) := C kγ χ[k;k+1)k∞ν (E ∩ [k; k + 1)) k∈Z with which we can conclude

X 2 kχ (A)xk2 = χ x E H [k;k+1)∩E H k∈Z X 2 ≤ γ−1χ χ γ(A)x [k;k+1) ∞ [k;k+1)∩E H k∈Z X ≤ γ−1χ kxk2 Cν(E ∩ [k; k + 1)) [k;k+1) ∞ + k∈Z 2 = µ(E)kxk+ for all Borel sets E ⊂ σ(A). And according to theorem 2.7.4 this implies the existence of an eigenfunction expansion for A on the rigging X+ ⊂ H ⊂ X−.

4 Operator Relations

We have seen that the weighted spaces L2,−r for which we could build an eigen- n function expansion for the Laplacian on R using (conditional) Hilbert-Schmidt riggings are those with r > n/2. But in the explicit eigenfunction expansion derived in [Str89][p. 64] it is possible to see that an eigenfunction expansion is al- 1 ready possible for r > 2 . In the next section an investigation of the eigenfunction expansion of vector-valued multiplication operators will demonstrate why (conditional) Hilbert-Schmidt riggings, in some cases, fail to give the optimal rigging. The latter will motivate the use of operator relations

BA − AB = h(A) leading to an eigenfunction expansion on the riggings

+ × − X ⊂ [D(B); H]1/2,1 ⊂ H ⊂ [D(B); H]1/2,1 ⊂ X . Before we apply this result, we will take a brief detour onto semi groups, where n we will ﬁnd one possible corresponding operator B for the Laplacian on L2(R ). Leading to an eigenfunction expansion for a homogeneous selfadjoint elliptic diﬀ- ential operator w.r.t. the rigging

n n n L2,s(R ) ⊂ L2(R ) ⊂ L2,s(R ) for all s > 1/2.

4.1 Vector valued multiplication operators

Let us consider the vector-valued multiplication operator

(Af)(t) = xf(t) deﬁned on the space L2(R; H) where H is a non-trivial Hilbert space. The operator A is self-adjoint, has continuous spectrum σ(A) = R and has no eigenfunctions in the usual sense. The functional eλ,w for a w ∈ H and λ ∈ R deﬁned by

hx, eλ,wi := hx(λ), wi (4.1) looks like an eigenvector but it is not continuous on L2(R; H). However it is s,2 continuous on the (vector valued) Sobolev spaces W (R; H) (for s > 1/2), i.e. −s,2 eλ,w ∈ W (R; H). This follows from the fact that there is a Sobolev embedding

31 4 Operator Relations

s,2 from W (R; H) to the bounded continuous functions CB(R; H). (see for example [AF07]) Moreoever hAx, eλ,wi = λhx, eλ,wi s,2 if x, Ax ∈ W (R; H). Indeed it is not hard to see that there is a continuous map

−s,2 F : L2(R; H) → L2(R; W (R; H)) where the F(x)(λ) are generalized eigenfunctions deﬁned by

F(x)(λ) = eλ,x(λ)

s,2 −s,2 To get back to our point: W (R; H) ⊂ L2(R; H) ⊂ W (R; H) is a rigged 1 Hilbert space s.t. there is an eigenfunction expansion for A, for any s > 2 . However the rigging is a conditional Hilbert-Schmidt-rigging only in the ﬁnite-dimensional case (i.e. if dim H < ∞). Otherwise there is a countable sequence of orthonormal vectors in hn ∈ H and for example the functions

− 2 fn(t) = e (t )hn form an orthogonal sequence in Ws,2(R; H) for any s. On the other hand kfnkL2(R;H) = kfmkL2(R;H) which implies that the embedding is not Hilbert-Schmidt. 1,2 Let us consider W (R,H) as the domain of a second operator d (Bf) = f. dx It fulﬁlls the following relation with the operator A (whoose eigenvectors we are looking for) BA − AB = I. (4.2) We can observe that the relation (4.2) implies

BAn − AnB = BAn − ABAn−1 + ABAn−1 + AAn−1B = An−1 + A[BAn−1 − An−1B] and by induction we get BAn − AnB = nAn−1 which means

Bf(A) − f(A)B = f 0(A) (4.3) at least for polynomials f. This is all informal and we will do this in great detail in the next sections. So let us assume we have the above for functions f with bounded measurable derivatives. By rewriting the above equation a bit we get

0 ∗ hf (A)x, xi = |hf(A)x, B xi| + |hBx, f(A)xi| ≤ 2kfk∞kBxkkxk.

0 If we take f (λ) = χ[a,b] then it has an integal f which is bounded by (b − a). (see also picture 4.1). Thus

32 4.2 Preliminaries

Figure 4.1: The function χa,b and its anti derivative.

f 0(λ) f(λ) b − a

a b a b

2 2 kχ[a,b]xk ≤ 2(b − a)kBxkkxk ≤ (b − a)kxkB which looks alot like the condition of theorem 2.7.4

2 2 kχE(A)xk ≤ µ(E)kxk+. The next section will start with some preliminaries on sectorial operators on Ba- nach spaces, for which we will derive the equation (4.3). After that we will introduce interpolation spaces, since we want to use rigged Hilbertspaces X+ ⊂ H ⊂ − + X where X is densely embedded in the interpolation space [D(B); H]1/2,1. The eigenfunction expansion theorem will be derived for the operator relation

BA − AB = h(A) where h is an analytic function on a sector Σω with ω > 0.

4.2 Preliminaries

Deﬁnition 4.2.1 For an ω ∈ (0, π) we denote the sector with angle ω by

Σω := {λ | λ 6= 0 and |arg λ| < ω} and the path along the boundary of the sector by Γω

isgn(t)ω Γω(t) := te where t ∈ (−∞, ∞)

Deﬁnition 4.2.2 [Wer05][p. 255] For a closed operator A on a Banach space X, we will denote the resolvent of A at λ ∈ ρ(A) by the symbol

R(λ, A)

It is a bounded injective operator on X. ∞ Deﬁnition 4.2.3 [Haa06][p. 28] We denote by H (Σω) the set of bounded analytic ∞ functions on Σω and by H0 (Σω) the set of analytic functions on Σω s.t. there exist C > 0, ε > 0 with f(λ) ≤ C min(|λ|ε, |λ|−ε)

33 4 Operator Relations

Deﬁnition 4.2.4 Let A be a closed injective operator on a Banach space X such that D(A) and R(A) are dense in X, then A is called sectorial of angle ω ∈ (0, π) if σ(A) ⊂ Σω and

0 sup kλR(λ, A)k|λ ∈ C \ (Σω ∪ {0}) ≤ ∞ for all ω0 > ω. The inﬁmum over all ω for which A is sectorial is called sectoriality angle ω(A). A sectorial operatar with ω(A) = 0 is called 0-sectorial. Deﬁnition 4.2.5 [KP04][p. 202,203] Let A be a sectorial operator on X, we say A has a bouded functional calculus with angle ω > ω(A), if there exists a linear multiplicative map ∞ ΦA : H (Σω) → B(X) s.t.

• ∞ There exists a constant C > 0 s.t. kΦA(f)kX ≤ CkfkH (Σω).

−1 • For all µ ∈ C \ Σω and tµ(λ) = (µ − λ) it is true that ΦA(tmu) = R(µ, A).

∞ • For a uniformly bounded sequence fn ∈ H (Σω) s.t. fn(λ) → f(λ) also fn(A)x → f(A)x.

In this case we will write f(A) instead of ΦA(f). We will denote the inﬁmum over all ω for which A has a bounded functional calculus by ω∞(A). Remark 4.2.6 The above map is the closed exension of

0 ∞ ΦA : H0 (Σω) → B(X) deﬁned by choosing an ω(A) < ω0 < ω: Z 0 ΦA(f)x = f(λ)R(λ, A)dλ. 0 Γω

Proposition 4.2.7 [KP04][p.206] Let A have a bounded functional calculus, ω > ω∞(A), and let f : (0, ∞) × Σω → C. Assuming that

• f(·, λ) is continuous on (0, ∞)

∞ • f(t, ·) ∈ H (Σω)

−1−ε • ∞ kf(t, ·)kH (Σω) ≤ C(1 + t) for a C > 0 and an ε > 0 then for Z ∞ h(λ) := f(t, λ)dt 0 ∞ R ∞ it is true that h ∈ H (Σω) and h(A)x = 0 f(A)x dt.

34 4.2 Preliminaries

Proposition 4.2.8 Let X be a Banach space, A a sectorial operator on X, ω > 0 ω(A) and f an analytic function on Σω s.t. f, f ∈ H0(Σω) then Z Z f 0(λ) R(λ, A) dλ = f(λ) R(λ, A)2 dλ (4.4) Γω Γω 0 Proof. The fact that f, f ∈ H0(Σω) means that there is a constant C > 0 and a ε > 0 s.t. |f(λ)| < C min(|λ|1+ε, |λ|−ε). The latter implies that both the right hand side, and the left hand side of (4.4) are well-deﬁned. Let us observe that d f (Γ (t)) R (Γ (t),A) x = f 0 (Γ (t)) R (Γ (t),A)Γ0 (t) dt ω ω ω ω ω 2 0 − f (Γω(t)) R (Γω(t),A) Γω(t) which means Z t=b 0 [f (Γω(t)) R (Γω(t),A) x]t=a = f (λ)R(λ, A)x dλ Γω|(a,b) Z − f(λ)R(λ, A)2x dλ. Γω|(a,b) We can proof the equality in two steps: First, we let a → 0 and b → +∞, and in the second one we let a → −∞ and b → 0. In both cases the values on the left hand side of the equation converge to zero. Hence the diﬀerence of the integrals in (4.4) must be zero as well.

Proposition 4.2.9 Let X be a Banach space, A a sectorial operator with a bounded ∞ functional calculus on X, ω > ω∞(A), h ∈ H (Σω) and B a closed operator. Assuming R(µ, A)D(B) ⊂ D(B) for all µ ∈ C \ Σω and assuming that BR(µ, A)x − R(µ, A)Bx = h(A)R(µ, A)2x for all x ∈ D(B). (4.5)

0 ∞ 0 0 ∞ 0 Then for an ω > ω and an f ∈ H (Σω) s.t. also f ∈ H (Σω) it is true that f(A)D(B) ⊂ D(B) and

Bf(A)x − f(A)Bx = h(A)f 0(A)x for all x ∈ D(B).

Proof. We can estimate the norm of R(µ, A) as an operator on the Banach space (D(B), k · kB) using

2 kR(µ, A)kB ≤ kR(µ, A)kX + kh(A)kX kR(µ, A) kX and since A was sectorial there is a C > 0 s.t.

−1 2 −2 kR(µ, A)kB ≤ C|µ| + C |µ| (4.6) for all arg µ = ω.

35 4 Operator Relations

0 ∞ 0 Step 1 We will start with the case f, f ∈ H0 (Σω), i.e. there exist C > 0 and ε > 0 s.t. |f(λ)| ≤ C min(|λ|1+ε, |λ|ε) which means that Z f(A)x = f(λ)R(λ, A)x dλ Γω is also a Bochner integral on D(B) (using 4.6). Thus f(A)D(B) ⊂ D(B). Moreover we can fold-in the operator B into the integral i.e. Z Bf(A)x − f(A)Bx = f(µ)(BR(µ, A) − R(µ, A)B) x dµ. Γω But according to the equation (4.5) this is equal to Z Z f(µ)h(A)R(µ, A)2 x dµ = h(A) f(µ)R(µ, A)2 x dµ Γω Γω and using proposition 4.2.9 we get Z Bf(A)x − f(A)Bx = h(A) f 0(µ)R(µ, A)dµ Γω which is equal to h(A)f 0(A)x.

0 ∞ Step 2 The general case i.e. f, f ∈ H (Σω) can be reduced to the former case using the approximation property of the functional calculus (4.2.5). Consider the sequence fn(λ) := gn(λ)f(λ) where 2 nλ − λ g (λ) := e n n 1 + nλ2 0 ∞ then we have that fn, fn ∈ H0 (Σω) and thus (by the ﬁrst step)

0 Bfn(A)x − fn(A)Bx = h(A)fn(A)x

0 for all x ∈ D(B). But we also have that fn, fn are uniformly bounded and that they are converging pointwise to f respectively f 0. And by the approximation 0 property of the functional calculus we have that fn(A)x, fn(A)(Bx), fn(A)x are converging. The closedness of B implies now

f(A)x = lim fn(A)x ∈ D(B) n→∞ and Bf(A)x − f(A)Bx = h(A)f 0(A)x

36 4.3 Interpolation spaces

4.3 Interpolation spaces

Deﬁnition 4.3.1 Let X, Y be Banach spaces, both continuously embedded in a topological vector space V . Then the spaces X + Y := {x + y | x ∈ X, y ∈ Y } and X ∩ Y are Banach space w.r.t to the norms

kvkX+Y := inf{kxkX + kykY | x ∈ X,y ∈ Y ,v = x + y}

kvkX∩Y := max(kxk, kyk) We need a small abbreviation before we can deﬁne the interpolation spaces. Deﬁnition 4.3.2 Let X be a Banach space, then we denote

∗ Lp(X) := Lp([0; ∞]; µ; X) where µ1 is the Haar measure, i.e. Z dt µ(E) := . E t We are going to use the J-method to deﬁne the real interpolation spaces. [AF07], [BL76] Deﬁnition 4.3.3 Let X, Y be Banach spaces, both continuously embedded in a topological vector space V . Let also θ ∈ [0; 1] and p ∈ [1; ∞]. For an u ∈ X ∩ Y , let J(t, u) := max (tkukX , kukY ) then x ∈ [X,Y ]θ;p if and only if there exists a ∗ measurable function u ∈ L1(X + Y ) s.t. Z ∞ dt x = u(t) (4.7) 0 t and s.t. Z dt1/p Φ(u) = t−θ (J(t, u(t)))p < ∞ t

The space [X,Y ]θ;p is a Banach space normed by ∗ kxkθ,p := inf{Φ(u)|u ∈ L1(X + Y ) s.t. 4.7 holds }

In the case that p ∈ (1, ∞) and θ ∈ (0, 1) the interpolation space [X,Y ]θ,p is reﬂexive provided that X,Y are reﬂexive themselves. This doesn’t hold true for p = 1 in general. [BL76]. Example 4.3.4 Let H be a Hilbert space, M a normal operator on H then for all ε > 0 it is true that

1 1 2 +ε 2 −ε D(|M| ) ⊂ [D(M); H]1/2,1 ⊂ D(|M| ) where D(M) denotes the domain of operator equipped with the graph norm, i.e. kxkM := kxkH + kMxkH .

1 dt sometimes denoted by t

37 4 Operator Relations

4.4 Eigenfunction expansion

Using the result of proposition 4.2.9 i.e. that

Bf(A) − f(A)B = h(A)f 0(A) we can again ﬁnd an estimate

2 2 + kχE(A)vkH ≤ kvk+µ(E) for all v ∈ X (4.8)

+ This time however X is the interpolation space [D(B); H]1/2,1 and µ is a spectral measure which is absolutely continuous w.r.t. to the Lebesgue measure (outside points where h is zero). This actually implies that all spectral measures of A are absolutely continuous w.r.t. to the spectral measure (upto the zeros of h). We will do this in two steps: The first is the analytic part, and the second is the - maybe less pleasent - measure theoretic part. In the following we will denote the ∗ norm of the interpolation space [D(B) ∩ D(B ); H]1/2,1 by k · k[B∩B∗;H] (provided that D(B) ∩ D(B∗) is dense in H). Proposition 4.4.1 Let A be a non-negative self adjoint operator, B a closed operator on a Hilbert space H s.t. D(B) ∩ D(B∗) is dense in H. Assuming A, B, + h are fulfilling the conditions of proposition (4.2.9). Let [a, b] ⊂ R be a finite ∗ non-empty interval, then for all x ∈ [D(B) ∩ D(B ); H]1/2,1 it is true that

2 hhfa,b(A)x, xi ≤ 2(b − a)kxk[B∩B∗,H] where  1 if λ ∈ (a, b),  1 fa,b(λ) = 2 if λ = a or λ = b 0 otherwise

Proof. Let x ∈ D(B) ∩ D(B∗) and let

1 f (λ) = log(1 + en(λ−a)) − log(1 + en(λ−b)) n n

0 ∞ 0 then fn, fn ∈ H (Σωn ) (where ωn → 0) , the supremum of fn on R is (b − a), hfn is bounded on R and converging pointwise to hfa,b. Hence,

0 hhfa,b(A)x, xi = lim hhf (A)x, xi n→∞ n = lim hBfn(A)x − fn(A)Bx, xi n→∞ ∗ = lim hfn(A)x, B xi − hBx, fn(A)xi n→∞ ≤ 2(b − a) max(kxkB, kxkB∗ )kxkH .

38 4.4 Eigenfunction expansion

∗ ∗ For an x ∈ [D(B) ∩ D(B ); H]1/2,1 and an ε > 0 there is a u(t) ∈ L1(H) s.t.

Z dt x = u(t) t with Φ(u) ≤ kxk[B∩B∗;H] + ε (cf. 4.3.3) which implies

Z dt kf (A)xk ≤ kf (A)u(t)k a,b H a,b H t Z ∞ 1/2 1/2 1/2 dt ≤ (2(b − a)) ku(t)kB∩B∗ ku(t)kH 0 t Z ∞ dt ≤ (2(b − a))1/2 t−1/2J(t; u(t)) 0 t 1/2 1/2 ≤ (2(b − a)) Φ(u) ≤ (2(b − a)) (kxk[B∩B∗,H] + ε) where we used that

1 1 −1 1 −1 2 2 2 2 − 2 ku(t)kB∩B∗ ku(t)kH ≤ max(t ku(t)kB∩B∗ , t ku(t)kH ) = t J(t; u(t)).

Proposition 4.4.2 Under the conditons of the previous proposition and that h is non-zero on (a, b) it is also true that

Z b 2 −1 0 2 kχ(a,b)(A)xkH ≤ |h (λ)| 2 + |h (λ)| dλ kxk[B∩B∗,H] a

Proof. For a partitioning of the interval [a, b] i.e. a = a1 < a2 < ··· < an = b we have (using the previous proposition):

2 kχ(a,b)(A)xk = hχ(a,b)(A)x, xi 2 ≤ hfa,b(A)x, xi n X = fak,ak+1 (A)x, x k=1

n−1 X −1 = h (ak) hfak,ak+1 (A)x, x k=1 X −1 + fak,ak+1 − h (ak)hfak,ak+1 (A)x, x k

X −1 2 ≤ 2h (ak)(ak+1 − ak) kxk[B∩B∗;H] k X 2 −1 + kxkH sup 1 − h (ak)h(λ) fak,ak+1 (λ) . k λ∈[ak;ak+1]

39 4 Operator Relations

If we let supk |ak+1 − ak| → 0 then the ﬁrst sum will converging to the integral of h−1 between a and b (since h−1 is Riemann integrable). Similarly, the limit of the second sum can be estimated by Z b 2 −1 0 kxkH |h (λ)||h (λ)|dλ. a

Proposition 4.4.3 Under the conditons of the previous proposition for all Borel sets E ⊂ σ(A) it is true that 2 2 kχE(A)xkH ≤ µ(E)kxk[B∩B∗,H] (4.9) provided that µ is the σ-finite measure defined by Z −1 0 µ(E) := |E ∩ Nh| + |h (λ)| 2 + |h (λ)| dλ E\Nh where Nh denotes the set of zeros of h on σ(A). Proof. We will use the abbreviation g(λ) = |h−1(λ)| (2 + |h0(λ)|). Let E ⊂ σ(A). If µ(E) is infinite, the proposition is valid anyway. In the case that it is finite, we can split E into disjoint components i.e. [ E = E∞ ∪ Ek k∈N −1 s.t. E∞ ⊂ Nh and Ek ⊂ Nk where Nk := σ(A) ∩ g ([k, k + 1)). Let ε > 0, since the Lebesgue measure vol1 on R is outer regular, we will find for all k ∈ Z an ε open set Uk containing Nk with vol1(Uk) ≤ vol1(Ek) + |k+1|2k . Without loss of −1 generality we are assuming that Uk ⊂ g (k − 1, k + 1) (the intersection with the latter set is still open and still contains Ek). Then we have ε µ(U ) ≤ µ(E ) + k k 2k S Any of the Uk can be written as a disjoint union of open intervals Uk := (ai, bi) with U ⊂ R\Nh. For each interval we have the estimate from the last proposition, thus 2 2 2 ε 2 kχ (A)xk ≤ kχ (A)xk ≤ µ(U )kxk ∗ ≤ µ(E ) + kxk ∗ Ek H Uk H k [B∩B ;H] k 2k [B∩B ;H] and we also have if µ(E∞) 6= 0 2 2 2 kχE∞ (A)xkH ≤ kxkH ≤ µ(E∞)kxk[B∩B∗;H] hence 2 2 kχE(A)xk ≤ kxk[B∩B∗;H](µ(E) + ε)

2 1 Follows from hfa,b(A)x, xi = hχ(a,b)(A)x, xi + 2 hχ{a,b}(A)x, xi where the latter term is a non- negative value.

40 4.5 Semigroups

Thus we found a σ-ﬁnite measure s.t. the estimate (4.9) and using theorem 2.7.4 we have: Theorem 4.4.4 Let X+ ⊂ H ⊂ X− be rigged Hilbert space, A a non-negative self-adjoint operator on H, B a closed operator on H, h an analytic function on Σω for some ω > 0. If for all µ ∈ C \ Σω the domain D(B) is R(µ, A) - invariant and

BR(µ, A)x − R(µ, A)Bx = h(A)R(µ, A)2x for all x ∈ D(B). (4.10)

+ ∗ and if X is embedded in the interpolation space [D(B) ∩ D(B ); H]1/2,1, there is an eigenfunction expansion (cf. deﬁnition 2.7.1) for A w.r.t. to the rigging X+ ⊂ H ⊂ X−.

4.5 Semigroups

It is not so easy to ﬁnd operators B with the necessary relations above, however if B is the generator of a semigroup for which 4.11 below holds then all the necessary conditions follow, i.e. let A be a non-negative self-adjoint operator on a Hilbert space H and let Tt be a normal C0-semigroup on H s.t. D(A) is Tt-invariant with

t TtAx = e ATtx for all x ∈ D(A). (4.11) then we have that

t TtR(µ, A)x = R(µ, A)Ttx + (1 − e )AR(µ, A)TtR(µ, A)x

+ for all x ∈ H and µ ∈ C \ R0 . The generator T of the semi-group is also normal (cf. [HP57, p. 595, 22.4.2]) and we can derive for an x ∈ D(T ) that 1 (T R(µ, A)x − R(µ, A)x) = t t 1 (1 − et)AR(µ, A)T R(µ, A)x + R(µ, A)T x − R(µ, A)x t t t which implies R(µ, A)D(T ) ⊂ D(T ) and

TR(µ, A) = R(µ, A)T + AR(µ, A)2 which can be rewritten to

TR(−1,A)R(µ, A)x = R(µ, A)TR(−1,A)x + AR(µ, A)2R(−1,A) i.e. for the closure B of TR(−1,A) we have

BR(µ, A)x − R(µ, A)Bx = h(A)R(µ, A)2x where h(λ) = λ(1 + λ)−1 for all x ∈ D(T ). Since B is closed this extends to all x ∈ D(B). The adjoint B∗ restricted to D(T ) = D(T ∗) is R(−1,A)T ∗, and we can show that the identity is a

41 4 Operator Relations

continuous embedding from the Banach space (D(B); k · kB) to the Banach space ∗ (D(B ); k · kB∗ ), i.e. there is a constant C > 0 s.t.

∗ ∗ ∗ 2 kB xkH = kR(−1,A)T xkH = kT R(−1,A)x + AR(−1,A) xkH 2 = kTR(−1,A)xkH + kAR(−1,A) xkH ≤ CkxkB for all x ∈ D(T ) which is dense in D(B). Hence Proposition 4.5.1 Let X+ ⊂ H ⊂ X− be a rigged Hilbert space, A be a non- negative self-adjoint operator on H, Tt a C0-semigroup with TtD(A) ⊂ D(A) s.t.

t TtAx = e TtAx for all x ∈ D(A). and let B be the closure of the operator deﬁned by B = R(−1,A)T where T is the generator of the semi-group Tt. + If X is a subset of the interpolation space [D(B); H]1/2,1, then A admits an eigenfunction expansion w.r.t. to the rigging X+ ⊂ H ⊂ X−.

Proof. We saw that the operators B,A and the function h are fulﬁlling the conditions of theorem 4.4.4. Since we have D(B) ⊂ D(B∗) we also have X+ ⊂ ∗ [D(B) ∩ D(B ); H]1/2,1.

4.6 Expansion for Homogeneous Elliptic Selfadjoint Diﬀerential Operators

n Let A be a homegeneous (of degree m > 0) diﬀerential operator on L2(R ) i.e.

X α Ax = aαD x |α|=m

n α where for an α ∈ N the operator D is the combination of the α(k)-th partial derivatives along the coordinated axis k i.e.

d α(1) d α(n) (Dαx)(u) = ··· du1 dun If the symbol a of A deﬁned by

X α n a(u) = aα(iu) for u ∈ R |α|=m is real valued then A is self-adjoint, and if there is also a c > 0 s.t. a(u) ≥ c|u|m then A is elliptic. Proposition 4.6.1 [Haa06][p. 229] Let A be a homogeneous diﬀerential operator n 0 of degree m on R and let Tt be the C -semi group deﬁned by t/m (Ttx)u = x(e u).

42 4.6 Expansion for Homogeneous Elliptic Selfadjoint Diﬀerential Operators

∗ −nt/m t We have that Tt = e T−t (i.e. Tt is normal) and ATt = e TtA. The generator of the semi-group is the operator T deﬁned by n X d T x = uk x duk k=1 Proof. For the partial diﬀerential operator d it is true that duk

d t/m d Ttx = e Tt x duk duk α t|α|/m α Using induction we can derive that D Tt = e TtD thus t ATtx = e TtAx for all x ∈ D(A)

According to the last section we have an eigenfunction expansion for a rigging + − + X ⊂ H ⊂ X with X ⊂ [D(B); H]1/2,1, where B is the closure of R(−1,A)C. It is possible to verify that the weighted square integrable functions (i.e. L2,1) are 2 in D(B). Therefore, let us recall that x ∈ L2,1 if kxk2,1 = k 1 + |u| x(u)k2 < ∞. Proposition 4.6.2 Let A be a homogeneous self-adjoint elliptic diﬀerential oper- n −1 ator on L2(R ) then for B = T (I + A) , there is a C > 0 s.t. for all Schwarz function x with kxk2,1 < 1 it is true that kBxk2 < C. Proof. The function y = (I + A)−1x is a Schwarz function. Using the Fourier transform we have \d d ui y(u) = − uiyb(u) dui dui d u = − i xˆ(u) dui 1 + a(u) d = g(u)ˆx(u) + h(u) xˆ(u) dui where d 1 ui a(u) g(u) := − dui 1 + a(u) (1 + a(u))2 u h(u) := i 1 + a(u) That h(u) is bounded is clear using a(u) ≥ c|u|m similarly the ﬁrst term of g(u) for the second term we have to verify. Its nominator is

d X α X m ui a(u) ≤ |aαα(i)u | ≤ |aα| α(i)|u| dui |α|=m |α|=m

43 4 Operator Relations which is smaller than a constant times (1 + a(u))2. Hence there is a C > 0 s.t.

Z 2 !1/2 −1 d kT (I + A) xk = kT yk = ui y(u) du Rn dui Z 2 !1/2 d = g(u)ˆx(u) + h(u) xˆ(u) du Rn dui Z 1/2 Z 2 !1/2 2 d ≤ C |xˆ(u)| du + C xˆ(u) du Rn Rn dui ≤ kxk2 + kxk2,1

And since L2,1 ⊂ D(B) implies for s > 1/2 that L2,s ⊂ [D(B); H]1/2,1, we have Theorem 4.6.3 There is an eigenfunction expansion (cf. 2.7.1) for a homoge- n n neous selfadjoint elliptic diﬀential operator on L2(R ) w.r.t. the rigging L2,s(R ) ⊂ n n L2(R ) ⊂ L2,s(R ) for all s > 1/2.

4.7 Conclusions

Using operator relations to derive eigenfunction expansions seems to be a new approach. At least in the case of the Laplacian on L2, it leads to a result which is approximately close to the optimal result. As we have seen this is not possible with Hilbert-Schmidt-Riggings. However the boundary case should be investigated further. As well as the possibility to derive a Plancharel formula. We were also able to identify a condition namely the existence of a dense reﬂexive Banach + 2 2 space X s.t. there is a measure µ with kχE(x)k ≤ µ(E)kxk+. While beeing more general, it subsumes the various results found through Hilbert-Schmidt- Riggings. We restricted our investigation to self-adjoint operator, but it is worth inverstigating whether a similar approach can be applied to normal operators (see also [HPY07]). It is also worth investigating, whether an inequality of the form

kBA − ABxk ≥ Ckh(A)xk leads to similar results. Moreover the operator relation itself makes also sense in the Banach space setting, which might lead to an eigenfunction expansion in this case. In the following we have compiled a short list of eigenfunction expansion methods with some examples:

44 4.7 Conclusions

Operator (class) Rigging and Method s,r n n −s,−r n n Any self-adjoint operator H (R ) ⊂ L2(R ) ⊂ H (R ) with s, r > 2 , unconditional Hilbert-Schmidt-Rigging (cf. section 3.2)

n n n n ∆ + V where V is a locally L2,s(R ) ⊂ L2(R ) ⊂ L2,−s(R ) with s > 2 and bounded potential n ≤ 3, conditional Hilbert-Schmidt-Rigging (cf. [PSW89, p. 407])

n n n 1 Homogeneous elliptic s.a. L2,s(R ) ⊂ L2(R ) ⊂ L2,−s(R ) with s > 2 , Op- diﬀerential operator erator Relations (cf. section 4.6)

n n n Laplace Operator L2,1/2(R ) ⊂ L2(R ) ⊂ L2,−1/2(R ), Harmonic Analyis (cf. [Str89])

5 Appendix

5.1 Acknowledgements

Thanks are going to my friends for their support and patience. Also to Professor Schnaubelt and Dr. Kunstmann for several suggestions. And of course to Professor Weis, who was a remarkable advisor. Usually he was bubbling all over with ideas, yet beeing open for new ones. He deﬁnitively has his way of continuously encouraging people.

5.2 Notation Symbol Description See also N The set of non-negative integers. Z The set of integers. R The set of real numbers. C The set of complex numbers. sgn Sign of a real variable. λ Conjugate of λ B(X,Y ) The space of bounded linear maps between [Wer05, p. 46] the normed linear spaces X and Y . Lp(Ω; µ) The space of p-integrable complex valued [Wer05, p. 18] measurable functions. Lp(Ω; µ; X) The spaces of Bochner integrable functions. [HP57, p. 78] Lp,s(Ω; µ) Weighted version of the above, i.e. f ∈ Lp,s ⇔ wsf ∈ Lp s 2,s H Sobolev spaces with p = 2, i.e. Hs = W [HP57, p. 78] Hs,r Weighted version of the above i.e. f ∈ [HP57, p. 78] s,r s H ⇔ (wrf) ∈ H X0 The dual of the Banach space X [Wer05, p. 58] X× The space of antilinear functionals on X A0 The adjoint of a closed operator A on X σ(A) Denotes the spectrum of a closed operator [Wer05, p. 352] A. ρ(A) Denotes the resolvent set of a closed operator [Wer05, p. 352] A. ω(A) Smallest sectoriality angle of the operator A ω∞(A) Smallest angle s.t. the sectorial operator A has a bounded functional calculus

47 5 Appendix

Symbol Description See also D(A) Denotes the domain of a closed operator A. [Wer05, p. 341] R(A) Denotes the range of a closed operator A. R(λ, A) Resolvent of A (i.e. (λI − A)−1) [Wer05, p. 257] Σω Sector on the complex plane with angle ω [KP04, p. 69] ∞ H (Σω) Set of bounded analytic functions on the sec- [KP04, p. 78] tor ∞ HA (Σω) Set of analytic functions in the functional [KP04, p. 201] calculus of the operator A C(Ω) Set of continuous complex valued functions on Ω. C(Ω; X) Set of continuous X-valued functions on Ω, where X is a Banach space. I The identity operator [Wer05, p. 514] χE Denotes the indicator function of E i.e. [Wer05, p. 490] χE(x) is 1 if x ∈ E and 0 otherwise. χE(A) Spectral projection [Wer05, p. 330] [X,Y ]θ,p Interpolation spaces [BL76], [AF07] f (k) The k-th derivative of f n voln The standard Lebesgue measure on R [Wer05, p. 493] d Derivative with resp. to the variable u dui i

48 5.3 Conventions

5.3 Conventions Symbol Description p, q Real value in [1, ∞] A, B, T, M (Closed) Operators on Banach spaces X,X+,X− Banach space i Injection map between Banach spaces (e.g. X → X−) (·, ·) Inner product of the Banach space with its dual H Hilbert space G Subspace of a Hilbert space h·, ·i Inner product of a Hilbert space x, y Elements of a Banach space X or Hilbert Space H v, w Elements of a Banach spaces X+ or X− k · kH , k · kX Norm of the space H resp. X + k · k+ Norm of the Banach space X − k · k− Norm of the Banach space X k · k2 Norm of a the space of square integrable functions i.e. L2(Ω; µ) k · k2, r Norm of the weighted space of square integrable functione i.e. L2,r(Ω; µ) λ, µ Element of the spectrum of an Operator f, g, h Element of the functional calculus of an Operator, resp. a mesurable function on the spectrum of the operator µ, ν Measure on a σ-Algebra (usually the Borel σ-algebra of a spectrum) E Element of a σ-Algebra ω Angle of a sector or elemente of Ω Ω Abstract domain, (usually equipped with a σ−-algebra n, m, k, i, j natural number ΦA Functional calculus F Eigenfunction expansion ek(λ) Generalized eigenvector for the eigenvalue λ n u Vector in R α Multi index

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