One dimensional ergodic schrödinger operators i general theory(pg 31 215)

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(a) Prove Hăolders inequality: If 1p + 1q= 1, φ∈ p(Z), and ψ ∈ q(Z),

then φψ∈ 1(Z) and

Hints: If either φ or ψ vanishes identically, the claim is trivial.

Otherwise, reduce to the case of unit vectors and apply Young’s inequality.

(b) Prove that · pis a norm. Hint : First show thatφp= supφ, ψwhere ψ ranges over all unit vectors in qandφ, ψ =φ(n)ψ(n).(c) Prove that (p(Z), · p) is a Banach space.

Exercise 1.2.10. Given Hausdorﬀ topological spaces X and Y , let C(X, Y )

denote the space of continuous functions from X to Y , C(X) = C(X,C) the

space of continuous complex-valued functions on X, and Cc(X) the set ofall f∈ C(X) such that supp(f) := {x : f(x) = 0} is compact.

(a) If X is compact, show that

deﬁnes a norm on C(X) that makes C(X) into a Banach space.(b) If X is not compact, show that Cc(X) is not complete. Identify

the completion of Cc(X) as an explicit subspace of C(X). Hint :Consider the one-point compactiﬁcation X := X∪ {∞}.

Exercise 1.2.11. Show that every Hilbert space has a maximal orthonormal

subset and that a Hilbert space is separable if and only if every orthonormal subset of H is countable. Hint: Zorn.

deﬁnes an inner product with respect to whichH is a Hilbert space.

(b) Give necessary and suﬃcient conditions on Γ and (Hγ)γ∈Γ for H

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1.3. Bounded Linear Operators 15

in the sense that there is a linear bijection between them that pre-serves inner products.

γ∈ΓHγthe (external) direct sum (or orthog-onal sum). If Γ ={1, 2, . . . , n} for some n ∈ N, we write the direct sum asH1⊕ · · · ⊕ Hn.

Notes and Remarks. Banach spaces received their name in the

mono-graph of Fr´echet [96], in honor of foundational work of Banach on normedlinear spaces [19]. Hilbert spaces are named for fundamental contributionsof Hilbert to spectral theory, such as the proof of the spectral theorem [122].

Schoenﬂies [221] used the term “Hilbertscher Raum” to refer to the unit ball

of 2(N).

The name of Theorem 1.2.4 comes from contributions of Cauchy [53],who proved the ﬁnite-dimensional version, and Schwarz [224], who proved it

for functions in C([0, 1]). It turns out that Bouniakowsky had the inequality

on C([0, 1]) before Schwarz [43]. Bessel’s inequality (formula (1.2.24) from

Theorem 1.2.14) gets its name from Bessel’s work [34] on trigonometric

The proof of Proposition 1.2.9 is an argument due to F. Riesz [205] andRellich [200]. Their arguments discuss complementation of closed subspaces

in a Hilbert space, but the changes needed to discuss convex sets are merely cosmetic.

Completeness of Lp is proved in most graduate-level real analysis books,

so we omit the proof here; see, e.g., Royden [210], Rudin [211], Simon [244],

or Stein and Shakarchi [250]. Even though completeness of 2 is a corollary of this statement (by considering a space equipped with the counting mea-sure), we elected to present the proof, since it is central to the text, whereas

completeness of general Lp spaces does not ﬁgure as prominently here.

1.3. Bounded Linear Operators

We now introduce linear operators on normed spaces. The primary goal of this section is to set notation and review background results to set the stage for the spectral theorem.

from V to W is a pair (D(A), A), where D(A) is a subspace of V, calledthe domain of A, and A is a linear map from D(A) toW, that is, we haveA(aφ + bψ) = aA(φ) + bA(ψ) for φ, ψ∈ D(A) and a, b ∈ C. For linearoperators, we will generally drop the parentheses and write Aφ for A(φ).We will often just write A for the linear operator (D(A), A), especially if

the domain is clear from context.

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The range of A is

(1.3.1) Ran(A) ={Aφ : φ ∈ D(A)},and the kernel of A is

(1.3.2) Ker(A) ={φ ∈ D(A) : Aφ = 0}.A linear operator is called bounded if D(A) =V and

(1.3.3) A := sup{Aφ : φ ∈ V, φ = 1} < ∞.

The quantity in (1.3.3) is called the operator norm of A. The space of

all bounded operators from V to W is denoted by B(V, W), in the event

that V = W, we say that A is a bounded operator in V and we writeB(V) := B(V, V).

The space B(V, W) inherits natural vector space operations. To wit:given A, B∈ B(V, W) and a ∈ C, one can deﬁne A + B, aA ∈ B(V, W) via

(1.3.4) (A + B)v = Av + Bv,(aA)v = a(Av).

In addition to vector space operations, composition of functions endows spaces of bounded operators with a multiplicative structure. Concretely, if V, W, and X are normed spaces, A ∈ B(V, W), and B ∈ B(W, X ),then BA∈ B(V, X ) (where (BA)v = B(Av)); indeed, since BAv ≤BAv ≤ BAv for all v ∈ V, one has

We denote by IVthe identity operator IVψ = ψ (and drop the subscriptwhen the underlying space is clear). An operator A∈ B(V, W) is calledinvertible if there is an element B∈ B(W, V) such that AB = IW and

BA = IV; we write B = A−1and call B the inverse of A.

collection of m× n matrices with complex entries.

It is often of interest to extend a linear operator from a subspace to the whole space. If the operator is bounded on the subspace and the subspace is dense, there is always a unique bounded extension.

(D(A), A) is a linear operator inV such that D(A) is dense in V and

(1.3.6) AD(A):= sup{Ax : x ∈ D(A) and x = 1} < ∞.

There is a unique A∈ B(V) such that Ax = Ax for all x∈ D(A). Moreover, A = AD(A).

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1.3. Bounded Linear Operators 17

D(A) such that ϕn→ ψ. Since

Aϕn− Aϕm = A(ϕn− ϕm) ≤ AD(A)ϕn− ϕm,η = lim Aϕnexists by completeness. Moreover, if (ϕ

n=1 is another

se-quence converging to ψ, the calculationAϕn− Aϕn = A(ϕn− ϕ

n) ≤ AD(A)ϕn− ϕnimplies that Aϕ

n→ η as well, so the limit is independent of the choice ofthe sequence converging to ψ. Thus, we deﬁne Aψ = η. One can check that

A∈ B(V, W) in norm if An− A → 0 as n → ∞. We say that An→ Astrongly, denoted

A = s-limn→∞An,

if Anψ→ Aψ for every ψ, that is (A − An)ψ → 0 as n → ∞ for every ψ.

The proof of the following proposition is left to the reader.

(a) B(V, W) is a vector space and (1.3.3) deﬁnes a norm on B(V, W)satisfying (1.3.5).

(b) IfW is a Banach space, so too is B(V, W).

(c) IfW is a Banach space and A1, A2, . . . is a sequence of elements ofB(V, W) that is absolutely summable in the sense that

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Proof. Exercise 1.3.2. Throughout the remainder of this section,V denotes a Banach space. Abounded operator is continuous, that is, if φn→ φ, then Aφn→ Aφ. As a

sample application of Proposition 1.3.5, we have the following.

ofB(V). If A ∈ G (V) and B < A−1−1, then A + B is invertible. InparticularG (V) is an open subset of B(V).

I + A−1B. By assumptionA−1B ≤ A−1B < 1, so one can check that

converges to the inverse of I + A−1B with the help of Proposition 1.3.5.

complex numbers z for which A− zI is invertible. The spectrum of A is

(1.3.11) σ(A) =C \ ρ(A) = {z ∈ C : A − zI is not invertible}.

A scalar z∈ C is called an eigenvalue of A if Av = zv for some nonzerov∈ V; equivalently, z is an eigenvalue of A if and only if Ker(A − zI) isnontrivial. Let us note that every eigenvalue of A necessarily belongs toσ(A), but the converse need not hold in inﬁnite-dimensional spaces, as we

will see later.

The discrete spectrum of A consists of all isolated eigenvalues of A havingﬁnite multiplicity. That is, z∈ σdisc(A) if and only if

(1.3.12) 0 < dim Ker(A− zI) < ∞

and B(z, δ)∩ σ(A) = {z} for some δ > 0 (recall that B(z, δ) denotes theopen ball of radius δ centered at z). The complement of σdisc(A) in σ(A) iscalled the essential spectrum of A and is denoted σess(A) = σ(A)\ σdisc(A).The spectral radius of A is given by

(1.3.13) spr(A) = sup{|z| : z ∈ σ(A)}.

Later on, we will see that σ(A) is compact, so sup can be replaced by max

in the deﬁnition of the spectral radius.

The resolvent of A is the following map

(1.3.14) R(A,·) : ρ(A) → B(V), z → (A − z)−1.

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1.3. Bounded Linear Operators 19

Here and throughout the text, we adopt a standard convention from the mathematical physics literature and suppress occurrences of the

iden-tity operator, e.g., writing (A− z)−1instead of (A− zI)−1. The following

elementary identities will be very useful.

Proposition 1.3.8 (Resolvent identities).

(a) If A∈ B(V) and z, z∈ ρ(A), then

R(A, z)− R(A, z) = (z− z)R(A, z)R(A, z) = (z− z)R(A, z)R(A, z).

(b) If A, B∈ B(V) and z ∈ ρ(A) ∩ ρ(B), then

R(A, z)− R(B, z) = R(A, z)(B − A)R(B, z)= R(B, z)(B− A)R(A, z).

Let us note the following pleasant consequence of Proposition 1.3.8. The

function z→ R(A, z) deﬁnes an analytic map from the open set ρ(A) ⊂ C

toB(H). Concretely, (1.3.15) implies thatd

dzR(A, z) := limw→z

R(A, w)− R(A, z)

w− z= R(A, z)2

for any z∈ ρ(A). Many facts about analytic functions carry over from the

scalar case to the case of Banach-space-valued functions; we make use of this in the following proofs.

and σ(A) is a nonempty compact subset ofC.

One can check that R∈ B(V) and R(A − z) = (A − z)R = I, whencez∈ ρ(A). Thus, σ(A) is bounded and (1.3.17) holds. By Theorem 1.3.6,ρ(A) is open. Thus, σ(A) is compact.

It remains to see that σ(A) is nonempty. To that end, suppose to thecontrary that σ(A) =∅ and deﬁne g(z) = (A − z)−1for each z∈ ρ(A) = C.

By (1.3.18), one has g(z) = O(1/|z|) as |z| → ∞. Thus, by Liouville’stheorem (Exercise 1.3.15), g(z) = 0 for all z∈ C, a contradiction.

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If A∈ B(V) and p(z) = cnzn+· · ·+c1z +c0 is a polynomial, it is natural to deﬁne

(1.3.19) p(A) = cnAn+· · · + c1A + c0I.

It is helpful to note that p(A) and q(A) commute with one another for allpolynomials p and q and that the spectrum of A transforms in a natural

fashion under this operation.

p is a polynomial, then

(1.3.20) σ(p(A)) = p(σ(A)) :={p(z) : z ∈ σ(A)}.

σ(p(A)) = σ(c0I) ={c0} = p(σ(A)).

Henceforth, assume deg(p)≥ 1. First, suppose w ∈ p(σ(A)), so w = p(E)with E∈ σ(A). One can factor

p(z)− w = (z − E)q(z)

for a nonzero polynomial q. Since A− E is not invertible, neither isp(A)− p(E) = (A − E)q(A) = q(A)(A − E),

proving w = p(E)∈ σ(p(A)) and thus p(σ(A)) ⊆ σ(p(A)).Conversely, let w∈ C \ p(σ(A)), and write

for each n, proving

n→∞An1/n.

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1.3. Bounded Linear Operators 21

Then, it remains to show

n→∞An1/n≤ spr(A).

To that end, consider the function f (z) = (I− zA)−1, which is analytic on D ={z ∈ C : z spr(A) < 1}, and which enjoys the series representation

thereupon. Denoting the radius of convergence of this series by R, Cauchy’sformula for R (Exercise 1.3.15) yields

lim sup

n→∞An1/n = 1

operations are continuous in B = B(V, W). That is, if an→ a ∈ C,An→ A ∈ B, and Bn→ B ∈ B, then An+ Bn→ A + B, AnBn→ AB,and anAn→ aA.

Exercise 1.3.2. Prove Proposition 1.3.5.Exercise 1.3.3. Prove Proposition 1.3.8.

n=1is a sequence of bounded linear operators in a Banach space V. If supnAn < ∞, show that (An)n≥1 converges strongly if there is a dense subset V0⊆ V such that Anψ is convergent inVfor all ψ∈ V0.

n=1 be a bounded sequence of complex

num-bers, and deﬁne a linear operator D : 2(N) → 2(N) by [Dψ](n) = dnψ(n).Characterize σ(D) in terms of the sequence d.

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Exercise 1.3.7. Let (Ω,B, μ) be a σ-ﬁnite measure space, and let g ∈L∞(μ). Deﬁne a linear operator M = M

g: L2(μ)→ L2(μ) by [M f ](ω) =g(ω)f (ω). Show that M is a bounded operator. Characterize σ(M ) in termsof g.

I− AB is invertible if and only if I − BA is invertible. Hint: Looking at

the case when AB < 1 might be helpful (though it is not necessary).

(a) Show that Ker(A) and Ran(A) are linear subspaces ofV and W,

(b) True or false: If V and W are Banach spaces and A is bounded,then Ker(A) and Ran(A) are closed subspaces. (Give a proof or

counterexample in each case.)

such that the following hold.

(a) A has an isolated eigenvalue of inﬁnite multiplicity.

(b) A has eigenvalues of ﬁnite multiplicity that are dense in [0, 1].(c) A has eigenvalues of inﬁnite multiplicity that are dense in [0, 1].

Show that

dist(z, σ(A)).

Recall that the distance from a point to a set is given by dist(z, σ(A)) :=

inf{|z − s| : s ∈ σ(A)}. Hint: Theorem 1.3.6.

γ∈ Γ, let H =γ∈ΓHγ denote their direct sum (cf. Deﬁnition 1.2.15), and

suppose we are given bounded operators Aγ∈ B(Hγ) for each γ∈ Γ suchCharacterize σ(A) in terms of σ(Aγ).

Deﬁnition 1.3.12. We denote the operator constructed in Exercise 1.3.12

γ∈ΓAγand call it the direct sum of the Aγ.

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1.3. Bounded Linear Operators 23

real vector space V, and suppose p : V → R≥0 is positive homogeneous and

subadditive. Recall that p is positive homogeneous if and only ifp(av) = ap(v) for all a≥ 0 and v ∈ V

and is subadditive if and only if

p(v + w)≤ p(v) + p(w) for all v, w ∈ V.

Suppose 0 : U → R is a linear functional such that 0(u)≤ p(u) for allu∈ U. Show that 0 enjoys an extension to all ofV that is also bounded byp in two steps:

(a) If w∈ V \ U, let W denote the subspace generated by w and U.Show that there exists :W → R linear such that |U= 0 and

(x)≤ p(x) for all x ∈ W. Hints: The extension is uniquely deﬁnedby (w). Show that one can choose (w) in such a manner that

(u1+ w)≤ p(u1+ w) for all u1∈ U,

(u2− w) ≤ p(u2− w) for all u2∈ U,

and that this deﬁnes the desired .

(b) Show that there exists an extension of 0 to all of V that is alsobounded by p. Hint : Zorn.

of a complex vector space V, and suppose p : V → R≥0 is subadditive and

satisﬁes p(av) =|a|p(v) for all a ∈ C and v ∈ V. Suppose 0 :U → C is a

linear functional such that |0(u)| ≤ p(u) for all u ∈ U. Show that 0 enjoys an extension to all ofV that is also bounded by p. Hint: Write 0 = 1+ i2 for real linear functionals j, and show that 2 uniquely determines 1 via (1.3.34) 2(v) =−1(iv),v∈ V.

set. A function f : D→ V is said to be continuously diﬀerentiable on D if

f (w)− f(z)w− z

exists for all z∈ D and deﬁnes a continuous map D → V.

(a) Formulate and prove a Banach-space-valued version of Liouville’s

theorem. Hint : If f :C → V is bounded and continuously diﬀeren-tiable, the same is true of ◦ f for any ∈ B(V, C).

(b) Formulate and prove a Banach-space-valued version of the Cauchy

integral formula. Hint : Reduce toC via linear functionals as in the previous part.

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formula for the radius of convergence of a power series. Hint :

Start-ing from the Cauchy integral formula, this should look fairly similar to the standard proof from complex analysis.

Notes and Remarks. Theorem 1.3.11 is due to Gelfand [100]. The

ex-tension result in Exercise 1.3.13 is a version of the Hahn–Banach theorem,

named for work of Hahn in 1927 [113] and Banach in 1929 [20, 21]. The

extension to complex functionals in Exercise 1.3.14 is generally credited to

Bohnenblust and Sobczyk [39], Murray [179], Sukhomlinov [255]; the key

observation (1.3.34) appeared earlier in work of Lăowig [170].1.4. Self-Adjoint Operators and the Spectral Theorem

We now arrive at the fundamental result of the ﬁrst portion of the chapter: the spectral theorem for bounded self-adjoint operators. In fact, we will see several theorems, each of which might be considered a facet of the spectral theorem. Each version of the spectral theorem will be used in an essential manner throughout the text. The multiplicity of results and perspectives will, in sum, comprise our understanding of “the spectral theorem”.

We begin by deﬁning fundamental objects, namely the dual space and adjoint of a bounded operator.

dual space ofV; elements of V∗are called bounded linear functionals onV. In view of Proposition 1.3.5, V∗ is a Banach space with respect to the

operator norm.

Throughout the remainder of this section,H denotes a separable Hilbertspace. For each φ∈ H,

deﬁnes an element of H∗. In fact, every element ofH∗ is of this form.

= φ. The map T : φ→ φis an isometric anti-linear1 bijection fromHontoH∗.

subspace of H. Choose a unit vector χ ∈ [Ker ]⊥and take φ = (χ)χ. Wehave = φand φ is the unique element ofH with this property

(Exer-cise 1.4.1). Isometry and anti-linearity follow from the properties deﬁning an inner product, which we leave to the reader.

1By “anti-linear”, we mean T (φ + ψ) = T φ + T ψ and T (aφ) = aT φ for all φ, ψ∈ H and alla∈ C.

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1.4. Self-Adjoint Operators and the Spectral Theorem 25

there is a unique element ψ∈ H such that

for all φ∈ H. The map A∗: ψ→ ψ is called the adjoint of A.

B(Cn) = Cn×n. In this case, one can check that the adjoint operation coincides with the conjugate transpose. That is, (A∗)

jk= Akjfor A∈ Cn×n.

Let us collect a few of the main properties of the adjoint operation. The following proposition is left to the reader.

unitary if AA∗= A∗A = I, normal if AA∗= A∗A, and idempotent ifA2= A.

self-adjoint, then σ(A) is a compact subset ofR. Moreover, for any normalA∈ B(H),

Proof. Suppose A is self-adjoint. In view of Proposition 1.3.9, we need only

show that σ(A)⊆ R to prove the ﬁrst statement. Given z = x + iy withy= 0, notice that, for any ψ, one has

(1.4.7) (A − z)ψ2 =(A − x)ψ2+|y|2ψ2≥ |y|2ψ2.

In particular, this implies that (A− z) is one-to-one. Since Im ¯z = 0also, this implies Ker(A− ¯z) = {0} as well.

Additionally, (1.4.7) implies that Ran(A− z) is closed. For, if φn∈Ran(A− z) and φn→ φ, writing φn= (A− z)ψn and making use of (1.4.7) gives us

ψn− ψm2≤ |y|−2(A − z)(ψn− ψm)2=|y|−2φn− φm2,

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showing that ψn→ ψ for some ψ ∈ H, and hence φ = (A − z)ψ belongs toRan(A− z).

Since Ran(A− z) is closed, we may employ Theorem 1.2.11 to arrive atRan(A− z) = Ran(A − z) =Ran(A− z)⊥⊥

Ker(A∗− ¯z)⊥={0}⊥=H.

Thus, A− z is bijective. Finally, note that (1.4.7) implies (A − z)−1 is

bounded: R(A, z) ≤ |y|−1. Thus, z∈ ρ(A), as desired.2

Now, if A is normal, using (1.4.4) gives us

A2 = (A∗)2A21/2=(A∗A)21/2 =A∗A = A2.

Inductively, A2n

= A2n

for all n∈ Z+. Thus, A = spr(A) follows

Orthogonal projections (see Deﬁnition 1.2.10) play an important role in the spectral theorem. Let us collect a few important properties here.

orthogonal projection ontoV.

where we explicitly allow the case in which both sides are∞. Inparticular, the sum does not depend on the choice of basis.

is precisely the orthgonal projection onto the closed subspace V⊥). For any ψ, since Qψ∈ V⊥, we have

ψ2 =P ψ2+Qψ2 ≥ P ψ2,

showingP ≤ 1, and hence P ∈ B(H), since P is readily seen to be linear,

which proves (a).

For any φ∈ V, P φ = φ by deﬁnition. In particular, given ψ ∈ H,P ψ∈ V, so the previous observation implies P2ψ = P ψ, showing P2 = P .Next, using P ψ⊥ Qφ and P φ ⊥ Qψ, one notes

φ, P ψ = P φ + Qφ, P ψ = P φ, P ψ = P φ, P ψ + Qψ = P φ, ψ,showing P = P∗ and concluding the proof of (b).

2By the closed graph theorem, one does not actually need to check boundedness of the inversehere. We elected to directly check it since we have not stated or proved the closed graph theoremin the text.

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1.4. Self-Adjoint Operators and the Spectral Theorem 27

To prove (1.4.8), ﬁrst note that φ, P φ = P φ2 ≥ 0 for any φ, so the

sum on the right-hand-side of (1.4.8) consists solely of nonnegative terms,

and hence exists as an element of [0,∞]. Let N denote the dimension of V

(which may be ﬁnite or inﬁnite), let {ηk}N

k=1 be an orthonormal basis ofV,

and let {φn} be an orthonormal basis for H. Using Theorem 1.2.14 in the

ﬁrst line and Parseval’s formula in the penultimate line, one has

Deﬁnition 1.4.9. The quantity on the right-hand-side of (1.4.8) is called

the trace of P and is denoted Tr P .

With the basic facts in hand, we now move on to discuss the spectral the-orem for self-adjoint operators on a separable, inﬁnite-dimensional Hilbert space. The situation is more subtle than the ﬁnite-dimensional setting. In-deed, we will discuss three results that one can think of as versions of the spectral theorem: the functional calculus, the existence of spectral mea-sures, and the equivalence between self-adjoint operators and multiplication operators.

To motivate what follows and elucidate the subtleties, let us brieﬂy recall the spectral theorem for self-adjoint operators on Cn (i.e., Hermitian

ma-trices). If A∈ Cn×nis self-adjoint, then there is a unitary matrix U and adiagonal matrix D such that U∗AU = D. The columns of U are normalizedeigenvectors of A, and the diagonal entries of D are the corresponding

eigen-values. Now, if H is an inﬁnite-dimensional Hilbert space and A ∈ B(H) isself-adjoint, A may have no eigenvalues whatsoever! The reader is invitedto verify that the discrete Laplacian Δ : 2(Z) → 2(Z), given by

(1.4.9) [Δψ](n) = ψ(n + 1) + ψ(n− 1), ψ ∈ 2(Z), n ∈ Z,

is an example of such an operator (Exercise 1.4.7). That is, Δ∈ B(2(Z)), Δ∗ = Δ, and Δ has no eigenvalues.

Thus, we need to reformulate the ﬁnite-dimensional spectral theorem in a fashion that is amenable to the present situation. To that end, given a

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self-adjoint matrix A∈ Cn×n, write P

λ for the orthogonal projection onto

Vλ:= Ker(A− λ). Then, the spectral theorem asserts that

λ∈σ(A)λPλ

and that PλPμ= PμPλ= 0 whenever λ= μ. This allows one to see that,for a polynomial function f , f (A) =

f (λ)Pλ, that is, f (A) depends onlyon the values assumed by f on the spectrum of A.

This is the ﬁrst point of view that we develop. Suppose that A∈ B(H)is self-adjoint. We wish to apply a function f to A in order to form a newoperator f (A). Recall that if f (z) = cnzn+· · · + c1z + c0 is a polynomial,

we deﬁned f (A) as follows:

(1.4.11) f (A) = cnAn+· · · + c1A + c0.

In order to pass from polynomials to more general continuous functions, one would like to understand the behavior of this construction with respect to uniform approximation on compact sets. The following proposition supplies the key input.

(1.4.12) p(A) = sup{|p(z)| : z ∈ σ(A)}.

Proof. This follows from Propositions 1.3.10 and 1.4.7, and the observation

Thus, if A is self-adjoint, it is then possible to deﬁne f (A) for functions fthat are continuous on σ(A). Concretely, given f∈ C(σ(A)), let p1, p2, . . . bea sequence of polynomials that converges uniformly to f on σ(A). Naturally,

one would like to deﬁne

The ﬁrst version of the spectral theorem states that this is well-deﬁned and has the kinds of properties one might desire from such an apparatus.

Theorem 1.4.11 (Spectral theorem: continuous functional calculus). Let

A∈ B(H) be self-adjoint, and let A ⊆ B(H) denote the algebra generatedby A (that is, the smallest closed subalgebra ofB(H) containing A).

(a) The association from (1.4.13) is well-deﬁned, i.e., the limit existsand does not depend on the choice of approximating polynomials.

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1.4. Self-Adjoint Operators and the Spectral Theorem 29

(b) The map C(σ(A)) f → f(A) ∈ A is a ∗-homomorphism. Thatis to say, for all continuous f and g and all a∈ C, one has

(f + g)(A) = f (A) + g(A),

where denotes the function that is identically equal to 1 on σ(A).(c) For all f∈ C(σ(A)), f(A) = f∞.

n=1is a sequence of polynomials with pn→ f uniformly onσ(A), then (pn(A))∞

n=1 is a Cauchy sequence inB(H) by Proposition 1.4.10,

and hence the limit in (1.4.13) exists by completeness ofB(H). If (qn)∞n=1is

another sequence of polynomials that converges uniformly to f on σ(A), thenqn(A) and pn(A) converge to the same limit (again by Proposition 1.4.10).

The proofs of (b) and (c) are left to the reader. Having constructed the continuous functional calculus, we are now able

to describe the spectral measures of A. Once again, the analogy with the

ﬁnite-dimensional case is a useful guide. Let A be a self-adjoint n× ncomplex matrix with spectral decomposition as in (1.4.10). For v∈ Cn and

f a continuous function, to calculate the corresponding matrix element off (A), we have

f (λ)Pλv2.

Thus, we can think of the matrix element v, f(A)v as being obtained byintegrating f against the Dirac measure supported on σ(A) with the weightat λ∈ σ(A) given by Pλv2. In the inﬁnite-dimensional setting, one can no longer guarantee the existence of a basis of eigenvectors, but one can still speak of the measures deﬁned by the left-hand side of (1.4.19).

Theorem 1.4.12 (Spectral theorem: existence of spectral measures). Let

A∈ B(H) be self-adjoint. For all φ, ψ ∈ H, there is a complex Borelmeasure μ = μAφ,ψsuch that

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for all f∈ C(σ(A)). In particular, if ψ = φ, then μ = μA

φ:= μAφ,φis ameasure.3

φ, f(A)ψ deﬁnes a bounded linear functional on C(σ(A)), and hence the

existence of a complex Borel measure satisfying (1.4.20) follows from the Riesz–Markov–Kakutani representation theorem.

Moreover, if φ = ψ and f∈ C(σ(A)) is positive, there is a uniquecontinuous g such that g2= f and g≥ 0. One then has

(f ) =φ, f(A)φ = φ, g(A)2φ = g(A)φ2≥ 0,

and hence is also a positive functional. Thus, in this case μ is also a

The measures constructed in Theorem 1.4.12 are called the spectral mea-sures of A. Whenever A is clear from context, we suppress it from thenotation, writing, e.g., μφ,ψand μφfor μAφ,ψand μAφ.

At this point, let us note a helpful consequence of Theorem 1.4.12 re-garding the norm of the resolvent operator.

dist(z, σ(A)).

dist(z, σ(A)), and let ψ be a unit vector. By Theorem 1.4.12, we have

proving the desired upper bound on R(A, z). The opposite inequality

holds more generally and follows from Theorem 1.3.6 (see Exercise 1.3.11). The construction of spectral measures in turn allows one to extend the

map f→ f(A) and the formula (1.4.20) to all bounded Borel measurablefunctions on σ(A). Concretely, given

f∈ B(σ(A)) := {g : σ(A) → C : g is bounded and Borel measurable} ,we deﬁne f (A) by specifying its matrix elements,

(1.4.22) φ, f(A)ψ =

f (E) dμAφ,ψ(E).

3We adopt the convention in this text that “measure” always means “nonnegative regularBorel measure”, using modiﬁers (e.g., “signed” or “complex”) to distinguish variants.

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1.4. Self-Adjoint Operators and the Spectral Theorem 31

B(H) be self-adjoint.

(a) For each f∈ B(σ(A)), (1.4.22) deﬁnes an element f(A) ∈ B(H).(b) This mapping is a∗-homomorphism from B(σ(A)) to B(H) (cf.

Theorem 1.4.11(b)).

(c) Additionally, if fn, n≥ 1 is a sequence in B(σ(A)) that is uni-formly bounded in the supremum norm and fn→ f pointwise onσ(A), then fn(A)→ f(A) strongly (i.e., fn(A)ψ→ f(A)ψ for ev-ery ψ∈ H).

By Theorem 1.4.14(b), let us note that if S⊆ R is a Borel set, then χS =

χ2S= χS implies that P(S) := χS(A) is an orthogonal projection, knownas the spectral projection of A on S. Here, χSdenotes the characteristicfunction of S, given by χS(x) = 1 for x∈ S and 0 otherwise. Using the

Borel functional calculus and the Cauchy–Schwarz inequality, one can check that

|μφ,ψ(S)| = |φ, χS(A)ψ| ≤ μφ(S)1/2μψ(S)1/2,

for all Borel S⊆ R, so μφ,ψ is absolutely continuous with respect to both

μφand μψ. The family of these projections comprises a projection-valued-measure (or a resolution of the identity) in thatP(∅) = 0, P(R) = I,P(S ∩ T ) = P(S)P(T ), and, if {Sn: n∈ N} is a countable collectionof pairwise disjoint Borel sets and S =

Let us discuss one ﬁnal manifestation of the spectral theorem. To

mo-tivate this, consider again a self-adjoint A∈ Cn×n. For simplicity, assume that every eigenvalue of A is simple (this is not necessary, but it makes this

discussion more transparent). Consider the space

H = 2(σ(A)) ={f : σ(A) → C}

and map v∈ Cnto f∈ H via f(λ) = eλ, v, where eλ is a normalized

eigenvector corresponding to the eigenvalue λ. Then, the spectral theoremfor ﬁnite matrices establishes that the map U taking v to f as above isunitary and [U AU∗g](λ) = λg(λ). Our ﬁnal instance of the spectral theorem

generalizes this perspective.

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Theorem 1.4.15 (Spectral theorem: equivalence to multiplication

opera-tors). Suppose A∈ B(H) is self-adjoint. Then, there exist at most countablymany φn∈ H such that with μn= μφn, there is a unitary map

Here we write g(E) ={gn(E)}nfor an element g∈nL2(R, μn).

The central notion here is the decomposition ofH into cyclic subspaces.

generated by A and φ is the smallest closed subspace ofH containing{Anφ : n = 0, 1, 2, . . .}. An element φ of H is called cyclic for the oper-ator A if the corresponding cyclic subspace is all ofH.

and φ is a cyclic vector for A. We wish to show that (1.4.24) can be deﬁnedwith just the measure μ = μφ. To deﬁne a map U :H → L2(R, μφ) with the

desired property, one needs to invert the map f→ f(A) discussed above.Namely, if f∈ C(σ(A)), set U(f(A)φ) = f; the reader should check thatthis is well-deﬁned (i.e., if f (A)φ = g(A)φ for f, g∈ C(σ(A)), then f = g).This deﬁnes U on a dense set. One can then verify that U preserves normsand that (U AU−1f )(E) = Ef (E) for such f ’s and then extend the deﬁnition

and the relation to the whole space.

This gives the statement of the spectral theorem if there is a cyclic vector. If not, a Zorniﬁcation shows that one can decompose H = Hn

where each Hnis a cyclic subspace generated by A and some φn∈ H.Denote μn= μAφ

nthe spectral measure and Un:H → L2(R, μn) the unitary

equivalence f (A)φn→ f. Then U =Un is the desired unitary map. The multiplication operators encountered in Theorem 1.4.15 can be thought of as generalizations of direct sums. Let us make this explicit. For this discussion, we only state theorems and leave proofs to the exercises.

a ﬁxed separable Hilbert space. A function A : Ω→ B(H) will be calledweakly measurable if for all φ, ψ∈ H, the function gφ,ψ : Ω→ C,

(1.4.25) gφ,ψ: ω→ φ, Aωψ

is measurable; following a convention for operator-valued functions that will

be standard later in the text, we write Aωrather than A(ω) for the value

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1.4. Self-Adjoint Operators and the Spectral Theorem 33

of A at ω. Similarly, a function ϕ : Ω→ H is said to be measurable4 if

ψ, ϕ(ω) is measurable for all ψ ∈ H.

The (constant-ﬁber) direct integral , denoted by L2(Ω, μ;H) or sometimes

consists of measurable maps f : Ω→ H (as always, modulo the subspace of

measurable functions that vanish almost everywhere) for which

L2(Ω, μ;H) is a Hilbert space. An operator A ∈ B(L2(Ω, μ;H)) is calleddecomposable if there is a weakly measurable family{Aω} such that

A·∞:= esssup

ω∈ΩAω < ∞

(recall that esssup denotes the μ-essential supremum, deﬁned in (1.2.11))

(1.4.29) [Af ](ω) = Aω(f (ω)),μ-a.e. ω∈ Ω.

In analogy to the notation from (1.4.26), we will sometimes write this as

⊕

Let us make a few observations about this construction. First, if μ is

the counting measure on Ω, then we see that

the direct sum of copies of H (cf. Deﬁnition 1.2.15), and the decomposable

operators are precisely those that can be expressed as direct sums of opera-tors inH (cf. Deﬁnition 1.3.12). Thus, the direct integral can be viewed as

a generalization of the direct sum.

4For consistency, we should also call this notion “weak measurability” in order to distinguishit from the ostensibly stronger notion of measurability given by equippingH with the σ-algebra

generated by the metric topology thereupon. However, these two notions are equivalent (Exer-cise 1.12.1).

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Given a measurable family {Aω}ω∈Ω with A·∞<∞, one can check(Exercise 1.4.13) that there is a unique decomposable operator A for whichone has (1.4.29) and the operator norm of A satisﬁes

Let us conclude with a versatile criterion for detecting the spectrum of

a self-adjoint operator. On the one hand, every eigenvalue of an operator Abelongs to the spectrum of A. On the other hand, as we noted earlier, there

are self-adjoint operators (such as Δ, the discrete Laplacian) that have no eigenvalues at all (Exercise 1.4.7). However, for self-adjoint operators, one has the following extremely helpful replacement: one can characterize the

spectra of self-adjoint operators as precisely those z for which the operatorenjoys a sequence of approximate eigenvectors. Let us make this precise.

a sequence (ψn)∞

n=1where ψn∈ H for all n, ψn = 1 for all n, and

n→∞(A − z)ψn = 0.

z∈ σ(A). If A is self-adjoint, the converse holds as well.

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1.4. Self-Adjoint Operators and the Spectral Theorem 35

1 =ψn = (A − z)−1(A− z)ψn ≤ (A − z)−1(A − z)ψn.

Thus, (A − z)ψn ≥ R(A, z)−1and hence (ψn)∞

n=1 cannot be a Weyl

sequence for A at z.

Conversely, if E∈ σ(A) and A = A∗, then E∈ R and E + iδ ∈ ρ(A)for all δ > 0. Choose a sequence δn↓ 0. By Theorem 1.4.13, we can ﬁnda sequence (φn)n≥1 of unit vectors such that (A − E − iδn)−1φn → ∞ as

Let us note the following helpful reformulation.

(1.4.36) dist(z, σ(A)) = inf

v
=1(A − z)v.

right-hand side vanishes by Theorem 1.4.20. If z∈ ρ(A), the claim follows by

not-ing that the left-hand side is equivalent to R(A, z)−1 by Theorem 1.4.13

and the right-hand side is equivalent to the same quantity by a direct

Additionally, we can use the Weyl sequence formalism to bound spectra of strong limits from above.

is a unit vector ψ for which(A − E)ψ < ε. By strong convergence, we get(An− E)ψ < ε

for all large n. For such n, this means that dist(E, σ(An)) < ε by Corol-lary 1.4.21. Consequently, E belongs to the right-hand side of (1.4.37). We caution the reader that the previous bound is one-sided. In

gen-eral, σ(A) can be much smaller than the right-hand side of (1.4.37)

(Exer-cise 1.4.12).

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Let us note that there is a characterization of the essential spectrum

via Weyl sequences comprised of orthonormal vectors similar to that of

Theorem 1.4.20.

A at z is an orthonormal sequence of vectors (ψn)∞

n=1, where ψn∈ H for alln,ψn = 1 for all n, and

n→∞(A − z)ψn = 0.

is a singular Weyl sequence for A at E.

multiplic-ity, then one can simply take (ψn)∞

n=1 to be an orthonormal basis of the corresponding eigenspace. Otherwise, with

one has Tr (Pn)≥ 1 for inﬁnitely many n ≥ 1. Thus, the desired orthonormalsequence can be extracted by choosing a normalized vector from Ran(Pn)for each n with Tr (Pn)≥ 1.

Conversely, assume that (ψn)∞

n=1is a singular Weyl sequence for A at E.By Theorem 1.4.20, E∈ σ(A), so it suﬃces to show that E /∈ σdisc(A). As-sume to the contrary that E∈ σdisc(A), which implies Pε:= χ(E−ε,E+ε)(A)is a ﬁnite-rank projection for some ε > 0. By a direct calculation usingorthonormality, Pεψn→ 0 as n → ∞. On the other hand, writing μψn for

the spectral measure associated to the pair (A, ψn), one has

which is incompatible with ψn = 1 and Pεψn→ 0. Thus, we obtain a

This has the following consequence: the essential spectrum is invariant under ﬁnite-rank perturbations.5

Corollary 1.4.25. If A and R are self-adjoint and R is ﬁnite-rank, then

(1.4.39) σess(A + R) = σess(A).

5In fact, the essential spectrum is invariant under compact self-adjoint perturbations, butwe do not state that result since we have not discussed compact operators in the text. Moreover,there are further generalizations beyond the self-adjoint case.

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1.4. Self-Adjoint Operators and the Spectral Theorem 37

σess(A + R) when R is rank-one, say R = λφ, ·φ for some λ and someφ∈ H. Given E ∈ σess(A), Theorem 1.4.24 implies the existence of a singu-lar Weyl sequence (ψn)∞

n=1for A at E. By a direct calculation, we have(A + R− E)ψn= (A− E)ψn+ λφ, ψnφ → 0.

The ﬁrst term goes to zero because the ψncomprise a Weyl sequence for Aat E. The second term goes to zero by orthonormality. Thus, (ψn)∞

n=1 is

also a singular Weyl sequence for A + R at E and hence E∈ σess(A + R),

let φ = (χ)χ with χ a unit vector in Ker()⊥. Show that = φ. Conclude the proof of Proposition 1.4.2 by showing that ψ→ ψ is an anti-linear isometry that maps H onto H∗. Hint : For any ψ, (ψ)φ− (φ)ψ ∈ Ker().

Exercise 1.4.2. Prove Proposition 1.4.5.

e−itAis unitary for every t∈ R.

σ(S) = ∂D.

Exercise 1.4.7. Let Δ be the operator deﬁned in (1.4.9).

(a) Show that Δ is a bounded, self-adjoint operator with σ(Δ) =

[−2, 2].

(b) Show that Δ has no eigenvalues.

(c) Let μ = μΔδ0 denote the spectral measure associated with the pair Hint : With S as in Exercise 1.4.6, Δ = S + S−1.

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Exercise 1.4.8. If H is separable and A ∈ B(H) is self-adjoint, show thatA may have at most countably many eigenvalues.

and [Bψ]n= ψn−1(with the convention ψ0 = 0).

(a) Show A = B∗, AB = I, and BA= I.

(b) Show that every z∈ D := {w ∈ C : |w| < 1} is an eigenvalue of A.(c) Show that σ(A) = σ(B) =D.

(d) Show that there is no Weyl sequence for B at 0.

(e) Construct X, Y∈ B(H) such that XY = I and Y X is an

orthog-onal projection onto a subspace of inﬁnite codimension.

Exercise 1.4.10. Prove Theorem 1.4.14. Hint : For part (c), use dominated

Exercise 1.4.11. Following the notation in Exercise 1.3.6, when is D

self-adjoint? Unitary? Normal? An orthogonal projection?

K, show that there exist self-adjoint operators Anand A such that An→A strongly, σ(An) = K for all n, and σ(A) ={a}. Hint: Try diagonaloperators on 2(N).

show that there is a unique decomposable operator A for which one has

(1.4.29) and (1.4.32).

Exercise 1.4.14. Prove Theorem 1.4.18.

Exercise 1.4.15 (Unbounded operators). Let (D(A), A) be a linear

opera-tor in a Hilbert space H for which D(A) is dense in H (in this case, we sayA is densely deﬁned ). Notice that we do not assume that A is bounded.

(a) Deﬁne

D(A∗) ={φ ∈ H : ψ → φ, Aψ

extends to a bounded linear functional onH}.Prove that, for φ∈ D(A∗), there is a unique element A∗φ∈ H with

the property that

A∗φ, ψ = φ, Aψ

and that this determines a linear operator (D(A∗), A∗) in H.(b) Show by example that D(A∗) ={0} is possible. Hint: Heuristically,

A∗ functions like the conjugate transpose of a matrix. Use this as

a starting point for a precise construction.

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1.5. Dual Spaces and Locally Convex Topologies 39

(c) Consider the linear operator A deﬁned by D(A) ={φ ∈ 2(Z) :

supp φ is compact} and [Aφ](n) = nφ(n). Calculate A∗.

(d) One says that A is self-adjoint if (D(A), A) = (D(A∗), A∗) (typi-cally abbreviated simply as A = A∗). Find a self-adjoint extension of the operator A from part (c), that is, a self-adjoint operator(D(B), B) such that D(B)⊃ D(A) and B|D(A)= A.

Notes and Remarks. Proposition 1.4.2 is due to F. Riesz [205] and is

sometimes called the Riesz representation theorem. The spectral theorem

for bounded operators goes back to Hilbert [122], and the extension tounbounded operators is due to von Neumann [262].

The exposition in this section draws inspiration from Reed and Simon

[196], Simon [243], and Teschl [257].

1.5. Dual Spaces and Locally Convex Topologies

We conclude the discussion of functional analysis in general spaces with a brief excursion into dual spaces and alternative topologies that one can place on vector spaces. For ease of exposition and since it suﬃces for our purposes,

we restrict our attention to real vector spaces and real linear functionals in

the present section.

We say that V is a topological vector space if the topology is Hausdorﬀ and

the natural vector space operations are continuous. That is,

We callV a locally convex topological vector space if in addition there is

a neighborhood basis for the topology at 0 comprised of open convex sets. If V is a normed linear space, the collection N = {B(0, r) : r > 0} is a

neighborhood basis for the norm topology at the origin. In particular, since balls are convex, every normed vector space is locally convex in the norm topology.

Let us brieﬂy recall some terminology and notions from topology. If

T1 and T2 are topologies on a set X, we say thatT1 is weaker thanT2

(equivalently, T2 is stronger thanT1) if every set open with respect to

T1 is open with respect to T2. Since the intersection of any collection of

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topologies is itself a topology, we can in certain situations precisely speak of

the weakest topology enjoying speciﬁed properties. For instance, if X is a

set and {fα: α∈ A } is a collection of functions from X into a topologicalspace Y , one can construct on X the weakest topology for which every fα

is continuous. The locally convex topologies of interest are those that are generated in precisely this fashion by families of linear functionals.

set of linear functions from V to R. We deﬁne T (V, L) to be the weakest

topology on V in which every ∈ L is continuous; equivalently, T (V, L) is

the intersection of all topologies for which every element of L is continuous.

To construct the topology T = T (V, L), we note that a set U ⊆ V is open

with respect to T if and only if for every x ∈ U, there are ﬁnitely many1, . . . , n∈ L and intervals (aj, bj) such that

We say that L is a separating family of linear functionals if for anyu= v in V, there is some ∈ L such that (u) = (v). Naturally, if L is

a separating family of linear functionals on V, then T (V, L) is a Hausdorﬀ

topology on V.

In the particular case in which V is a Banach space and L = V∗, we

callT (V, V∗) the weak topology onV. In the case V = U∗ for some normed

space U, each u ∈ U induces a linear functional on V via Eu(v) = v(u).

Identifying U with the collection of evaluation functionals {Eu: u∈ U}, we

callT (U∗,U) the weak-∗ topology on V.

weak-∗ topology on V∗ are locally convex.

functionals, then the topologyT (V, L) makes V a locally convex topologicalvector space.

n∈ N} be an orthonormal basis of H. Sinceψ2=

|φn, ψ|2<∞,

we have φn, ψ → 0. In particular, φn→ 0 weakly. However,

orthonor-mality implies that φn− φm =√2 whenever n= m, so (φn)∞

n=1 is not convergent in the norm topology.

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1.5. Dual Spaces and Locally Convex Topologies 41

xn→ 0, equipped with the supremum norm. One can identify U∗ with V = 1(N) and V∗with ∞(N) (Exercise 1.5.2). One can check that δnconverges to 0 as n→ ∞ in the weak-∗ topology on 1(N), but not in the

weak topology on 1(N). In fact, one can show that a sequence (yn)∞n=1 of

elements of 1(N) converges in the weak topology if and only if it converges in the norm topology (Exercise 1.5.3).

C(X,R) be the space of continuous functions equipped with the supremum norm. By the Riesz–Markov–Kakutani representation theorem, V∗ may be

isometrically identiﬁed withM(X), the space of signed measures on X with

the total variation norm. In view of these identiﬁcations, the weak-∗

topol-ogy on M(X) is then precisely the weakest topology for which every linear

functional of the form

f dμ

is continuous.

One advantage aﬀorded by the weak-∗ topology is that it makes the

closed unit ball of the dual space a compact space that is metrizable when-ever V is separable. First, we prove compactness.

is compact in the weak-∗ topology.

∈ B to an element x = (xv)v∈V∈ S = !v∈V[−v, v], namely bychoosing the element x∈ S whose vth coordinate is (v). Equipping S withthe product topology, S is compact by Tychonoﬀ’s theorem. The mappingf taking ∈ B to x ∈ S is a homeomorphism from B onto its image, by

the deﬁnitions of weak-∗ topology and product topology. One can conﬁrmthat the image of B under f is closed. Thus, B is homeomorphic to a closed

subset of a compact space and hence is itself compact.

Next, we discuss metrizability of the unit ball, which holds whenever the initial Banach space is separable.

ofV∗is metrizable in the weak-∗ topology.

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Proof. Let S ={vj: j∈ N} be a dense subset of V. The reader can verifythat the function d given on the closed unit ball ofV∗ by

is a metric that generates the weak-∗ topology thereupon.

is a compact metrizable space in the weak topology.

Proof. This follows by combining Theorems 1.5.8 and 1.5.9 with

Proposi-tion 1.4.2 (which implies that the weak and weak-∗ topologies on H

The next fundamental result we want to prove is the Krein–Milman theorem, which shows that a compact convex set is precisely the closed convex hull of its extreme points. First, we need a deﬁnition.

convex. A nonempty subset E ⊆ K is called an extreme subset of K if it isconvex and whenever ax + (1−a)y ∈ E with 0 < a < 1 and x, y ∈ K, one hasx, y∈ E. A point x ∈ K is an extreme point of K if {x} is an extreme subset

of K. The convex hull of a set S ⊆ V is the smallest convex set containingS, that is, the intersection of all convex sets containing S. Similarly, theclosed convex hull ofS is the smallest closed convex set containing S.

locally convex spaceV.

(a) K has at least one extreme point.

(b) K is the closed convex hull of its extreme points.

To prove this, it will be helpful to develop a bit of theory regarding continuous linear functionals on a locally convex space.

If K ⊆ V contains 0 and is nonempty, open, and convex, the gauge of K is

deﬁned by

(1.5.5) pK(v) = inf{c ∈ (0, ∞) : v/c ∈ K}.

One can view this as a generalization of the notion of a norm, for, ifV is

a normed space andK is the open unit ball in V, then one has pK(x) =xfor all x∈ V.

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1.5. Dual Spaces and Locally Convex Topologies 43

convex, and let p = pKdenote its gauge. Then p has the following properties.

(a) 0≤ p(v) < ∞ for all v ∈ V.(b) p(x) < 1 for all x∈ K.

(c) p(z)≥ 1 for all z /∈ K.

(d) p(av) = ap(v) for all a≥ 0 and v ∈ V.(e) p(u + v)≤ p(u) + p(v) for all u, v ∈ V.

(f) If :V → R is a linear functional, then is continuous if and onlyif ≤ pKfor some nonempty, open, convexK containing 0.

any open, convexK ⊆ V and any y ∈ V \ K, there is a continuous linearfunctional onV such that

Since y /∈ K, we have p(y) ≥ 1. In particular, if we deﬁne a linearfunctional on the span of y by (ay) = a, we have

By Lemma 1.5.14, p is positive homogeneous and subadditive, so we may

apply the Hahn–Banach theorem (cf. Exercise 1.3.13) to construct a linear

functional for which (y) = 1 and (v)≤ p(v) for all v ∈ V. In particular,(x)≤ p(x) < 1 = (y) for every x ∈ K. By Lemma 1.5.14, is continuous,

so the ﬁrst statement is proved.

For the second statement, suppose x= y. By local convexity, there isan open convex neighborhood of x that does not meet y. Apply the ﬁrstpart of the lemma to this set and y yields a continuous linear functional

We can push just a bit further and also get a stronger separation

state-ment for closed convex sets.

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Corollary 1.5.16. SupposeV is a locally convex topological vector space.For any closed, convexK ⊆ V and any y ∈ V \ K, there is a continuouslinear functional onV and a scalar α such that

(1.5.8) (x) < α < (y)for all x∈ K.

topology of V is locally convex, there is an open convex set G containing

0 that is disjoint from K. In particular, the set K1 = K − G is open and

convex (cf. Exercise 1.5.4) and does not contain 0, so Lemma 1.5.15 implies

that there is a linear functional with (x) < (y) = (0) = 0 for all x∈ K1.

Since does not vanish identically andG is a neighborhood of 0, there existsz∈ G with (z) = α < 0. Since (x) < 0 for all x ∈ K1, we get for all w∈ K

(1.5.9) (w) = (z) + (w− z) < α + 0 < 0 = (y),

extreme subsets of K. By compactness, the intersection of any totally

or-dered subcollection ofE is nonempty. Thus, by a Zorniﬁcation, there exists

a minimal elementS of E . We claim that S consists of a single point. Sup-pose on the contrary that x1 = x2 belong to S. By Lemma 1.5.15, thereexists a continuous linear functional deﬁned onV such that (x1)= (x2).

By compactness, attains a maximal value M onS. One can check thatthe set of x∈ S for which (x) = M is an extreme subset of K that must be

strictly smaller than S, since is nonconstant on S; a contradiction. Thus,S consists of a single point and therefore K has at least one extreme point.

(b) Let E ⊆ K denote the collection of all extreme points of K, and

let K denote the closed convex hull of E. Naturally, K ⊆ K, so it remainsto show the opposite inclusion. To that end, assume that u∈ V \ K. ByCorollary 1.5.16, there is a continuous linear functional such that (v) <(u) for all v∈ K. By compactness, attains a maximum value M on K.The set of x∈ K for which (x) = M is a closed extreme subset of K andcontains an element y ofE by a repetition of the Zorniﬁcation in part (a).In particular, for all z∈ K, one has (z) ≤ M = (y) < (u), showing u /∈ K,

Exercise 1.5.1. Prove Theorem 1.5.4.

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1.5. Dual Spaces and Locally Convex Topologies 45

n=1: xn∈ R, xn→ 0 asn→ ∞} consist of all real-valued sequences converging to zero, equippedwith the ∞ metric.

(a) Show that every y∈ 1(N) deﬁnes a bounded linear functional on

(b) Show that this gives an isometric bijection of c0(N)∗with 1(N).

(c) In a similar fashion, show that ∞(N) may be isometrically

identi-ﬁed with 1(N)∗.

(d) Show that δn→ 0 in the weak-∗ topology on 1(N) but not in the weak topology.

n=1 denote a sequence of elements of V. Prove that yn→ y in the weak topology on V if and only if yn→ y

in the norm topology on V. Hints: For the hard direction, assume to thecontrary that yn→ 0 weakly in 1 and yn1 → 0. Pass to a subsequencezk= ynksuch that “most” of the norm of zkis concentrated on a set Sk in

such a way that the Sk’s are disjoint.

(a) If K1,K2 ⊆ V are convex, show that K1− K2={x1− x2 : xj∈ Kj}

is convex. Moreover, if V is a topological vector space, show thatK1− K2 is open wheneverK1 is open.

(b) If T :V → W is linear and K ⊆ V is convex, show that T (K) is

(c) If T :V → W is linear and K ⊆ W is convex, show that T−1(K) is

Exercise 1.5.5. Prove Lemma 1.5.14.

set of linear functions fromV to R. If U ⊆ V is open with respect to T (V, L)

and 0∈ U, show that there are linearly independent 1, . . . , n∈ L and δ > 0

set of linear functions fromV to R. Show that a linear functional : V → R

is continuous with respect to the topology T (V, L) if and only if is a

ﬁnite linear combination of elements of L. Hint: Start by showing that if

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{1, . . . , n} is a linearly independent subset of L, then the range of the mapv→ (1(v), . . . , n(v)) is all of Rn.

Notes and Remarks. The exposition in this section is inspired by Lax

[166] and Simon [244].

Theorem 1.5.8 is usually named the Banach–Alaoglu theorem after

con-tributions of those authors [5,21]. The result was discovered in various formsindependently by many authors. Hilbert [122], Riesz [203], and Helly [120]had special cases before the general result, while Bourbaki [44], Shmulyan[228], and Kakutani [137] proved general results independently around the

same time as Alaoglu. Discussing the number of authors with theorems related to this result, Pietsch remarks:

To be historically complete, one should speak of the Ascoli–Hilbert–Fr´echet–Riesz–Helly–Banach–Tychonoﬀ– Alaoglu–Cartan–Bourbaki–Shmulyan–Kakutani theorem.

Theorem 1.5.12 is due to Krein and Milman [157].

1.6. Spectral Decompositions and Quantum Dynamics

Quantum mechanics suggests the study of the Schrăodinger equation, which is a time-dependent problem. A classical and very important approach to this problem is based on a detailed spectral analysis of the Schrăodinger operator appearing in the Schrăodinger equation, which is in fact a time-independent problem. This has led to the development of the study of suitable spectral decompositions, which may be investigated for any self-adjoint operator. These decompositions can then be connected to corresponding behaviors in the time-dependent setting. In this section we develop the traditional aspects of this approach and present the decomposition of spectral measures and spectra into their absolutely continuous, singular continuous, and pure point parts, and we furthermore discuss how each piece of the decomposition gives information about the behavior of the solution of the time-dependent Schrăodinger equation via the so-called RAGE theorem. After this section, the next two sections address reﬁnements in the pure point and singular continuous subspaces.

1.6.1. Decompositions of Measures and Operators. Recall that any

measure μ on R has a unique Lebesgue decomposition (1.6.1) μ = μpp+ μsc+ μac,

where μppis a pure point measure, μscis a singular continuous measure, and

μac is an absolutely continuous measure. This means that μpp(R \ C) = 0

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1.6. Spectral Decompositions and Quantum Dynamics 47

for some countable set C, μsc({E}) = 0 for every E ∈ R, μsc(R \ S) = 0for some set S of zero Lebesgue measure, and μac(N ) = 0 for every set Nof zero Lebesgue measure. We also write μc = μsc+ μac for the continuouspart and μs = μpp+ μsc for the singular part of μ.

In general, we write μ ν to mean μ is absolutely continuous withrespect to ν, that is, μ(S) = 0 whenever ν(S) = 0. If μ and ν are mutuallyabsolutely continuous, that is, μ ν and ν μ, we say that μ and ν areequivalent. We write μ⊥ ν to mean μ and ν are mutually singular, that is,there are disjoint sets A and B such that μ(R \ A) = ν(R \ B) = 0.

is self-adjoint. Deﬁne the following spaces:

Whenever we can make statements about multiple pieces of the spectral decomposition simultaneously, we may employ• as a stand-in for ac, sc, pp,

c, and s. Concretely, we could rephrase the deﬁnitions above by declaring (1.6.7) H• ={φ : μφ= μφ,•}, • ∈ {ac, sc, pp, c, s}.

(a) The setsH•,• ∈ {ac, sc, pp, c, s}, are A-invariant closed subspaces

{ac, sc, pp, c, s} is closed and A-invariant, so, for each •, we can deﬁne alinear operator A•:= A|H•∈ B(H•) by restricting A toH•. Deﬁne

(1.6.8) σ•(A) = σ(A•),• ∈ {ac, sc, pp, c, s}.

Respectively, the sets σac(A), σsc(A), σpp(A), σc(A), and σs(A) are called

the absolutely continuous, singular continuous, pure point, continuous, and

singular spectra of A. Finally, letP•=P•(A) be the orthogonal projection

onto H• for• ∈ {ac, sc, pp, c, s}.

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We say that A has purely absolutely continuous spectrum ifHac = H,purely singular continuous spectrum ifHsc = H, pure point spectrum6 if

Hpp =H, purely continuous spectrum if Hc=H, and purely singular spec-trum ifHs =H.

or-thogonal, σac(A), σsc(A), and σpp(A) need not be pairwise disjoint

(Exer-cise 1.6.3).

A nonzero vector φ∈ H is called an eigenvector of an operator A ifthere is a so-called eigenvalue E∈ C with the property Aφ = Eφ. We writeσp(A) for the set of eigenvalues of A. Eigenvalues are intimately connected

to point spectrum:

(a) Every eigenvector of A belongs toHpp.(b) Every eigenvalue of A belongs to σpp(A).

(c) Aψ = Eψ if and only if ψ∈ Ran χ{E}(A).

(d) Hppis the smallest closed linear subspace ofH containing all eigen-vectors of A.

(e) The pure point spectrum of A is given by

E. By induction one has f (A)ψ = f (E)ψ for any polynomial f and henceany continuous f by approximation. Thus, for any continuous f ,

f dμψ =ψ, f(A)ψ = ψ, f(E)ψ = f(E)ψ2,

showing that μψ is ψ2 times a Dirac δ measure at E. Thus, μψ is a point

measure, implying ψ∈ Hpp, and in particular that (App− E)ψ = 0, whichproves (a) and (b). Moreover, since μψ =ψ2δE,

(1.6.11) χ{E}(A)ψ2=ψ, χ{E}(A)ψ =

proving that ψ∈ Ran χ{E}(A) in this case and one direction of (c). For theother direction, assume ψ∈ Ran χ{E}(A) and note that for any φ, one has

(1.6.12) φ, ψ = φ, χ{E}(A)ψ =

χ{E}dμφ,ψ= μφ,ψ({E}),

6Strict parallelism would insist that we refer to this as “purely pure point spectrum”, butthe usage here is standard.

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1.6. Spectral Decompositions and Quantum Dynamics 49

which yields in turn

φ, Aψ = φ, Aχ{E}(A)ψ =

xχ{E}(x) dμφ,ψ(x)= E· μφ,ψ({E})

=φ, Eψ.

Since φ, Aψ = φ, Eψ for all φ, Aψ = Eψ, concluding the proof of (c).

By (a), Hpp contains every eigenvector, hence it contains their closed

linear span. Conversely, given ψ∈ Hpp, μψ is a point measure, so we can

which, in view of (c), shows that ψ belongs to the closed linear span of the

eigenvectors, concluding the proof of (d). Then (e) follows from (b) and (d)

since by the spectral theorem, σpp(A) is the smallest closed set that supports

If the operator A has a cyclic vector φ, then the spectral theorem assertsthat A is unitarily equivalent to multiplication by the independent variableon L2(R, μφ) via the canonical unitary map U : g(A)φ→ g. By choosingdisjoint supports of μφ,pp, μφ,sc, and μφ,ac, we obtain the orthogonal sum

(1.6.14) L2(μφ) = L2(μφ,pp)⊕ L2(μφ,sc)⊕ L2(μφ,ac).

Moreover, this decomposition corresponds to the orthogonal splitting H =Hac⊕ Hsc⊕ Hpp in the sense that the unitary equivalence U mapsH• onto

L2(R, μφ,•) for • ∈ {ac, sc, pp}. In particular, we can recover the point,singular continuous, and absolutely continuous spectra of A from supportsof the decomposition of μφ. In general, a support of a measure μ is anyset whose complement has zero μ-measure. We also deﬁne the topologicalsupport of μ, supp μ, to be the intersection of all closed supports of μ, orequivalently, the complement of the union of all open sets of zero μ-measure.

(1.6.15) σ(A) = supp(μφ),σ•(A) = supp(μφ,•),• ∈ {ac, sc, pp}.

Proof. We write the proof of the ﬁrst statement and leave the modiﬁcations

for the other three to the reader. By the spectral theorem, A is unitarilyequivalent to the operator M : L2(μφ)→ L2(μφ) given by [M g](x) = xg(x)

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and hence it suﬃces to show σ(M ) = supp(μφ). If E /∈ supp(μφ), one can

check that M− E is invertible with inverse

sogn−1gnis a Weyl sequence for M at E, showing E∈ σ(M). Let us note that Theorem 1.6.6 suggests how one can go about construct-ing operators with prescribed spectral properties. For instance, to ﬁnd an operator with singular continuous spectrum equal to a given compact set

K⊆ R, it suﬃces to construct a ﬁnite Borel measure μ on R such thatsupp(μsc) = K and to consider the operator [Af ](x) = xf (x) in L2(μ).

1.6.2. The RAGE Theorem. The Lebesgue decomposition of a spectral

measure μφ is also of importance in the study of the long-time asymptotics of solutions to the Schrăodinger equation

(1.6.16) it(t) = A(t), (0) = φ.

Equation (1.6.16) has a unique solution given by

which implies that the evolution is unitary (compare Exercise 1.4.3). The following theorem establishes precise relations between spectral measures

and solutions of (1.6.16) for operators in 2(Z).

φ∈ 2(Z), and φ(t) is a solution of the Schrăodinger equation (1.6.16).(a) We have = ,ppif and only if for every ε > 0, there is N∈ Z+

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1.6. Spectral Decompositions and Quantum Dynamics 51

The RAGE theorem has its roots in the following celebrated theorem of

Wiener regarding Fourier transforms of measures. The Fourier transformof a ﬁnite Borel measure μ on R is deﬁned by

so (1.6.22) follows by the dominated convergence theorem. We now prove a technical result that will be useful in the proof of the RAGE theorem and also later on in the text. In fact, we will prove the “only

if” direction of (b) in the course of proving this lemma. For N∈ Z+, let

PN denote projection onto coordinates in [−N, N]. That is, PNδn= δn for

|n| ≤ N and PNδn= 0 otherwise.

denoteP• =P•(A). Then, we have

proof of (1.6.24). Let φ, ψ be given, and abbreviate φ• =P•φ and ψ•=P•ψ

for• ∈ {c, pp}.

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If φ = φpp and ψ = ψpp, then, by Proposition 1.6.5, we may write φ andψ as convergent sums of eigenvectors of A. Having done so, we deduce

from a calculation that we leave to the reader (Exercise 1.6.6).

Now, suppose φ = φc, that is, φ∈ 2(Z)c. Denote by Πk projection onto

the coordinate k∈ Z so that PN = Since μδk,φis absolutely continuous with respect to μφ= μφ,c, Wiener’stheorem implies that the ﬁnal line tends to zero as T→ ∞. The sameargument holds true if ψ∈ 2(Z)c (and φ is arbitrary). Consequently, for

whenever at least one of φ or ψ is in 2(Z)c. (Notice that this already proves

the “only if” direction of Theorem 1.6.7(b) by taking ψ = φ.)At last, given general φ, ψ, we have

by (1.6.27) and (1.6.26), which completes the proof of (1.6.24).

Proof of Theorem 1.6.7. (a) For the “only if” direction, suppose we are

given φ∈ 2(Z)pp, and δ > 0. We can choose a ﬁnite sum of eigenfunctionsη = η1+· · · + ηk so that

Aηj= λjηj, andφ − η ≤ δ.

(I − PN)φ(t) = (I − PN)e−itAφ

≤ (I − PN)e−itAη + (I − PN)e−itA(φ− η)

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1.6. Spectral Decompositions and Quantum Dynamics 53

For all N suﬃciently large, the ﬁrst term of the bottom line is no largerthan δ for all t. Consequently,

For the converse, if for every ε > 0 there exists an N so that (1.6.18)holds, then setting ψ = φ in (1.6.25) yieldsPcφ = 0, whence μφ= μφ,pp.

(b) Taking φ = ψ in (1.6.27) proves the “only if” implication, so itremains to show the “if” direction. If (1.6.19) holds for all N , then (1.6.24)

implies Pppφ = 0, which implies μφ= μφ,c.

(c) This follows from the same argument as in the forward direction of (b), but with the Riemann–Lebesgue lemma replacing Wiener’s theorem.

In the mathematical formulation of quantum mechanics, A is given by

a so-called Schrăodinger operator and for a solution (Ã) of the associated

Schrăodinger equation, |δn, φ(t)|2 corresponds to the probability of ﬁnding

the system at site n at time t. Thus, in this situation, we can interpret the

results of the RAGE theorem as follows. The spectral measure of the initial state is pure point if and only if with arbitrarily high probability, the system can be found in some ﬁxed compact region in space at any given time; the spectral measure of the initial state is purely continuous if and only if the system leaves any ﬁxed compact region in space in a time-averaged sense as time goes to inﬁnity; and if the spectral measure of the initial state is purely absolutely continuous, then the system leaves any ﬁxed compact region in space as time goes to inﬁnity.

1.6.3. Dynamical Averages via Resolvent Averages. We conclude

this section with a useful lemma that connects the time-averaged quantum evolution to an integral involving the resolvent function at energies with

constant nonzero imaginary part. For the proof, let us deﬁne the Fourier

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One dimensional ergodic schrödinger operators i general theory(pg 31 215)

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