Vector-valued Holomorphic Functions

Appendix A

Vector-valued HolomorphicFunctions

Let X be a Banach space and let Ω ⊂ C be an open set. A function f : Ω→ X isholomorphic if

f ′(z0) := limh→0

h∈C\{0}

f(z0 + h)− f(z0)

h(A.1)

exists for all z0 ∈ Ω.If f is holomorphic, then f is continuous and weakly holomorphic (i.e. x∗ ◦ f

is holomorphic for all x∗ ∈ X∗). If Γ := {γ(t) : t ∈ [a, b]} is a finite, piecewisesmooth contour in Ω, we can form the contour integral

∫Γf(z) dz. This coincides

with the Bochner integral∫ b

af(γ(t))γ′(t) dt (see Section 1.1). Similarly we can

define integrals over infinite contours when the corresponding Bochner integral isabsolutely convergent. Since⟨∫

Γ

f(z) dz, x∗⟩=

∫Γ

〈f(z), x∗〉 dz,

many properties of holomorphic functions and contour integrals may be extendedfrom the scalar to the vector-valued case, by applying the Hahn-Banach theorem.For example, Cauchy’s theorem is valid, and also Cauchy’s integral formula:

f(w) =1

2πi

∫|z−z0|=r

f(z)

z − wdz (A.2)

whenever f is holomorphic in Ω, the closed ball B(z0, r) is contained in Ω andw ∈ B(z0, r). As in the scalar case one deduces Taylor’s theorem from this.

Proposition A.1. Let f : Ω → X be holomorphic, where Ω ⊂ C is open. Letz0 ∈ Ω, r > 0 such that B(z0, r) ⊂ Ω. Then

W. Arendt et al., Vector-valued Laplace Transforms and Cauchy Problems: Second Edition, 461Monographs in Mathematics 96, DOI 10.1007/978-3-0348-0087-7, © Springer Basel AG 2011

462 A. VECTOR-VALUED HOLOMORPHIC FUNCTIONS

f(z) =∞∑

n=0

an(z − z0)n

converges absolutely for |z − z0| < r, where

an :=1

2πi

∫|z−z0|=r

f(z)

(z − z0)n+1dz.

We also mention a special form of the identity theorem.

Proposition A.2 (Identity theorem for holomorphic functions). Let Y be a closedsubspace of a Banach space X. Let Ω be a connected open set in C and f : Ω→ Xbe holomorphic. Assume that there exists a convergent sequence (zn)n∈N ⊂ Ω suchthat limn→∞ zn ∈ Ω and f(zn) ∈ Y for all n ∈ N. Then f(z) ∈ Y for all z ∈ Ω.

Note that for Y = {0}, we obtain the usual form of the identity theorem.

Proof. Let x∗ ∈ Y 0 := {y∗ ∈ X∗ : 〈y, y∗〉 = 0 (y ∈ Y )}. Then x∗ ◦f(zn) = 0 for alln ∈ N. It follows from the scalar identity theorem that x∗ ◦ f(z) = 0 for all z ∈ Ω.Hence, f(z) ∈ Y 00 = Y for all z ∈ Ω.

In the following we show that every weakly holomorphic function is holomor-phic. Actually, we will prove a slightly more general assertion which turns out tobe useful. A subset N of X∗ is called norming if

‖x‖1 := supx∗∈N

|〈x, x∗〉|

defines an equivalent norm on X. A function f : Ω → X is called locally boundedif supK ‖f(z)‖ <∞ for all compact subsets K of Ω.

Proposition A.3. Let Ω ⊂ C be open and let f : Ω → X be locally bounded suchthat x∗ ◦ f is holomorphic for all x∗ ∈ N , where N is a norming subset of X∗.Then f is holomorphic.

In particular, if X = L(Y, Z), where Y, Z are Banach spaces, and if f : Ω→X is locally bounded, then the following are equivalent:

(i) f is holomorphic.

(ii) f(·)y is holomorphic for all y ∈ Y .

(iii) 〈f(·)y, z∗〉 is holomorphic for all y ∈ Y, z∗ ∈ Z∗.

Proof. We can assume that ‖x‖1 = ‖x‖ for all x ∈ X. In order to show holomorphyat z0 ∈ Ω we can assume that z0 = 0, replacing Ω by Ω− z0 otherwise. For smallh, k ∈ C\{0}, let

u(h, k) :=f(h)− f(0)

h− f(k)− f(0)

k.

463

We have to show that for ε > 0 there exists δ > 0 such that ‖u(h, k) ‖ ≤ εwhenever |h| ≤ δ and |k| ≤ δ. Let r > 0 such that B(0, 2r) ⊂ Ω and

M := supz∈B(0,2r)

‖ f(z) ‖ <∞.

Then by Cauchy’s integral formula, for |z| < r, |h| ≤ r, |k| ≤ r, h, k �= 0, x∗ ∈ N ,

〈u(h, k), x∗〉 =1

2πi

∫|z|=2r

〈f(z), x∗〉{1

h

(1

z − h− 1

z

)− 1

k

(1

z − k− 1

z

)}dz

=h− k

2πi

∫|z|=2r

〈f(z), x∗〉z(z − h)(z − k)

dz.

Hence, |〈u(h, k), x∗〉| ≤ |h− k|M/r2. Since N is norming, we deduce that

‖u(h, k) ‖ ≤ |h− k| Mr2

.

This proves the claim.

Corollary A.4. Let Ω ⊂ C be a connected open set and Ω0 ⊂ Ω be open. Leth : Ω0 → X be holomorphic. Assume that there exists a norming subset N ofX∗ such that for all x∗ ∈ N there exists a holomorphic extension Hx∗ : Ω → Cof x∗ ◦ h. If supx∗∈N

z∈Ω|Hx∗(z)| < ∞, then h has a unique holomorphic extension

H : Ω→ X.

Proof. Again we assume that ‖ · ‖1 = ‖ · ‖. Let

Y :={y = (yx∗)x∗∈N ⊂ C : ‖y‖∞ := sup

x∗∈N|yx∗ | <∞

},

and let H : Ω→ Y be given by H(z) := (Hx∗(z))x∗∈N . It follows from PropositionA.3 that H is holomorphic. By x ∈ X �→ (〈x, x∗〉)x∗∈N , one defines an isometricinjection from X into Y . Since H(z) ∈ X for z ∈ Ω0, it follows from the identitytheorem (Proposition A.2) that H(z) ∈ X for all z ∈ Ω.

We will extend Proposition A.3 considerably in Theorem A.7. Before that weprove Vitali’s theorem.

Theorem A.5 (Vitali). Let Ω ⊂ C be open and connected. Let fn : Ω → X beholomorphic (n ∈ N) such that

supn∈N

z∈B(z0,r)

‖fn(z)‖ <∞

whenever B(z0, r) ⊂ Ω. Assume that the set

Ω0 :={z ∈ Ω : lim

n→∞ fn(z) exists}

A. VECTOR-VALUED HOLOMORPHIC FUNCTIONS

464 A. VECTOR-VALUED HOLOMORPHIC FUNCTIONS

has a limit point in Ω. Then there exists a holomorphic function f : Ω→ X suchthat

f (k)(z) = limn→∞ f (k)

n (z)

uniformly on all compact subsets of Ω for all k ∈ N0.

Proof. Let l∞(X) := {x = (xn)n∈N ⊂ X : ‖x‖∞ := sup ‖xn‖ < ∞}. Then l∞(X)is a Banach space for the norm ‖·‖∞ and the space c(X) of all convergent sequencesis a closed subspace of l∞(X). Consider the function F : Ω → l∞(X) given byF (z) = (fn(z))n∈N. It follows from Proposition A.3 that F is holomorphic. (Onemay take N to be the space of all functionals on l∞(X) of the form (xn)n∈N �→〈xk, x

∗〉 where k ∈ N, x∗ ∈ X∗, ‖x∗‖ ≤ 1). Since F (z) ∈ c(X) for all z ∈ Ω0,it follows from the identity theorem (Proposition A.2) that F (z) ∈ c(X) for allz ∈ Ω. Consider the mapping φ ∈ L(c(X), X) given by φ((xn)n∈N) = limn→∞ xn.Then f = limn→∞ fn = φ ◦ F : Ω→ X is holomorphic.

Finally, we prove uniform convergence on compact sets. Let B(z0, r) ⊂ Ω andk ∈ N0. It follows from (A.2) that

1

k!f (k)n (z) =

1

2πi

∫|w−z0|=r

fn(w)

(w − z)k+1dw.

Now the dominated convergence theorem implies that f(k)n (z) converges uniformly

on B(z0, r/2) to f (k)(z). Since every compact subset of Ω can be covered by afinite number of discs, the claim follows.

If in Vitali’s theorem (fn) is a net instead of a sequence, the proof showsthat f(z) = lim fn(z) exists for all z ∈ Ω and defines a holomorphic functionf : Ω→ X.

Next we recall a well known theorem from functional analysis.

Theorem A.6 (Krein-Smulyan). Let X be a Banach space and W be a subspaceof the dual space X∗. Denote by B∗ the closed unit ball of X∗. Then W is weak*closed if and only if W ∩B∗ is weak* closed.

For a proof, see [Meg98, Theorem 2.7.11].

Now we obtain the following convenient criterion for holomorphy.

Theorem A.7. Let Ω ⊂ C be open and connected, and let f : Ω → X be a locallybounded function. Assume that W ⊂ X∗ is a separating subspace such that x∗ ◦ fis holomorphic for all x∗ ∈W . Then f is holomorphic.

Here, W is called separating if 〈x, x∗〉 = 0 for all x∗ ∈ W implies x = 0(x ∈ X).

Proof. Let Y := {x∗ ∈ X∗ : x∗ ◦ f is holomorphic}. Since W ⊂ Y , the subspaceY is weak* dense. It follows from Vitali’s theorem (applied to nets if X is not

465

separable) that Y ∩ B∗ is weak* closed. Now it follows from the Krein-Smulyantheorem that Y = X∗. Hence, f is holomorphic by Proposition A.3.

Notes: Usually, Vitali’s theorem is proved with the help of Montel’s theorem which isonly valid in finite dimensions. A vector-valued version is proved in the book of Hille andPhillips [HP57] by a quite complicated power-series argument going back to Liouville.The very simple proof given here is due to Arendt and Nikolski [AN00] who also provedTheorem A.7 (see also [AN06]).

A. VECTOR-VALUED HOLOMORPHIC FUNCTIONS

Appendix B

Closed Operators

Let X be a complex Banach space. An operator on X is a linear map A : D(A)→X, where D(A) is a linear subspace of X, known as the domain of A. The rangeRanA, and the kernel KerA, of A are defined by:

RanA := {Ax : x ∈ D(A)},KerA := {x ∈ D(A) : Ax = 0}.

The operator A is densely defined if D(A) is dense in X.An operator A is closed if its graph G(A) is closed in X ×X, where

G(A) := {(x,Ax) : x ∈ D(A)}.Thus, A is closed if and only if

Whenever (xn) is a sequence in D(A), x, y ∈ X,‖xn − x‖ → 0 and ‖Axn − y‖ → 0, then x ∈ D(A) and Ax = y.

It is immediate from this that if A is closed and α, β ∈ C with α �= 0, then theoperator αA+ β with D(αA+ β) = D(A) is closed.

An operator A is said to be closable if there is an operator A (known as theclosure of A) such that G(A) is the closure of G(A) in X ×X. Thus A is closableif and only if

Whenever (xn) is a sequence in D(A), y ∈ X,‖xn‖ → 0 and ‖Axn − y‖ → 0, then y = 0.

When A is closable,

D(A) =

{x ∈ X : there exist xn ∈ D(A) and y ∈ X

such that ‖xn − x‖ → 0 and ‖Axn − y‖ → 0

},

Ax = y.

468 B. CLOSED OPERATORS

For an operator A, D(A) becomes a normed space with the graph norm

‖x‖D(A) := ‖x‖+ ‖Ax‖.

The operator A : D(A) → X is always bounded with respect to the graph norm,and A is closed if and only if D(A) is a Banach space in the graph norm. Notethat if A is replaced by αA + β where α �= 0, then the space D(A) is unchangedand the graph norm is replaced by an equivalent norm.

Let A be a closed operator on X. A subspace D of D(A) is said to be a coreof A if D is dense in D(A) with respect to the graph norm. Thus, D is a core ofA if and only if A is the closure of A|D, or equivalently for each x ∈ D(A) there isa sequence (xn) in D such that ‖xn − x‖ → 0 and ‖Axn −Ax‖ → 0.

An operator A on X is said to be invertible if there is a bounded operator A−1

on X such that A−1Ax = x for all x ∈ D(A) and A−1y ∈ D(A) and AA−1y = yfor all y ∈ X.

Proposition B.1. Let A be an operator on X. The following assertions are equiv-alent:

(i) A is invertible.

(ii) RanA = X and there exists δ > 0 such that ‖Ax‖ ≥ δ‖x‖ for all x ∈ D(A).

(iii) A is closed, RanA is dense in X, and there exists δ > 0 such that ‖Ax‖ ≥δ‖x‖ for all x ∈ D(A).

(iv) A is closed, RanA = X and KerA = {0}.Proof. The equivalence of (i) and (ii) is an easy consequence of the definition. Sinceany bounded operator has closed graph, and since

G(A−1) = {(y, x) : (x, y) ∈ G(A)},

any invertible operator is closed. Thus, (i) and (ii) imply (iii) and (iv). When (iii)holds, G(A) is complete, and the map (x,Ax) �→ Ax is an isomorphism of G(A)onto RanA, so RanA is complete and (ii) follows. When (iv) holds, the inversemapping theorem can be applied to the map A from D(A) (with the graph norm)to X, showing that A−1 exists as a bounded map from X to D(A) and hence toX.

Let λ ∈ C. Then λ is said to be in the resolvent set ρ(A) of A if λ − Ais invertible, and we write R(λ,A) := (λ − A)−1. The remarks in the previousparagraphs show that if ρ(A) is non-empty, then A is closed. The function R(·, A) :ρ(A)→ L(X) is the resolvent of A. The spectrum of A is defined to be:

σ(A) := C \ ρ(A),

469

and the spectral bound is:

s(A) := sup{Reλ : λ ∈ σ(A)}if the supremum exists (s(A) := −∞ if σ(A) is empty). The point spectrum σp(A),and approximate point spectrum σap(A), of A are defined by:

σp(A) := {λ ∈ C : Ker(λ−A) �= {0}} ,σap(A) :=

{λ ∈ C : there exist xn ∈ D(A) such that

‖xn‖ = 1 and limn→∞ ‖(λ−A)xn‖ = 0

}.

Thus, σp(A) and σap(A) consist of the eigenvalues and approximate eigenvalues ofA, respectively. It is clear that σp(A) ⊂ σap(A) ⊂ σ(A).

Proposition B.2. Suppose that A has non-empty resolvent set, and let μ ∈ ρ(A).Let λ ∈ C, λ �= μ. Then

a) λ ∈ ρ(A) if and only if (μ− λ)−1 ∈ ρ(R(μ,A)). In that case,

R(λ,A) = (μ− λ)−1((μ− λ)−1 −R(μ,A)

)−1R(μ,A). (B.1)

b) λ ∈ σp(A) if and only if (μ− λ)−1 ∈ σp(R(μ,A)).

c) λ ∈ σap(A) if and only if (μ− λ)−1 ∈ σap(R(μ,A)).

d) The topological boundary of σ(A) is contained in σap(A).

Proof. Parts a), b) and c) follow immediately from the identity

λ−A = (μ− λ)((μ− λ)−1 −R(μ,A)

)(μ−A).

Part d) follows from a), c) and the corresponding result for bounded operators.Alternatively, d) may be proved directly in exactly the same way as for boundedoperators.

Corollary B.3. For any operator A, ρ(A) is open and σ(A) is closed in C. Moreover,if μ ∈ ρ(A), λ ∈ C and |λ− μ| < ‖R(μ,A)‖−1, then λ ∈ ρ(A), and

R(λ,A) =

∞∑n=0

(μ− λ)nR(μ,A)n+1,

where the series is norm-convergent. Hence,

‖R(λ,A)‖ ≤ ‖R(μ,A)‖1− |λ− μ| ‖R(μ,A)‖ .

Moreover, R(·, A) is holomorphic on ρ(A) with values in L(X) and

R(μ,A)(n)

n!= (−1)nR(μ,A)n+1 (n ∈ N).

B. CLOSED OPERATORS


Proof. This is immediate from (B.1) and the Neumann expansion, (I − T )−1 =∑∞n=0 T

n, when T is a bounded operator with ‖T‖ < 1.

Proposition B.4. Let A be an operator on X, and let λ, μ ∈ ρ(A). Then

R(λ,A)−R(μ,A) = (μ− λ)R(λ,A)R(μ,A). (B.2)

Proof. The identity (B.2) follows by rearranging (B.1).

Proposition B.5. Let A be an operator on X, and U be a connected open subset ofC. Suppose that U ∩ ρ(A) is nonempty and that there is a holomorphic functionF : U → L(X) such that {λ ∈ U ∩ ρ(A) : F (λ) = R(λ,A)} has a limit point inU . Then U ⊂ ρ(A) and F (λ) = R(λ,A) for all λ ∈ U .

Proof. Let V = {λ ∈ U ∩ ρ(A) : F (λ) = R(λ,A)}, μ ∈ ρ(A), x ∈ D(A), y ∈ X.For λ ∈ V ,

F (λ)(λ−A)x = x, (B.3)

F (λ)y = R(μ,A)y − (λ− μ)R(μ,A)F (λ)y, (B.4)

using (B.2). By uniqueness of holomorphic extensions (Proposition A.2), (B.3) and(B.4) are valid for all λ ∈ U . Now, (B.4) implies that F (λ)y ∈ D(A) and

R(μ,A)(λ−A)F (λ)y = F (λ)y + (λ− μ)R(μ,A)F (λ)y

= R(μ,A)y.

Since R(μ,A) is injective, (λ − A)F (λ)y = y for all λ ∈ U . This and (B.3) implythat λ ∈ ρ(A) and F (λ) = R(λ,A).

The equation (B.2) is known as the resolvent equation or resolvent identity. Afunction R : U → L(X), defined on a subset U of C, is said to be a pseudo-resolventif it satisfies the resolvent equation; i.e., if

R(λ)−R(μ) = (μ− λ)R(λ)R(μ) (λ, μ ∈ U).

The following proposition is easy to prove.

Proposition B.6. Let R : U → L(X) be a pseudo-resolvent. Then

a) KerR(λ) and RanR(λ) are independent of λ ∈ U .

b) There is an operator A on X such that R(λ) = R(λ,A) for all λ ∈ U if andonly if KerR(λ) = {0}.An operator A is said to have compact resolvent if ρ(A) �= ∅ and R(λ,A) is

a compact operator on X. Since the compact operators form an ideal of L(X), itis immediate from (B.2) that this property is independent of λ ∈ ρ(A). When Ahas compact resolvent, then σ(A) is a discrete subset of C. This follows from (B.1)and the fact that the spectrum of a compact operator has 0 as its only limit point.

The following is easy to prove.

471

Proposition B.7. Let A be an operator on X with non-empty resolvent set, and letT ∈ L(X). The following are equivalent:

(i) R(λ,A)T = TR(λ,A) for all λ ∈ ρ(A).

(ii) R(λ,A)T = TR(λ,A) for some λ ∈ ρ(A).

(iii) For all x ∈ D(A), Tx ∈ D(A) and ATx = TAx.

For an operator A, the powers An (n ≥ 2) are defined recursively:

D(An) :={x ∈ D(An−1) : An−1x ∈ D(A)

},

Anx := A(An−1x).

Note that D((λ−A)n) = D(An) for all λ ∈ C, n ∈ N. It is easy to see that An isinvertible if and only if A is invertible, and then (An)−1 = (A−1)n.

If A is densely defined and ρ(A) �= ∅, then D(An) is a core for A, for eachn ∈ N. To see this, let λ ∈ ρ(A). Then R(λ,A) has dense range D(A). It followsthat the range D(An−1) of R(λ,A)n−1 is dense in X. Let x ∈ D(A). There isa sequence (ym)m∈N in D(An−1) converging to (λ − A)x. Let xm := R(λ,A)ym.Then xm ∈ D(An), ‖xm − x‖ → 0 and ‖Axm −Ax‖ → 0.

Let A be an operator on X, and let Y be a closed subspace of X. The partof A in Y is the operator AY on Y defined by

D(AY ) := {y ∈ D(A) ∩ Y : Ay ∈ Y },AY y := Ay.

The following results are easy to prove.

Proposition B.8. Let A be an operator on X, and let Y be a closed subspace of X.

a) If D(A) ⊂ Y , then ρ(A) ⊂ ρ(AY ) and R(λ,AY ) = R(λ,A)|Y for all λ ∈ρ(A).

b) Suppose that ρ(A) �= ∅ and there is a projection P of X onto Y such thatPR(λ,A) = R(λ,A)P for some λ ∈ ρ(A). Then A maps D(A)∩Y into Y , AY

is the restriction of A to D(A) ∩ Y , λ ∈ ρ(AY ) and R(λ,AY ) = R(λ,A)|Y .One situation where the conditions of Proposition B.8 b) are satisfied is

described in the following.

Proposition B.9. Let A be a closed operator on X with ρ(A) �= ∅, and suppose thatthere are a compact subset E1 and a closed subset E2 of C such that E1∩E2 = ∅ andE1∪E2 = σ(A). Then there is a bounded projection P on X such that R(λ,A)P =PR(λ,A) for all λ ∈ ρ(A), P (X) ⊂ D(A), σ(AY ) = E1 and σ(AZ) = E2, whereY := P (X), Z := (I − P )(X). Moreover, P is unique, and A|Y ∈ L(Y ).

B. CLOSED OPERATORS


The projection P is known as the spectral projection of A associated with E1.

Proof. Take μ ∈ ρ(A) and consider R(μ,A) ∈ L(X). Then σ(R(μ,A)) = E′1 ∪ E′2,where

E′1 := {(μ− λ)−1 : λ ∈ E1}and

E′2 :=

{{(μ− λ)−1 : λ ∈ E2} if D(A) = X,

{(μ− λ)−1 : λ ∈ E2} ∪ {0} otherwise.

Then E′1 and E′2 are compact and disjoint. By the functional calculus for boundedoperators (see [DS59, p.573]), there is a unique bounded projection P on Xsuch that R(λ,A)P = PR(λ,A) for all λ ∈ ρ(A), σ(R(μ,A)|Y ) = E′1 andσ(R(μ,A)|Z) = E′2. Since 0 �∈ σ(R(μ,A)|Y ), Y ⊂ D(A) and A|Y is bounded bythe closed graph theorem. The remaining properties follow easily from PropositionB.2 a).

Suppose that A has compact resolvent, let λ ∈ ρ(A) and μ ∈ σ(A). Let Pbe the spectral projection of A associated with {μ}. Then there exists m ∈ Nsuch that (R(λ,A)P − (λ − μ)−1P )m = 0 (see [DS59, Theorem VII.4.5]). Hence,(A− μ)mP = 0.

Given an operator A on X, let

G(A∗) := {(x∗, y∗) ∈ X∗ ×X∗ : 〈Ax, x∗〉 = 〈x, y∗〉 for all x ∈ D(A)} ,which is a weak* closed subspace of X∗×X∗. If (and only if) A is densely defined,then G(A∗) is the graph of an operator A∗ in X∗, known as the adjoint of A. Forthe remainder of this appendix, we shall assume that A is densely defined, and weshall consider properties of A∗.

When A is closed, the operator A can be recovered from A∗ in the followingway.

Proposition B.10. Let A be a closed, densely defined operator on X, and let x, y ∈X. The following assertions are equivalent:

(i) x ∈ D(A) and Ax = y.

(ii) 〈x,A∗x∗〉 = 〈y, x∗〉 for all x∗ ∈ D(A∗).

Hence, D(A∗) is weak* dense in X∗.

Proof. The implication (i) ⇒ (ii) is immediate from the definition of A∗. For theconverse, suppose that (x, y) /∈ G(A). By the Hahn-Banach theorem, there exists(x∗, y∗) ∈ X∗×X∗ such that 〈x, x∗〉+ 〈y, y∗〉 �= 0 but 〈u, x∗〉+ 〈Au, y∗〉 = 0 for allu ∈ D(A). The latter condition implies that y∗ ∈ D(A∗) and A∗y∗ = −x∗. Thus,〈x,A∗y∗〉 = −〈x, x∗〉 �= 〈y, y∗〉, so (ii) is violated.

If D(A∗) is not weak* dense in X∗, then by the Hahn-Banach theorem thereexists y ∈ X such that y �= 0 and 〈y, x∗〉 = 0 for all x∗ ∈ D(A∗). By the previouspart, A0 = y, which is absurd.

473

If A is closable (and densely defined), it is easy to see that (A)∗ = A∗, soD(A∗) is weak* dense by Proposition B.10. Conversely, it is easy to see that ifD(A∗) is weak* dense, then A is closable.

Proposition B.11. Let A be a closed, densely defined operator on X. Then

a) A∗ is invertible if and only if A is invertible, and then (A∗)−1 = (A−1)∗.

b) σ(A∗) = σ(A), and R(λ,A∗) = R(λ,A)∗ for all λ ∈ ρ(A).

c) σ(A) = σap(A) ∪ σp(A∗).

Proof. a) If A is invertible, it is easy to verify that (A−1)∗A∗x∗ = x∗ for allx∗ ∈ D(A∗) and A∗(A−1)∗y∗ = y∗ for all y∗ ∈ X∗. Thus, A∗ is invertible and(A∗)−1 = (A−1)∗.

Now suppose that A∗ is invertible, and let δ =∥∥(A∗)−1

∥∥−1. Since KerA∗ =

{0}, RanA is dense in X, by a simple application of the Hahn-Banach theorem.For x ∈ X, there exists x∗ ∈ X∗ such that ‖x∗‖ = 1 and 〈x, x∗〉 = ‖x‖. Lety∗ = (A∗)−1x∗ ∈ D(A∗), so that ‖y∗‖ ≤ δ−1 and A∗y∗ = x∗. Hence,

‖Ax‖ ≥ δ |〈Ax, y∗〉| = δ |〈x,A∗y∗〉| = δ‖x‖.

It follows from Proposition B.1 that A is invertible.b) This follows from a) by replacing A by λ−A.c) This follows from applying Proposition B.1 and the fact that RanA is

dense in X if and only if KerA∗ = {0} (by the Hahn-Banach theorem), with Areplaced by λ−A.

Now, let H be a Hilbert space with inner product (·|·)H . Identifying H∗ withH by means of the Riesz-Frechet lemma, we obtain the following. If A is a denselydefined operator on H, the adjoint A∗ of A is defined by

D(A∗) :=

{x ∈ H : there exists y ∈ H such that

(Au|x)H = (u|y)H for all u ∈ D(A)

},

A∗x = y.

We say that A is selfadjoint if A = A∗.

Example B.12 (Multiplication operators). Let (Ω, μ) be a measure space, H :=L2(Ω, μ), m : Ω→ R a measurable function. Define the operator Mm on H by

D(Mm) := {f ∈ H : mf ∈ H},Mmf := mf.

It is easy to see that Mm is selfadjoint.

B. CLOSED OPERATORS


Let H, H be Hilbert spaces. Two operators A on H and A on H are calledunitarily equivalent if there exists a unitary operator U : H → H such that

D(A) = U−1D(A),

Ax = U−1AUx.

It is easy to see that, in that case, A is selfadjoint whenever A is.Now we can formulate the spectral theorem as follows; we refer to [RS72,

Theorem VIII.4] for a proof.

Theorem B.13 (Spectral Theorem). Each selfadjoint operator is unitarily equiva-lent to a real multiplication operator.

Thus, selfadjoint and real multiplication operators are effectively the samething. In proofs, we frequently regard an arbitrary selfadjoint operator as being areal multiplication operator.

A selfadjoint operator A is always symmetric; i.e., (Ax|y)H = (x|Ay)H forall x, y ∈ D(A). In particular, (Ax|x)H ∈ R for all x ∈ D(A). We say that A isbounded above if there exists ω ∈ R such that

(Ax|x)H ≤ ω(x|x)H (x ∈ D(A)).

In that case, ω is called an upper bound of A. If A is a multiplication operator Mm,then this is equivalent to saying that

m(y) ≤ ω for almost all y ∈ Ω.

It is easy to see (for example, from the spectral theorem) that for any selfadjointoperator A, we have σ(A) ⊂ R and ω is an upper bound for A if and only ifσ(A) ⊂ (−∞, ω]; i.e., ω ≥ s(A). Similarly, we say that A is bounded below by ω if

(Ax|x)H ≥ ω(x|x)H (x ∈ D(A)).

The definition of selfadjointness is not easy to verify in practice. Here is ahandy criterion, for a proof of which we refer to [RS72, Theorem X.1].

Theorem B.14. Let A be an operator on H and let ω ∈ R. The following areequivalent:

(i) A is selfadjoint with upper bound ω.

(ii) a) (Ax|y)H = (x|Ay)H (x, y ∈ D(A)),b) (Ax|x)H ≤ ω(x|x)H (x ∈ D(A)), andc) there exists λ > ω such that Ran(λ−A) = X.

Finally, we mention one or two topics concerning bounded operators. By theclosed graph theorem, an operator T on a Banach space X is bounded if T isclosed and D(T ) = X. Conversely, a densely defined, closed, bounded operator is

475

everywhere defined. By convention, a bounded operator T on a Banach space Xwill be assumed to be defined on the whole of X. The spectral radius of T will bedenoted by r(T ), so

r(T ) = sup{|λ| : λ ∈ σ(T )} = inf{‖Tn‖1/n : n ∈ N

}.

In order to allow a convenient citation in the book, we state the followingstandard fact whose proof is straightforward. Note that a family of bounded linearoperators is equicontinuous if and only if it is bounded.

Proposition B.15. Let X,Y be Banach spaces, Tn ∈ L(X,Y ) (n ∈ N) such thatsupn∈N ‖Tn‖ <∞. The following are equivalent:

(i) (Tnx)n∈N converges for all x in a dense subspace of X.

(ii) (Tnx)n∈N converges for all x ∈ X.

(iii) (Tnx)n∈N converges uniformly in x ∈ K for all compact subsets K of X.

Notes: The material of this appendix is standard, and can be found in various books, forexample [Kat66, Chapter 3].

B. CLOSED OPERATORS

Appendix C

Ordered Banach Spaces

Let X be a real Banach space. By a positive cone in X we understand a closedsubset X+ of X such that

X+ +X+ ⊂ X+; (C.1)

R+ ·X+ ⊂ X+; (C.2)

X+ ∩ (−X+) = {0}; and (C.3)

X+ −X+ = X. (C.4)

Then an ordering on X is introduced by setting

x ≤ y ⇐⇒ y − x ∈ X+.

The space X together with the positive cone is called a real ordered Banach space.The elements of X+ are called positive.

Remark C.1. Property (C.3) is frequently expressed by saying that X+ is a propercone, and (C.4) says thatX+ is generating. We assume these properties throughoutwithout further notice.

If x∗ ∈ X∗, then we say that x∗ is positive and write x∗ ≥ 0 if

〈x, x∗〉 ≥ 0 for all x ∈ X+.

The set X∗+ := {x∗ ∈ X∗ : x∗ ≥ 0} is closed and satisfies (C.1), (C.2) and (C.3).

For x, y ∈ X such that x ≤ y we denote by

[x, y] := {z ∈ X : x ≤ z ≤ y}

the order interval defined by x and y. One says that the cone X+ is normal if allorder intervals are bounded.

478 C. ORDERED BANACH SPACES

Proposition C.2. The cone X∗+ is normal. The cone X+ is normal if and only if

X∗+ −X∗

+ = X∗.

Thus, if X+ is normal then (X∗, X∗+) is also an ordered Banach space with

normal cone. We call X∗+ the dual cone of X+.

If the cone X+ is normal then there is a constant c ≥ 0 such that

y ≤ x ≤ z =⇒ ‖x‖ ≤ cmax(‖y‖, ‖z‖). (C.5)

Indeed, passing to an equivalent norm one can even arrange that c = 1.If X is a real ordered Banach space we tacitly consider the complexification

of X. So in this book an ordered Banach space is always the complexification of areal ordered Banach space. Thus, any C∗-algebra is an ordered Banach space withnormal cone.

Let X be an ordered Banach space. A linear mapping T : X → X is calledpositive if

Tx ∈ X+ for all x ∈ X+.

Then we write T ≥ 0. If S, T : X → X are linear, we write S ≤ T if T − S ≥ 0.If X+ is normal, every positive linear mapping T : X → X is continuous.

Moreover, there is a constant k ≥ 0 such that

±S ≤ T =⇒ ‖S‖ ≤ k‖T‖ (C.6)

if S, T : X → X are linear.A real ordered Banach space X is a lattice if for all x, y ∈ X there exists a

least upper bound x∨y of x and y (i.e., x∨y ∈ X, x∨y ≥ x, x∨y ≥ y and w ≥ x, yimplies w ≥ x ∨ y). In that case, there also exists a largest lower bound x ∧ y =−((−x) ∨ (−y)). One sets x+ = x ∨ 0, x− = (−x)+, |x| = x ∨ (−x) = x+ + x−.Then X is called a real Banach lattice if in addition the following compatibilitycondition is satisfied:

|x| ≤ |y| =⇒ ‖x‖ ≤ ‖y‖ (C.7)

for all x, y ∈ X. Thus, the cone of a Banach lattice is always normal.In this book, a Banach lattice is the complexification of a real Banach lattice.

Important examples of Banach lattices are the spaces Lp(Ω, μ) (1 ≤ p ≤ ∞), where(Ω, μ) is a measure space, and

C(K) := {f : K → C : f continuous},where K is a compact space.

Let X be a real Banach lattice. A subspace Y of X is called a sublattice if

x ∈ Y implies |x| ∈ Y.

The space Y is called an ideal if

x ∈ Y, y ∈ X, |y| ≤ |x| implies y ∈ Y.

479

Let (Ω, μ) be a σ-finite measure space and X = Lp(Ω, μ), where 1 ≤ p <∞.Then Y is a closed ideal of X if and only if

Y = {f ∈ X : f |S = 0 a.e.}

for some measurable subset S of Ω.If M ⊂ X is a subset, then

Md := {x ∈ X : |x| ∨ |y| = 0 for all y ∈M}

is a closed ideal of X. One says that M is a band if M = Mdd. In that case,M ⊕Md = X.

If X is a complex Banach lattice, then a subspace Y of X is called a sublattice(ideal, band) if

a) x ∈ Y =⇒ Rex ∈ X, and

b) Y ∩XR is a sublattice (ideal, band) of XR,

where XR denotes the underlying real Banach lattice.An ordered Banach space has order continuous norm if each decreasing pos-

itive sequence (xn)n∈N converges; i.e.,

If xn ≥ xn+1 ≥ 0 (n ∈ N), then limn→∞xn exists.

The spaces Lp(Ω, μ) (1 ≤ p <∞) have order continuous norm, but L∞(Ω, μ) andC(K) do not if they have infinite dimension. Also, the dual of a C∗-algebra hasorder continuous norm.

Let X be a Banach lattice. Then the following assertions are equivalent:

(i) If 0 ≤ xn ≤ xn+1 and supn∈N ‖xn‖ <∞, then (xn)n∈N converges.

(ii) X is a band in X∗∗.

(iii) c0 is not isomorphic to a closed subspace of X.

In assertion (ii), we identify X with a closed subspace of X∗∗ via the canonicalevaluation mapping.

A Banach lattice X satisfying the equivalent conditions (i), (ii), and (iii)is called a KB-space. Every reflexive Banach lattice and every space of the formL1(Ω, μ) are KB-spaces. Moreover, if X is a KB-space then X has order continuousnorm. The space c0 does have order continuous norm but is not a KB-space. Eachclosed ideal of a KB-space is a band.

Notes: We refer to the monograph [Sch74] by Schaefer and to the survey article [BR84]for all this and for further information.

C. ORDERED BANACH SPACES

Appendix D

Banach Spaces which Contain c0

We let c0 be the Banach space of all complex sequences a = (ar)r≥1 such thatlimr→∞ ar = 0, with ‖a‖ = supr |ar|. For n ≥ 1, let en := (δnr)r≥1, so ‖en‖ = 1and ∥∥∥∥∥

m∑n=1

αnen

∥∥∥∥∥ = maxn|αn|

for all m ∈ N and α1, . . . , αm ∈ C.A complex Banach space X is said to contain c0 if there is a closed linear

subspace Y of X which is isomorphic (linearly homeomorphic) to c0. This is equiv-alent to the existence of a sequence (xn)n≥1 in X and strictly positive constantsc1 and c2 such that

c1 maxn|αn| ≤

∥∥∥∥∥m∑

n=1

αnxn

∥∥∥∥∥ ≤ c2 maxn|αn| (D.1)

for all m ∈ N and α1, . . . , αm ∈ C. Then the map∑m

n=1 αnxn �→∑m

n=1 αnenextends to an isomorphism of the closed linear span of {xn} onto c0.

Since c0 is not reflexive, a reflexive Banach space cannot contain c0. Moreover,for any measure space (Ω, μ), the space L1(Ω, μ) does not contain c0.

A formal series∑∞

n=1 xn in X is said to be unconditionally bounded if thereis a constant M such that ∥∥∥∥ m∑

j=1

xnj

∥∥∥∥ ≤M (D.2)

whenever m ∈ N and 1 ≤ n1 < n2 < · · · < nm. The series∑

n en in c0 isunconditionally bounded (with M = 1), but it is divergent. It follows that anyBanach space which contains c0 has a divergent, unconditionally bounded series.In this appendix, we shall give a converse result showing that if X contains adivergent, unconditionally bounded series, then X contains c0.

482 D. BANACH SPACES WHICH CONTAIN C0

Lemma D.1. Suppose that∑

n xn is a divergent, unconditionally bounded series ina complex Banach space X, and let M be as in (D.2). Then∥∥∥∥∥

m∑n=1

αnxn

∥∥∥∥∥ ≤ 4M max1≤n≤m

|αn|

for all m ∈ N and α1, . . . , αm ∈ C.

Proof. First suppose that αn ≥ 0 for n = 1, 2, . . . ,m. By rearranging x1, x2, . . . , xm,we may suppose that 0 ≤ α1 ≤ α2 ≤ · · · ≤ αm. Then

m∑n=1

αnxn = α1

m∑n=1

xn + (α2 − α1)m∑

n=2

xn + · · ·+ (αm − αm−1)xm.

Hence,∥∥∥∥∥m∑

n=1

αnxn

∥∥∥∥∥ ≤ α1M + (α2 − α1)M + · · ·+ (αm − αm−1)M = αmM.

The general case follows by decomposing each complex number αn as∑3

j=0 αnjij

where αnj ≥ 0 and |αnj | ≤ |αn|.Lemma D.2. Suppose that X contains a divergent, unconditionally bounded series.Then there is a sequence (yj)j≥1 in X such that ‖yj‖ = 1 for all j and∥∥∥∥∥∥

m∑j=1

βjyj

∥∥∥∥∥∥ ≤ 3

2max

1≤j≤m|βj |

for all m ∈ N and β1, . . . , βm ∈ C.

Proof. Let∑

n xn be a divergent, unconditionally bounded series, and let

γk := sup

{∥∥∥∥∥m∑

n=k+1

αnxn

∥∥∥∥∥ : m > k, αn ∈ C, |αn| ≤ 1

}.

By Lemma D.1, γk is finite, and clearly (γk) is a decreasing sequence. Let γ :=limk→∞ γk. Then γ > 0, since

γk ≥ sup

{∥∥∥∥∥m∑

n=k+1

xn

∥∥∥∥∥ : m > k

}

and∑

n xn is divergent. Replacing xn by (5/4γ)xn, we may assume that γ = 5/4.

483

Choose k1 ≥ 1 such that γk1 < 3/2. Since γk1 > 1, there exist k2 > k1 andαn ∈ C (k1 < n ≤ k2) such that |αn| ≤ 1 and

ν1 :=

∥∥∥∥∥k2∑

n=k1+1

αnxn

∥∥∥∥∥ > 1.

Iterating this, we may choose k1 < k2 < . . . and αn ∈ C (n > k1) such that|αn| ≤ 1 and

νj :=

∥∥∥∥∥∥kj+1∑

n=kj+1

αnxn

∥∥∥∥∥∥ > 1.

Let

yj := ν−1j

kj+1∑n=kj+1

αnxn.

Then ‖yj‖ = 1. Moreover, if m ∈ N and βj ∈ C (j = 1, . . . ,m) and jn is chosen sothat kjn < n ≤ kjn+1 (n > k1), then∥∥∥∥∥∥

m∑j=1

βjyj

∥∥∥∥∥∥ =

∥∥∥∥∥∥km+1∑

n=k1+1

βjnν−1jn

αnxn

∥∥∥∥∥∥ ≤ 3

2max

k1<n≤km+1

∣∣βjnν−1jn

αn

∣∣ ≤ 3

2max

1≤j≤m|βj |.

Theorem D.3. Suppose that X contains a divergent, unconditionally bounded series∑n xn. Then X contains c0.

Proof. Let (yj) be as in Lemma D.2. Let m ∈ N and βj ∈ C (j = 1, . . . ,m). Then∥∥∥∑mj=1 βjyj

∥∥∥ ≤ 32 maxj |βj |. We shall establish that

∥∥∥∑mj=1 βjyj

∥∥∥ ≥ 12 maxj |βj |,

so that (yj) satisfies the condition (D.1), and therefore X contains c0.Choose k such that |βk| = maxj |βj |, and choose x∗ ∈ X∗ such that ‖x∗‖ = 1

and βk〈yk, x∗〉 = |βk|. Let

β′j :=

{βj (j �= k),

−βk (j = k).

Then∥∥∥∥∥∥m∑j=1

βjyj

∥∥∥∥∥∥ ≥ Re 〈m∑j=1

βjyj , x∗〉 = 2|βk|+Re 〈

m∑j=1

β′jyj , x∗〉

≥ 2|βk| −∥∥∥∥∥∥

m∑j=1

β′jyj

∥∥∥∥∥∥ ≥ 2|βk| − 3

2max

j|βj | = 1

2max

j|βj |.

D. BANACH SPACES WHICH CONTAIN C0

484 D. BANACH SPACES WHICH CONTAIN C0

This completes the proof.

Notes: Theorem D.3 is due to Bessaga and Pelczynski [BP58]. They also showed thatX contains c0 if (and only if) there is a sequence of unit vectors (yj) in X such that∑

j |〈yj , x∗〉| < ∞ for all x∗ ∈ X∗. Our proof, which is adapted from [LZ82], establishessuch a property but in a specific way which eliminates some of the cases considered in[BP58]. Moreover, this proof shows (when constants 5/4 and 3/2 are replaced by constantsarbitrarily close to 1) that X contains c0 “almost isometrically”, thereby establishinga positive solution to the “distortion problem” in c0. This was first proved by James[Jam64].

Another direct proof of Theorem D.3 is given in a paper of Eberhardt and Greiner[EG92]. There are numerous other characterizations of Banach spaces which contain c0,some of which may be found in the books of Guerre-Delabriere [Gue92], Lindenstraussand Tzafriri [LT77] and Megginson [Meg98]. Note in particular that a Banach lattice Xdoes not contain c0 (as a subspace, or equivalently as a sublattice) if and only if X isa KB-space, that is, every bounded increasing sequence in X converges [LT77, TheoremII.1.c.4], [Mey91, Theorem 2.4.12].

Appendix E

Distributions and FourierMultipliers

In this appendix we collect basic facts on distributions and Fourier multipliers.They are needed at various places in the book; those which are essential to un-derstanding Parts I and II are also explained at the appropriate point in the text,while other results from this appendix are needed only for examples in Chapter 3or for the applications in Part III.

First, we consider distributions on Rn. A multi-index is an element α =(α1, . . . , αn) ∈ Nn

0 . We write |α| for ∑nj=1 αj , Dj for ∂

∂xjand Dα for Dα1

1 · · ·Dαnn .

We denote by D(Rn) (or by C∞c (Rn) in other contexts) the space of all complex-valued C∞-functions on Rn with compact supports (the test functions), and byS(Rn) the Schwartz space of all smooth, rapidly decreasing functions on Rn, i.e.

S(Rn) := {ϕ ∈ C∞(Rn) : ‖ϕ‖m,α <∞ for all m ∈ N0, α ∈ Nn0} ,

where

‖ϕ‖m,α := supx∈Rn

(1 + |x|)m|Dαϕ(x)|.

When equipped with the topology defined by the family of all norms ‖·‖m,α, S(Rn)is a Frechet space, and D(Rn) is a dense subspace of S(Rn).

We denote by D(Rn)′ the space of all distributions, i.e., linear maps f : ϕ �→〈ϕ, f〉 of D(Rn) into C such that for each compact K ⊂ Rn there exist m ∈ N andC > 0 such that

|〈ϕ, f〉| ≤ C sup|α|≤m

supx∈Rn

|Dαϕ(x)|

for all ϕ ∈ D(Rn) with suppϕ ⊂ K. Let S(Rn)′ be the space of all temperate distri-butions, i.e., continuous linear maps from S(Rn) into C. Then S(Rn)′ is embeddedin D(Rn)′ in a natural way.

486 E. DISTRIBUTIONS AND FOURIER MULTIPLIERS

We considerD(Rn)′ to have the topology arising from the duality withD(Rn),so a net (fα) of distributions converges to 0 in D(Rn)′ if and only if 〈ϕ, fα〉 → 0for all ϕ ∈ D(Rn).

Any locally integrable f : Rn → C can be identified with a distribution by

〈ϕ, f〉 :=∫Rn

ϕ(x)f(x) dx (ϕ ∈ S(Rn)). (E.1)

We shall make such identifications freely.A function f : Rn → C is said to be absolutely regular if there exists k ∈ N0

such that x �→ (1 + |x|)−kf(x) is Lebesgue integrable on Rn. For an absolutelyregular function f , the corresponding distribution is temperate.

Any continuous linear map T : S(Rn) → S(Rn) induces an adjoint. Wenow describe how this enables operators of multiplication, differentiation, Fouriertransform and convolution to be extended from functions to distributions.

Let g : Rn → C be a C∞-function. Then ϕ · g ∈ D(Rn) for all ϕ ∈ D(Rn).Given a distribution f ∈ D(Rn)′, we can define g · f by

〈ϕ, g · f〉 := 〈ϕ · g, f〉 (ϕ ∈ D(Rn)). (E.2)

If, for each multi-index α, there exists mα ∈ N and cα > 0 such that

|(Dαg)(x)| ≤ cα(1 + |x|)mα (x ∈ Rn), (E.3)

then the map ϕ �→ ϕ · g is continuous from S(Rn) into S(Rn), and thereforeg · f ∈ S(Rn)′ whenever f ∈ S(Rn)′.

Given a distribution f ∈ D(Rn)′, the derivatives Djf (j = 1, . . . , n) aredefined in D(Rn)′ by

〈ϕ,Djf〉 := −〈Djϕ, f〉 (ϕ ∈ D(Rn)). (E.4)

Then Dj maps S(Rn)′ into itself. Integration by parts shows that this notationis consistent when differentiable functions are identified with distributions, andthe product law extends to derivatives of products of differentiable functions anddistributions as discussed above. For higher order derivatives, (E.4) becomes

〈ϕ,Dαf〉 = (−1)|α|〈Dαϕ, f〉 (ϕ ∈ D(Rn)). (E.5)

Next, we consider convolutions. For functions f and g, the convolution f ∗ gis defined by

(f ∗ g)(x) :=∫Rn

f(x− y)g(y) dy

whenever the integral exists. For ϕ, ψ ∈ S(Rn), ψ ∗ ϕ ∈ S(Rn) and the mapψ �→ ψ∗ϕ is continuous. Hence, the convolution ϕ∗f of ϕ ∈ S(Rn) and f ∈ S(Rn)′

can be defined by

〈ψ,ϕ ∗ f〉 := 〈ψ ∗ ϕ, f〉 (ψ ∈ S(Rn)), (E.6)

487

where ϕ(x) := ϕ(−x), and then ϕ ∗ f ∈ S(Rn)′. An easy calculation shows thatthis is consistent when functions are identified with distributions.

An alternative way to define ϕ ∗ f is as follows. For x ∈ Rn and ψ ∈ S(Rn),let τxψ(y) := ψ(y − x). For ϕ ∈ S(Rn), τxϕ ∈ S(Rn) and the map x �→ τxϕ iscontinuous on S(Rn). For f ∈ S(Rn)′, let

(ϕ ∗ f)(x) := 〈τxϕ, f〉 (x ∈ Rn). (E.7)

Then ϕ ∗ f is a continuous, bounded function.These two definitions of ϕ ∗ f are consistent when functions are identified

with distributions. Moreover,

Dj(ϕ ∗ f) = (Djϕ) ∗ f. (E.8)

For f ∈ L1(Rn), the Fourier transform Ff of f is defined by:

(Ff)(ξ) =∫Rn

e−ix·ξf(x) dx (ξ ∈ Rn), (E.9)

where x · ξ :=∑n

j=1 xjξj . The Fourier inversion theorem [Hor83, Theorem 7.1.5]shows that F is a linear and topological isomorphism of S(Rn), and

(F−1ϕ)(ξ) = (2π)−n(Fϕ)(−ξ) (ϕ ∈ S(Rn), ξ ∈ Rn).

The Fourier transform therefore induces an isomorphism of S(Rn)′, also denotedby F :

〈ϕ,Ff〉 := 〈Fϕ, f〉 (ϕ ∈ S(Rn), f ∈ S(Rn)′). (E.10)

A simple application of Fubini’s theorem shows that this notation is consistentwhen f ∈ L1(Rn) and f is identified with a distribution in S(Rn)′.

The following relations, which are elementary for functions, extend to distri-butions f :

F−1f = (2π)−n(Ff ) = (2π)−nF f , where 〈ϕ, f〉 := 〈ϕ, f〉, (E.11)

FDjf = iξj · Ff, (E.12)

F(ϕ ∗ f) = (Fϕ) · (Ff) (ϕ ∈ S(Rn)). (E.13)

Plancherel’s theorem states that

〈Fϕ,F ψ〉 = (2π)n〈ϕ, ψ〉 (ϕ, ψ ∈ S(Rn)), (E.14)

where ψ is the complex conjugate of ψ, and hence F extends by continuity to alinear operator F on the Hilbert space L2(Rn) such that (2π)−n/2F is unitary.This also says that, for each f ∈ L2(Rn), the distribution Ff belongs to L2(Rn).

Many of the concepts above can be extended to the case of distributions onan open subset Ω of Rn. Let D(Ω) be the space of test functions on Ω, i.e., C∞-functions of compact support in Ω, and D(Ω)′ be the space of distributions on Ω,

E. DISTRIBUTIONS AND FOURIER MULTIPLIERS


i.e., linear functionals f on D(Ω) such that for each compact K ⊂ Ω there existm ∈ N and C > 0 such that

|〈ϕ, f〉| ≤ C sup|α|≤m

supx∈Ω

|Dαϕ(x)|

for all ϕ ∈ D(Ω) with suppϕ ⊂ K. Locally integrable functions on Ω can beidentified with distributions, and the derivatives Dj of a distribution f are definedby

〈ϕ,Djf〉 := −〈Djϕ, f〉 (ϕ ∈ D(Ω)).For m ∈ N and 1 ≤ p ≤ ∞, the Sobolev space Wm,p(Ω) is defined by

Wm,p(Ω) := {f ∈ Lp(Ω) : Dαf ∈ Lp(Ω) for all α ∈ Nn0 with |α| ≤ m},

where Dαf is understood in the sense of distributions. Thus, f ∈Wm,p(Ω) if andonly if for each α ∈ Nn

0 with |α| ≤ m there exists fα ∈ Lp(Ω) such that∫Ω

ϕfα dx = (−1)|α|∫Ω

(Dαϕ)f dx (ϕ ∈ D(Ω)).

In the special case when n = 1, f ∈Wm,p(Ω) if and only if f ∈ Cm−1(Ω), f (m−1)

is absolutely continuous, and f (j) ∈ Lp(Ω) for j = 0, 1, . . . ,m. Equipped with thenorm

‖f‖Wm,p(Ω) :=∑|α|≤m

‖Dαf‖p,

Wm,p(Ω) becomes a Banach space. The closure of D(Ω) in Wm,p(Ω) is denotedby Wm,p

0 (Ω). For p = 2, we also use the notation

Hm(Ω) := Wm,2(Ω) and Hm0 (Ω) := Wm,2

0 (Ω).

Equipped with the equivalent norm

‖f‖Hm(Ω) :=

⎛⎝ ∑|α|≤m

‖Dαf‖22

⎞⎠1/2

,

Hm(Ω) is a Hilbert space with the inner product

(f |g)Hm(Ω) =∑|α|≤m

∫Ω

DαfDαg dx.

Note that Plancherel’s theorem and (E.12) show that

Hm(Rn) = {f ∈ L2(Rn) : ξα · Ff ∈ L2(Rn) for all α ∈ Nn0 with |α| ≤ m},

where ξα is the function ξ �→ ξα11 ξα2

2 · · · ξαnn . Hence, f ∈ Hm(Rn) if and only if

ξ �→ (1 + |ξ|2)m/2(Ff)(ξ) belongs to L2(Rn).

489

Now we consider Fourier multipliers. If g is a C∞-function satisfying theestimates (E.3), then the map ϕ �→ F−1gFϕ := F−1(g · (Fϕ)) is a continuouslinear map on S(Rn). It is a classical problem to seek conditions on a function suchthat such a map becomes continuous on a function space X = Lp(Rn) (1 ≤ p ≤ ∞)or C0(Rn). Let m : Rn → C be an absolutely regular function. For ϕ ∈ S(Rn), wedefine m · (Fϕ) ∈ S(Rn)′ by

〈ψ,m · (Fϕ)〉 :=∫Rn

ψm · (Fϕ) dx (ψ ∈ S(Rn)).

Then we consider the distribution F−1(m · (Fϕ)) ∈ S(Rn)′. We call m a Fouriermultiplier for X if F−1(m·(Fϕ)) ∈ X for all ϕ ∈ S(Rn) and there exists a constantC such that

‖F−1(m · (Fϕ))‖X ≤ C‖ϕ‖X (ϕ ∈ S(Rn)).

Then the map ϕ �→ F−1(m · (Fϕ)) extends to a bounded linear operator Tm :f �→ F−1mFf on X (in the case when X = L∞(Rn), the extension is weak*continuous). When m is a C∞-function, Tmf agrees with the distribution F−1(m ·Ff) defined earlier.

We denote the space of all Fourier multipliers for X by MX(Rn), or byMp(Rn) when X = Lp(Rn), with the usual identification of functions which coin-cide a.e. We put

‖m‖MX(Rn) := ‖Tm‖L(X).

Fourier multipliers are bounded functions, and ‖m‖MX(Rn) ≥ ‖m‖∞ (see Propo-sition E.2). It follows easily that MX(Rn) is a Banach space. Note also thatMC0

(Rn) ⊂M∞(Rn).

For a ∈ Rn, define τa ∈ L(X) by τaf(x) := f(x − a). If m ∈ MX(Rn), it iseasy to see that

Tmτa = τaTm (a ∈ Rn). (E.15)

Conversely, we have the following result.

Proposition E.1. Let X = Lp(Rn) (1 ≤ p ≤ ∞) or C0(Rn), and assume thatT ∈ L(X) satisfies (E.15). Then there exists m ∈MX(Rn) such that

Tf = F−1mFf (f ∈ X).

For a proof of Proposition E.1, see [Hor60].

For N ∈ N, we let MNp (Rn) be the space of all matrices m = (mij)1≤i,j≤N ,

where mij ∈ Mp(Rn). Each such matrix m defines a bounded operator F−1mFon Lp(Rn)N , where F : Lp(Rn)N → Lp(Rn)N acts on each coordinate function,and matrix multiplication operates as usual. The norm onMN

p (Rn) is taken to be

the norm of the operator F−1mF when Lp(Rn)N is given the norm of Lp(Rn ×{1, . . . , N}). Note that M1

p(Rn) =Mp(Rn).



Proposition E.2. Let 1 ≤ p ≤ ∞, N ∈ N. Then the following hold true:

a) MNp (Rn) is a Banach algebra.

b) MN2 (Rn) = L∞(Rn,L(CN )) := {(mij)1≤i,j≤N : mij ∈ L∞(Rn)N}.

c) MNp (Rn) =MN

p′ (Rn), where 1/p+ 1/p′ = 1.

d) MN1 (Rn) ⊂MN

p (Rn) ⊂MN2 (Rn). Moreover, for m ∈MN

1 (Rn),

‖m‖MNp (Rn) ≤ ‖m‖θMN

1 (Rn)‖m‖1−θMN

2 (Rn), (E.16)

where θ := 2∣∣ 1p − 1

2

∣∣.e) Given a ∈ MN

p (Rn) define at by at(ξ) := a(tξ) for t > 0, ξ ∈ Rn. Then

at ∈MNp (Rn) for all t > 0 and

‖at‖MNp (Rn) = ‖a‖MN

p (Rn) (t > 0).

f) Let (aj)j∈N ⊂ MNp (Rn). Assume that there exists a constant C > 0 such

that ‖aj‖MNp (Rn) ≤ C for j ∈ N. Let a ∈ L∞(Rn) such that aj(x) → a(x)

for almost all x ∈ Rn as j →∞. Then a ∈MNp (Rn) and ‖a‖MN

p (Rn) ≤ C.

Proof. We give only sketches of the proofs; details may be found in [Hor60] or[Ste93].

a) follows from the formal identity F−1(m1m2)F = (F−1m1F)(F−1m2F),b) is an easy consequence of Plancherel’s theorem and c) is easily proved by duality,showing even that the equalities are isometric.

For d), we can assume by c) that 1 ≤ p ≤ 2. Let m ∈ MNp (Rn). By c),

F−1mF is bounded on Lp(Rn)N and on Lp′(Rn)N . Moreover the two versionsof the map agree on Lp(Rn)N ∩ Lp′(Rn)N . By the Riesz-Thorin theorem [Hor83,Theorem 7.1.12], F−1mF extends to a bounded linear operator on L2(Rn)N . Thisshows that MN

p (Rn) ⊂ MN2 (Rn). The inclusion MN

1 (Rn) ⊂ MNp (Rn) and the

inequality (E.16) also follows from the Riesz-Thorin theorem.e) follows from the fact that F−1atF = J−1

t F−1aFJt, where Jt is the isom-etry, (Jtf)(ξ) := t−n/pf(tξ), on Lp(Rn), and f) is proved by taking limits throughthe definitions of Fourier multipliers.

An extremely useful sufficient condition for a functionm to belong toM1p(R

n)for 1 < p < ∞ is given by the Mikhlin multiplier theorem. Let j := min{k ∈ N :k > n

2 }. Define the Banach space MM by

MM :={m : Rn → K : m ∈ Cj(Rn \ {0}), |m|M <∞}

, (E.17)

where the norm | · |M is defined by

|m|M := max|α|≤j

supξ∈Rn\{0}

|ξ||α||Dαm(ξ)|. (E.18)

We then have the following result.

491

Theorem E.3 (Mikhlin). Let 1 < p <∞. Then MM ↪→Mp(Rn).

For a proof of Mikhlin’s theorem, we refer to [Ste93, Theorem VI.4.4].The following results on Fourier multipliers will be useful in Chapter 8.

Theorem E.4. Let 1 ≤ p ≤ ∞. Then the following hold true:

a) Let a be a real homogeneous polynomial on Rn of degree m > 1. Then eia ∈Mp(Rn) if and only if p = 2.

b) Let a ∈ C∞(Rn) satisfy

a(ξ) :=

{|ξ|−βe−i|ξ|α (|ξ| ≥ 2),

0 (|ξ| ≤ 1),

where α > 0 and β ≥ 0.

(i) If α �= 1 and 1 < p <∞ (respectively, p = 1), then a ∈ Mp(Rn) if and

only if n∣∣ 12 − 1

p

∣∣ ≤ βα (respectively, n

2 < βα ).

(ii) If α = 1 and 1 < p < ∞ (respectively, p = 1) then a ∈ Mp(Rn) if andonly if (n− 1)

∣∣ 12 − 1

p

∣∣ ≤ β (respectively, n−12 < β).

c) Define a1 : R3 → C and a2 : R3 → C by

a1(ξ) := (−i)(ξ1 + ξ22 + ξ23 − i),

a2(ξ) := ξ1 + ξ22 + ξ23 + i.

(i) If p �= 2, then a−11 �∈ Mp(R3).

(ii) Let a := a1a2. Define the operator Ap on Lp(R3) by Apf := F−1(aFf)with D(Ap) := {f ∈ Lp(R3) : F−1(aFf) ∈ Lp(R3)}. If ∣∣ 12 − 1

p

∣∣ > 38 ,

then σ(Ap) = C.

For a proof of the assertions of Theorem E.4 we refer to [Hor60] (assertiona)), [FS72], [Miy81] and [Per80] (assertion b)), [KT80] (assertion c)i)) and [IS70](assertion c)ii)).

Finally, we note one instance of Mikhlin’s Theorem.For x ∈ R, define

signx :=

⎧⎪⎨⎪⎩1 (x > 0),

0 (x = 0),

−1 (x < 0).

Then sign ∈MM . By Mikhlin’s theorem, sign ∈Mp(R) for 1 < p <∞.For ϕ ∈ S(R), one finds that F−1(−i sign)Fϕ is a function given by

(F−1(−i sign)Fϕ)(x) = limε↓0

1

π

∫|x−y|≥ε

ϕ(y)

x− ydy. (E.19)

This is known as the Hilbert transform of ϕ. Thus, we have the following.



Proposition E.5. Let 1 < p < ∞. Then the Hilbert transform is a bounded linearoperator on Lp(R).

Notes: The material on distributions is very standard and can be found in many books,for example [Hor83]. The basic material on Fourier multipliers can be found in [Ste93].

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Notation

Function and Distribution Spaces

AAP(R+, X) space of asymptotically almost periodic functions . . . . . 307

AP(I,X),AP(I) space of almost periodic functions . . . . . . . . . . . 292, 297, 307

BSV([a, b], X) space of functions of bounded semivariation . . . . . . . . . . . . 48

BSVloc(R+, X) space of functions of locally bounded semivariation . . . . . 48

BUC(I,X),BUC(I) space of bounded, uniformly continuous functions . . . . . 15

Lipω(R+, X) space of Lipschitz continuous functions . . . . . . . . . . . . . 63, 77

D(Ω)′ space of distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485, 488

D(Ω) space of test functions . . . . . . . . . . . . . . . . . . . . . . . . 15, 485, 487

E , E(R+, X) space of totally ergodic functions . . . . . . . . . . . . . . . . 301, 308

FL1(Rn) Fourier algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436

MM Mikhlin space of Fourier multipliers . . . . . . . . . . . . . . 436, 490

MNp (Rn) space of matrices of Fourier multipliers . . . . . . . . . . . . . . . 489

MX(Rn),Mp(Rn) space of Fourier multipliers on X or Lp(Rn) . . . . . . . . . . 489

Mε strong Mikhlin space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436

S(Rn)′ space of temperate distributions . . . . . . . . . . . . . . . . . . . . . . 486

S(Rn) Schwartz space of rapidly decreasing functions . . . 319, 485

E quotient of space of totally ergodic functions . . . . . 301, 308

C(I,X), C(Ω) space of continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . . 15

c(X) space of convergent sequences . . . . . . . . . . . . . . . . . . . . . . . . . . 41

526 NOTATION

Ck(I,X), Ck(Ω) space of k-times continuously differentiable functions . . . 15

C∞(I,X), C∞(Ω) space of infinitely differentiable functions . . . . . . . . . . . . . . 15

C0(I,X), C0(Ω) space of continuous functions vanishing at infinity . . . . . . 15

c0 space of null sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10, 481

Cc(I,X), Cc(Ω) space of continuous functions with compact support . . . 15

C∞c (I,X), C∞c (Ω) space of infinitely differentiable functions with compact sup-port . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

C∞W ((ω,∞), X) Widder space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64, 78

C1ω(R+, X) space of functions with continuous, exponentially bounded

derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

H2(C+, X), H2(C+) Hardy spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

Hm(Ω), Hm0 (Ω) Sobolev space of order (m, 2) . . . . . . . . . . . . . . . . . . . . . . . . . 488

Lp(Ω, μ) space of p-integrable functions on a measure space . . . . 175

Lp(I) space of Lebesgue p-integrable functions . . . . . . . . . . . . . . . 14

Lp(I,X) space of Bochner p-integrable functions . . . . . . . . . . . . 13, 14

lp space of p-summable sequences . . . . . . . . . . . . . . . . . . . . 10, 132

L∞(I,X), L∞(I) space of bounded measurable functions . . . . . . . . . . . . . . . . 14

l∞(X) space of bounded sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

L∞ω (I,X) space of exponentially bounded functions . . . . . . . . . 77, 226

L1loc(R+, X) space of locally Bochner integrable functions . . . . . . . . . . . 13

Smρ,0 space of symbols of pseudo-differential operators . . . . . . 430

Wm,p(Ω),Wm,p0 (Ω) Sobolev space of order (m, p) . . . . . . . . . . . . . . . . . . . . . . . . . 488

Dual Spaces and Subspaces

V ′ antidual of V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420

X∗ dual space of X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

X sun-dual of X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

X0 space of vectors converging to 0 . . . . . . . . . . . . . . . . . . . . . . . 360

NOTATION 527

Xe space of totally ergodic vector in X . . . . . . . . . . . . . . . . . . . . 267

Xap space of almost periodic vectors in X . . . . . . . . . . . . 290, 361

Xe0 space of totally ergodic vectors with means 0 . . . . . . . . . 267

Norms and Dualities

(·|·)H inner product on a Hilbert space H . . . . . . . . . . . . . . . . . . . . 45

(·|·) duality between a space and its antidual . . . . . . . . . . . . . . 421

〈·, ·〉 duality between a space and its dual . . . . . . . . . . . . . . . 7, 485

‖ · ‖D(A) graph norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468

‖ · ‖W Widder norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64, 78

|α| norm of multi-index α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485

| · |M Mikhlin norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436, 490

‖ · ‖p Lebesgue-Bochner norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13, 14

| · |Mεstrong Mikhlin norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436

‖ · ‖ω,∞ exponentially bounded norm . . . . . . . . . . . . . . . . . . . . . . 77, 226

|π| norm of partition π . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

Functions, Integrals and Abscissas

abs(f), abs(dF ) abscissa of convergence . . . . . . . . . . . . . . . . . . . . . . 27, 30, 56, 57

χΩ characteristic function of Ω . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Cos cosine function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

hol(f), hol(T ) abscissa of holomorphy of f or T . . . . . . . . . . . . . . . . . . . 33, 35

hol0(f) abscissa of boundedness of f . . . . . . . . . . . . . . . . . . . . . . . . . . . 33∫ b

ag(t) dF (t) Riemann-Stieltjes integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49∫ b

ag(t) dt Riemann integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50∫

If(t) dt Bochner integral of f over I . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

ω(f), ω(T ) exponential growth bound of f or T . . . . . . . . . . . . . . . . 29, 30

ω1(T ) exponential growth bound of classical solutions . . . . . . . 343

528 NOTATION

sign signum function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138, 491

Sin sine function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206, 218

En Newtonian potential on Rn . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404

eλ exponential function t �→ eλt . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

eλ ⊗ x the function t �→ eλtx . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292

f ∗ g, T ∗ f convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21, 24, 487

kt, kz Gaussian kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

S(g, π) Riemann sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

S(g, F, π) Riemann-Stieltjes sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

ux orbit of T through x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30, 337

V (π, F ) variation of F over π . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

V (F ), V[a,b](F ) total variation of F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Operators

Δ distributional Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

Δ0 Laplacian on C0(Ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406

Δp Laplacian on Lp(Rn) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

ΔX Laplacian on X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

Δmax Laplacian on C(Ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403

ΔL2(Ω) Dirichlet-Laplacian on L2(Ω) . . . . . . . . . . . . . . . . . . . . . . . . . 140

Ap system of differential operators on Lp(Rn) . . . . . . . . . . . . 449

AX pseudo-differential operator on X . . . . . . . . . . . . . . . . . . . . . 431

K(X) space of compact operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

L(X,Y ),L(X) space of bounded linear operators . . . . . . . . . . . . . . . . . . . . . . 24

OpX(a),Opp(a) pseudo-differential operator on X or Lp(Rn) . . . . . . . . . . 430

KerA kernel of A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261, 467

RanA range of A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261, 467

NOTATION 529

A closure of an operator A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467

Φ Riesz operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

ΦS Riesz-Stieltjes operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

A∗ adjoint of an operator A . . . . . . . . . . . . . . . . . . . . . . . . . . 472, 473

AH operator associated with quadratic form . . . . . . . . . . . . . . . 419

AY part of an operator A in Y . . . . . . . . . . . . . . . . . . . . . . . 136, 471

B−z fractional power of B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

B1/2 square root of B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

D(A) domain of an operator A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467

Dα higher order partial derivative . . . . . . . . . . . . . . . . . . . 485, 486

Dj partial derivative ∂/∂xj . . . . . . . . . . . . . . . . . . . . . . . . . . 485, 486

R(λ,A) resolvent of an operator A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468

Spectrum and Resolvent Set

spB(f) Beurling spectrum of f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322

spC(f) Carleman spectrum of f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295

sp(f) half-line spectrum of f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272

spw(f) weak half-line spectrum of f . . . . . . . . . . . . . . . . . . . . . . . . . . 325

ρ(A) resolvent set of an operator A . . . . . . . . . . . . . . . . . . . . . . . . . 468

ρu(A, x) imaginary local resolvent set . . . . . . . . . . . . . . . . . . . . . . . . . . 371

σ(A, x) local spectrum of A at x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299

σ(A) spectrum of A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468

σp(A) point spectrum of A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469

σu(A, x) imaginary local spectrum of A at x . . . . . . . . . . . . . . . . . . . 371

σap(A) approximate point spectrum of A . . . . . . . . . . . . . . . . . . . . . 469

r(T ) spectral radius of an operator T . . . . . . . . . . . . . . . . . . . . . . . 475

s(A) spectral bound of A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188, 469

530 NOTATION

s0(A) pseudo-spectral bound of A . . . . . . . . . . . . . . . . . . . . . . . . . . . 345

Subsets of Rn or C

C+ open right half-plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

C− open left half-plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362

T unit circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316

N set of positive integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

N0 set of non-negative integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

∂Ω topological boundary of Ω . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401

R+ set of non-negative real numbers . . . . . . . . . . . . . . . . . . . . . . . . 13

Σα sector of angle α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

Transformations

f reflection of f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

f , T Laplace or Carleman transform of f or T . . . . . . 27, 32, 295

F Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44, 487

L Laplace transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

LS Laplace-Stieltjes transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

F conjugate Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

dF Laplace-Stieltjes transform of F . . . . . . . . . . . . . . . . . . . . . . . 55

Cauchy Problems

(ACP0) homogeneous abstract Cauchy problem . . . . . . . . . . . . . . . . 108

(ACPf ) inhomogeneous abstract Cauchy problem . . . . . . . . . . . . . 117

(ACPk+1) (k + 1)-times integrated abstract Cauchy problem . . . . 129

ACP0(R) abstract Cauchy problem on the line . . . . . . . . . . . . . . . . . . 118

D(ϕ) Dirichlet problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401

P∞(u0, ϕ, f) inhomogeneous heat equation . . . . . . . . . . . . . . . . . . . . . . . . . 415

Pτ (u0, ϕ) heat equation with inhomogeneous boundary conditions 408,412

NOTATION 531

Other Notation

(Hr) growth hypothesis for symbols . . . . . . . . . . . . . . . . . . . . . . . . . 439

Freq(x),Freq(f) set of frequencies of vector x or function f . . 267, 293, 315

dN(x) subdifferential of the norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

supp support of a function or distribution . . . . . . . . . . . . . . . . . . 318

B(x, ε) closed ball, centre x, radius ε . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

D closure of a set D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

B(x, ε) open ball, centre x, radius ε . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

m(Ω) Lebesgue measure of Ω . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Mηx,Mηf mean of vector x or function f at η . . . . 266, 293, 308, 315

x · ξ scalar product of x and ξ in Rn . . . . . . . . . . . . . . . . . . 430, 487

x ≤ y ordering in a Banach space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477

X+ positive cone in X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477

Zd↪→ X continuous dense embedding . . . . . . . . . . . . . . . . . . . . . 184, 418

Z ↪→ X continuous embedding of Z in X . . . . . . . . . . . . . . . . . . . . . 184

Index

AAbel-convergence, 244, 257Abel-ergodic, 263Abelian theorem, 243, 245abscissa

of boundedness, 33, 285of convergence, 27, 31, 56, 58of holomorphy, 33

absolutelycontinuous, 15, 18convergent, 14regular, 486

adjoint, 472, 473almost periodic

function on the half-line, 307function on the line, 292orbits, 294vector, 290, 361

almost separably valued, 7analytic

Radon-Nikodym property, 61representation, 84

antiderivative, 15antidual, 420, 422antilinear, 420approximate

eigenvalue, 469point spectrum, 469unit, 23

approximation theorem, 41, 67asymptotically

almost periodic, 307, 365norm-continuous, 389

BB-convergence, 244Banach lattice, 478band, 479Bernstein, 90, 100, 435Beurling spectrum, 322

Bochner, 9integrable, 9integral, 9

boundarygroup, 172semigroup, 171

boundedabove, 214, 474holomorphic semigroup, 150semivariation, 48variation, 15, 48

Brenner, 450, 452

CCarleman

spectrum, 295spectrum and C0-groups, 295transform, 295

Cauchy problemabstract, 108inhomogeneous, 117on the line, 118

Cesaro-convergence, 244Cesaro-ergodic, 262character, 289classical solution, 108, 117, 203closable operator, 467closed operator, 467closure, 467Coifman-Weiss, 175compact resolvent, 470complete orbit, 120completely monotonic, 90, 106complex

inversion, 75, 259representation, 81Tauberian condition, 247

cone, 477converges

in the sense of Abel, 244in the sense of Cesaro, 244

534 INDEX

convex, 91convolution, 21, 24, 26, 486core, 468cosine function, 203countable spectrum, 374, 385countably valued, 6

DDa Prato-Sinestrari, 142Datko, 340densely defined, 467Desch-Schappacher, 161Dirac’s equation, 456Dirichlet

boundary conditions, 140, 423kernel, 257Laplacian, 424problem, 401regular, 402

dissipative, 137distribution, 485, 487

semigroup, 231dominated convergence, 11dual cone, 478Dunford-Pettis

property, 270theorem, 19

Eeigenvalue, 469elliptic

equation, 170maximum principle, 402operator, 425, 431polynomial, 431

ergodic vector, 266eventually differentiable, 284exponential growth bound, 29, 30, 338

FFattorini, 227feebly oscillating, 249Fejer, 257

kernel, 258

first order perturbation, 160form domain, 420Fourier

coefficients, 257inversion theorem, 45, 487multiplier, 489sums, 257transform, 44, 487type, 61, 387

fractional powers, 163frequencies, 267, 293, 311Fubini, 12function

absolutely continuous, 15, 18absolutely regular, 486almost separably valued, 7asymptotically almost periodic,

307Bochner integrable, 9completely monotonic, 90convex, 91countably valued, 6feebly oscillating, 249holomorphic, 461Laplace transformable, 28Lipschitz continuous, 15locally bounded, 462measurable, 6normalized, 100of bounded semivariation, 48of bounded variation, 15, 48of weak bounded variation, 48Riemann integrable, 50Riemann-Stieltjes integrable, 49simple, 6slowly oscillating, 247step, 6strongly continuous, 24test, 15, 485, 487totally ergodic, 296, 308, 315uniformly ergodic, 296, 308, 328weakly measurable, 7

fundamental theorem of calculus, 18

INDEX 535

GGaussian semigroup, 150, 153, 156,

170, 172, 183Gelfand, 280generating cone, 477generator

infinitesimal, 114of C0-group, 119of C0-semigroup, 112of cosine function, 205of integrated semigroup, 122of semigroup, 126of sine function, 218

Glicksberg-deLeeuw, 389graph norm, 468Grothendieck space, 270group

C0, 119, 295boundary, 172integrated, 179

HHormander, 173half-line spectrum, 272, 310, 315Hardy, 256Hardy-Littlewood, 253Hilbert transform, 198, 491Hille-Yosida

operator, 141theorem, 134

holomorphicfunction, 461semigroup, 148

hyperbolicequation, 427semigroup, 388system, 449

hypoelliptic, 431

Iideal, 192, 478identity theorem, 462

imaginary localresolvent set, 371spectrum, 371

improper integral, 13infinitesimal generator, 114Ingham, 327integral

absolutely convergent, 14Bochner, 9improper, 13Laplace, 27Laplace-Stieltjes, 55Riemann, 50Riemann-Stieljes, 49

integrated semigroup, 122integration by parts, 50intermediate points, 49interpolation property, 90inversion

complex, 75, 259Phragmen-Doetsch, 73Post-Widder, 42, 73

invertible, 468irreducible, 394

Jjump, 99

KKadets, 300Karamata, 251Katznelson-Tzafriri, 317, 391KB-space, 479kernel, 261, 394, 467Krein-Smulyan, 464

LL-space, 359Laplace

integral, 27, 32transform, 63, 110transformable, 28

536 INDEX

Laplace-Stieltjesintegral, 55transform, 63

Laplacianand boundary group, 173and boundary integrated group,

183and cosine functions, 448first order perturbation, 160generates Gaussian semigroup, 150on continuous functions, 403square root, 170with Dirichlet boundary condi-

tions, 140, 424with inhomogeneous boundary con-

ditions, 408largest lower bound, 478lattice, 478least upper bound, 478Lebesgue point, 16Lipschitz continuous, 15local

integrated semigroup, 232spectrum, 299, 371

locally bounded, 462Loomis, 297Lotz, 272Lumer-Phillips, 139

MMaxwell’s equations, 455mean-ergodic, 262measurable, 6Mikhlin, 491mild solution, 108, 117, 119, 203, 368,

408, 413, 415mollifier, 23, 319multi-index, 485multiplication operator, 419, 473

NNewtonian potential, 404non-resonance, 380normal cone, 477

normalization, 100normalized

antiderivative, 28function, 100

norming, 462

Ooperator

adjoint, 472, 473associated with form, 418closable, 467closed, 467elliptic, 425, 431Hille-Yosida, 141invertible, 468kernel, 394multiplication, 419, 473Poisson, 404positive, 478pseudo-differential, 430resolvent positive, 188Riesz, 72Riesz-Stieltjes, 66sectorial, 162selfadjoint, 150, 473symmetric, 474

ordercontinuous norm, 479interval, 477

ordered Banach space, 477

PPaley-Wiener, 46parabolic

domain, 412equation, 427maximum principle, 410problem, 408

part, 471partitioning points, 49period, 292periodic vector, 290

INDEX 537

perturbationcompact, 161first order, 160of C0-semigroup, 144of cosine function, 210, 213of Hille-Yosida operator, 143, 144of integrated semigroup, 187, 232of resolvent positive operator, 195of selfadjoint operator, 420, 423of sine function, 220relatively bounded, 159

Petrovskii correct systems, 232Pettis, 7phase space, 210Phragmen-Doetsch, 73Phragmen-Lindelof, 176Plancherel, 45point spectrum, 469Poisson

equation, 404operator, 404semigroup, 152, 170, 447

positivecone, 477element, 477functional, 477operator, 478

Post-Widder, 42, 73Pruss, 82primitive, 15principal

part, 431, 449value, 13

proper cone, 477pseudo-differential operator, 430pseudo-resolvent, 470pseudo-spectral bound, 345

RRadon-Nikodym property, 19, 72range, 261, 467real

Banach lattice, 478ordered Banach space, 477

representation, 69, 78Tauberian condition, 247

realization, 430regular point, 295, 310regularized semigroup, 232relatively compact orbit, 288relatively dense, 288, 310representation

analytic, 84complex, 81Paley-Wiener, 46real, 69, 78Riesz-Stieltjes, 66

resolvent, 335, 468compact, 470equation, 470identity, 470positive, 188set, 468

Riemannintegrable, 50integral, 50sum, 50

Riemann-Lebesgue, 45Riemann-Liouville semigroup, 175Riemann-Stieltjes

integrable, 49integral, 49sum, 49

Riesz operator, 72Riesz-Stieltjes

operator, 66representation, 66

Ssandwich theorem, 185Schwartz space, 318, 485sectorial operator, 162selfadjoint operator, 150, 473semigroup, 126

C-, 232C0, 111Abel-ergodic, 263

538 INDEX

asymptotically almost periodic,365

asymptotically norm-continuous,388

boundary, 171bounded holomorphic, 150Cesaro-ergodic, 262distribution, 231eventually differentiable, 284Gaussian, 150, 153, 156, 170, 172,

183holomorphic, 148hyperbolic, 388irreducible, 394k-times integrated, 122local integrated, 232norm-continuous, 201once integrated, 122Poisson, 152, 170, 447regularized, 232Riemann-Liouville, 175smooth distribution, 232totally ergodic, 266, 373

separating, 8, 262, 464sesquilinear form, 420similar operators, 144simple

function, 6pole, 269

sine function, 206, 218slowly oscillating, 247smooth distribution semigroup, 232smoothing effect, 158Sobolev space, 488spectral

projection, 472bound, 188, 343, 469radius, 475synthesis, 291, 293, 391theorem, 474

spectrum, 468approximate point, 469Beurling, 322Carleman, 295

half-line, 272, 310, 315imaginary local, 371local, 299point, 469weak half-line, 325

square root, 164step function, 6strong convergence, 31strong splitting theorem, 364strongly continuous, 24subdifferential, 137sublattice, 478sun-dual, 137support, 318symbol, 430symmetric, 474

TTauberian

condition, 243, 247theorem, 88, 243, 247

temperate distribution, 485tempered integrated semigroup, 232test function, 15, 485, 487Theorem

Abel, 247Analytic Representation, 84Approximation, 41, 67Bernstein, 100Bochner, 9Brenner, 450, 452Coifman-Weiss, 175Complex Inversion, 75Complex Representation, 81Countable spectrum, 374Da Prato-Sinestrari, 142Datko, 340Desch-Schappacher, 161Dominated Convergence, 11Dunford-Pettis, 19Fattorini, 227Fejer, 257Fubini, 12Gelfand, 280

INDEX 539

Glicksberg-deLeeuw, 389Hormander, 173Hardy, 256Hardy-Littlewood, 253Hille-Yosida, 134Identity, 462Ingham, 327Kadets, 300Karamata, 251Katznelson-Tzafriri, 317, 391Krein-Smulyan, 464Loomis, 297Lotz, 272Lumer-Phillips, 139Mikhlin, 491Non-resonance, 380Paley-Wiener, 46Pettis, 7Phragmen-Doetsch Inversion, 73Phragmen-Lindelof, 176Plancherel, 45Post-Widder Inversion, 42, 73Real Representation, 69, 78Riesz-Stieltjes Representation, 66Sandwich, 185Spectral, 474Splitting, 364, 368, 389Tauberian, 88Trotter-Kato, 146Uniqueness, 40, 294Vitali, 463

totally ergodicfunction, 296, 308, 315semigroup, 266, 373vector, 266, 290, 361

transference principle, 175trigonometric polynomial, 292, 365Trotter-Kato, 146

UUMD-space, 198unconditionally bounded, 304, 481uniform ellipticity, 425

uniformlyconvex, 303ergodic, 296, 308, 328

unimodular eigenvector, 361uniqueness

sequence, 40theorem, 40, 294

unitarily equivalent, 474

Vvariation of constants formula, 118,

158Vitali, 463

Wwave equation, 170, 332, 425, 455weak

bounded variation, 48half-line spectrum, 325splitting theorem, 368

weaklyalmost periodic, 294almost periodic in the sense of

Eberlein, 294asymptotically almost periodic,

334holomorphic, 461measurable, 7regular point, 325

Weierstrass formula, 216well-posed, 115

YYoung’s inequality, 22, 24

Vector-valued Holomorphic Functions

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