Integral Transform Functional Calculi

Chapter 6 Integral Transform Functional Calculi

In this chapter we continue our investigations from the previous one and en- counter functional calculi associated with various semigroup representations.

6.1 The Fourier–Stieltjes Calculus

Recall from the previous chapter that the Fourier transform

d d F : M(R ) → UCb(R ) is a contractive and injective unital algebra homomorphism. Hence, it is an isomorphism onto its image

d d d FS(R ) := F(M(R )) = {µb | µ ∈ M(R )}.

This algebra, which is called the Fourier–Stieltjes algebra1 of Rd, is en- dowed with the norm

d kµbkFS := kµkM (µ ∈ M(R )), which turns it into a Banach algebra and the Fourier transform F : M(Rd) → FS(Rd) into an isometric isomorphism. (The Fourier algebra of Rd is the closed ideal(!) A(Rd) := F(L1(Rd)) of Fourier transforms of L1-functions.) Let S ⊆ Rd be a closed subsemigroup. Then M(S) is a Banach subalgebra of M(Rd) and d FSS(R ) := F(M(S)),

1 The Fourier transform on the space of measures is sometimes called the Fourier–Stieltjes transform, hence the name “Fourier–Stietjes algebra” for its image. In books on Banach algebras one often ﬁnds the symbol “B(Rd)” for it.

85 86 6 Integral Transform Functional Calculi called its associated Fourier–Stieltjes algebra, is a Banach subalgebra of FS(Rd). Let T : S → L(X) be a strongly continuous and bounded representation with associated algebra representation Z M(S) → L(X), µ 7→ Tµ = Tt µ(dt). (6.1) S d Since the Fourier transform F : M(S) → FSS(R ) is an isomorphism, we can compose its inverse with the representation (6.1). In this way a functional calculus Ψ : FS ( d) → L(X),Ψ (µ) := T , T S R T b µ is obtained, which we call the Fourier–Stieltjes calculus for T . Note that by the deﬁnition of the norm on FS(Rd) and by (5.6) we have

d kΨT (f)k ≤ MT kfkFS (f ∈ FSS(R )), (6.2) where, as always, M = sup kT k. T t∈S t Example 6.1. Let Ω = (Ω, Σ, ν) be a measure space and a : Ω → Rd a measurable function. Fix p ∈ [1, ∞) and abbreviate X := Lp(Ω). For t ∈ Rd let Tt ∈ L(X) be deﬁned by

−it·a(·) Ttx := e x (x ∈ X).

Then (T ) d is a bounded and strongly continuous group and, by Exercise t t∈R 5.8, Tµx = (µb ◦ a) x (x ∈ X) for all µ ∈ M(Rd). This means that the Fourier-Stieltjes calculus for T is nothing but the restriction of the usual multiplication operator functional calculus to the algebra FS(Rd).

Example 6.2. Let T = τ be the regular (right shift) representation of Rd on 1 d d X = L (R ). Then, for f = µb ∈ FS(R ) one has −1 −1 1 d Ψτ (f)x = µ ∗ x = F (µb · xb) = F (f · xb)(x ∈ L (R )).

That is, the operator Ψτ (f) is the so-called Fourier multiplier operator with the symbol f: ﬁrst take the Fourier transform, then multiply with f, ﬁnally transform back.

In the following we shall examine the Fourier–Stieltjes calculus for the special cases S = Z, S = Z+, and S = R+. 6.1 The Fourier–Stieltjes Calculus 87

Doubly power-bounded operators

An operator T ∈ L(X) is called doubly power-bounded if T is invertible and M := sup kT nk < ∞. Such operators correspond in a one-to-one T n∈Z fashion to bounded Z-representations on X (cf. Example 5.1). The spectrum of a doubly power-bounded operator T is contained in the torus T = {z ∈ C : |z| = 1}. ∼ 1 By Exercise 5.9, M(Z) = ` (Z) and FSZ(R) consists of all functions

X −int 1 f = αne (α = (αn)n ∈ ` ). (6.3) n∈Z

These functions form a subalgebra of C2π(R), the algebra of 2π-periodic functions on R. By the Fourier–Stieltjes calculus, the function f as in (6.3) is mapped to

X n ΨT (f) = αnT . n∈Z Hence, this calculus is basically the same as a Laurent series calculus

X n X n ΦT : αnz 7−→ αnT (6.4) n∈Z n∈Z where the object on the left-hand side is considered as a function on T. We prefer this latter version of the Fourier–Stieltjes calculus because it works with functions deﬁned on the spectrum of T and has T as its generator. The algebra X n 1 A(T) := αnz | α ∈ ` (Z) ⊆ C(T) n∈Z is called the Wiener algebra and the calculus (6.4) is called the Wiener calculus. In order to make precise our informal phrase “basically the same” from above, we use the following notion from abstract functional calculus theory. Deﬁnition 6.3. An isomorphism of two proto-calculi Φ : F → C(X) and Ψ : E → C(X) on a Banach space X is an isomorphism of unital algebras η : F → E such that Φ = Ψ ◦ η. If there is an isomorphism, the two calculi are called isomorphic or equivalent.

Let us come back to the situation from above. The mapping e−it : R → T induces an isomorphism (of unital Banach algebras)

−it C(T) → C2π(R), f 7→ f(e ).

This restricts to an isomorphism η : A(T) → FSZ(R) by virtue of which the Fourier–Stieltjes calculus is isomorphic to the Wiener calculus. 88 6 Integral Transform Functional Calculi

Power-bounded operators

Power-bounded operators T ∈ L(X) correspond in a one-to-one fashion to ∼ 1 Z+-representations on X. By Exercise 5.9, M(Z+) = ` (Z+) and FSZ+ (R) consists of all functions

∞ X −int 1 f = αne (α = (αn)n ∈ ` ). (6.5) n=0

These functions form a subalgebra of C2π(R). Under the Fourier–Stieltjes calculus, the function f as in (6.5) is mapped to

∞ X n ΨT (f) = αnT . n=0 This calculus is isomorphic to the power-series calculus

∞ ∞ 1 X n X n ΦT :A+(D) → L(X),ΦT αnz = αnT . (6.6) n=0 n=0 The isomorphism of the two calculi is again given by the algebra homomor- −it 1 P∞ n phism f 7→ f(e ). (Observe that for α ∈ ` the function f = n=0 αnz can be viewed as a function on D, or on D or on T or on (0, 1) and in either interpretation α is determined by f.)

6.2 Bounded C0-Semigroups and the Hille–Phillips Calculus

We now turn to the case S = R+. A strongly continuous representation of 2 R+ on a Banach space is often called a C0-semigroup . Operator semigroup theory is a large ﬁeld and a thorough introduction would require an own course. We concentrate on the aspects connected to functional calculus theory. If you want to study semigroup theory proper, read [2] or [3].

Let T = (Tt)t≥0 be a bounded C0-semigroup on a Banach space X. Then we have the Fourier–Stieltjes calculus Z Ψ : FS ( ) → L(X),Ψ (µ) = T µ(dt). T R+ R T b t R+

2 The name goes back to the important monograph [1, Chap. 10.1] of Hille and Phillips, where diﬀerent continuity notions for semigroup representations are considered. 6.2 Bounded C0-Semigroups and the Hille–Phillips Calculus 89

However, as in the case of power-bounded operators, we rather prefer working with an isomorphic calculus, which we shall now describe. The Laplace transform (also called Laplace–Stieltjes transform) of a measure µ ∈ M(R+) is the function Z −zt Lµ : C+ → C, (Lµ)(z) = e µ(dt). R+

Here, C+ := {z ∈ C : Re z > 0} is the open right half-plane. By some standard arguments,

kLµk∞ ≤ kµkM and Lµ ∈ UCb(C+) ∩ Hol(C+), and it is easy to see that the Laplace transform

L : M(R+) → UCb(C+), µ 7→ Lµ is a homomorphism of unital algebras. Moreover,

(Lµ)(is) = (Fµ)(s) for all s ∈ R (6.7) and since the Fourier transform is injective, so is the Laplace transform. Let us call its range

LS(C+) := {Lµ : µ ∈ M(R+)} the Laplace-Stieltjes algebra and endow it with the norm

kLµkLS := kµkM (µ ∈ M(R+)). Then the mapping

LS(C+) → FSR+ (R), f 7→ f(is) is an isometric isomorphism of unital Banach algebras. Given a C0-semigroup (Tt)t≥0 one can compose its Fourier–Stieltjes calculus ΨT with the inverse of this isomorphism to obtain the calculus Z ΦT : LS(C+) → L(X),ΦT (Lµ) := Tt µ(dt). R+ This calculus is called the Hille–Phillips calculus3 for T . It satisﬁes the norm estimate

kΦT (f)k ≤ MT kµkM (f = Lµ, µ ∈ M(R+)). (6.8)

3 One would expect the name “Laplace–Stieltjes calculus” but we prefer sticking to the common nomenclature. 90 6 Integral Transform Functional Calculi

Similar to the discrete case, we note that an element f = Lµ ∈ LS(C+) can be interpreted as a function on C+, on C+, on iR, or on (0, ∞) and in either interpretation µ is determined by f. The function z is unbounded on the right half-plane and hence it is not contained in the domain of the Hille–Phillips calculus. Nevertheless, this functional calculus has a generator, as we shall show next.

The Generator of a Bounded C0-Semigroup

Suppose as before that T = (Tt)t≥0 is a bounded C0-semigroup on a Banach space X. For Re λ, Re z > 0 one has

1 Z ∞ = e−λte−zt dt. λ + z 0

1 In other words, the function λ+z (deﬁned on C+) is the Laplace transform 1 −λt −1 of the L -function e 1R+ . As such, (λ + z) ∈ LS(C+) for each λ ∈ C+.

Theorem 6.4. Let T = (Tt)t≥0 be a bounded C0-semigroup on a Banach space X with associated Hille–Phillips calculus ΦT . Then there is a (uniquely determined) closed operator A such that 1 (λ + A)−1 = Φ T λ + z for one/all λ ∈ C+. The operator A has the following properties: −1 a) [ Re z < 0 ] ⊆ ρ(A) and R(−λ, A) = −ΦT ((λ + z) ) for all Re λ > 0. b) dom(A) is dense in X. c) λ(λ + A)−1 → I strongly as 0 < λ % ∞. d) For all w ∈ C, t > 0 and x ∈ X: Z t Z t ws sw tw e Tsx ds ∈ dom(A) and (A − w) e Tsx ds = x − e Ttx. 0 0

e) ΦT (f)A ⊆ AΦT (f) for all f ∈ LS(C+). Proof. The operator family 1 1 R(w) := Φ = −Φ , Re w < 0, T w − z T (−w) + z is a pseudo-resolvent. As such, there is a uniquely determined closed linear relation A on X such that R(w) = (w − A)−1 for one/all Re w < 0 (Theorem A.13). In order to see that A is an operator and not just a relation, we need to show that R(w) is injective for one (equivalently: all) Re w < 0. As ker(R(w)) 6.2 Bounded C0-Semigroups and the Hille–Phillips Calculus 91 does not depend on w (by the resolvent identity), the claim follows as soon as we have proved part c). a) This holds by deﬁnition of A. b) Let t > 0, x ∈ X, and w, λ ∈ C. Then a little computation yields Z t Z t Z t (z + λ) ewse−sz ds = (z − w) ewse−sz ds + (w + λ) ewse−sz ds 0 0 0 Z t = 1 − ewte−tz + (w + λ) ewse−sz ds =: f(z) (Re z > 0). 0

That means that f = Lµ ∈ LS(C+) with

−wt ws µ = δ0 − e δt + (w + λ)1[0,t]e ds.

For each Re λ > 0 one can divide by z + λ and then apply the Hille–Phillips calculus to obtain Z t Z t ws sw −sz e Tsx ds = ΦT e e ds x = −R(−λ, A)ΦT (f)x ∈ dom(A). 0 0 R t For w = 0 we hence obtain 0 Tsx ds ∈ dom(A), and since

1 Z t Tsx ds → x as t & 0 t 0 by the strong continuity of T , we arrive at x ∈ dom(A). c) It follows from the norm estimate (6.8) that Z ∞ −1 − Re λt MT k(λ + A) k ≤ MT e dt ≤ 0 Re λ

−1 for all Re λ > 0. In particular, supλ>0 kλ(λ + A) k < ∞. Hence, for ﬁxed λ0 > 0 the resolvent identity yields

−1 −1 λ −1 −1 −1 λ(λ + A) (λ0 + A) = (λ0 + A) − (λ + A) → (λ0 + A) λ − λ0

−1 in operator norm as λ → ∞. Since dom(A) = ran((λ0 + A) ) is dense, assertion c) follows. R t ws −1 d) Let V := 0 e Ts ds. In b) we have seen that V = (λ + A) ΦT (f), hence

wt (λ + A)V = ΦT (f) = I − e Tt + (w + λ)V.

By adding scalar multiples of V we obtain the identity

wt (λ + A)V = I − e Tt + (w + λ)V 92 6 Integral Transform Functional Calculi for all λ ∈ C, in particular for λ = −w. e) As LS(C+) is commutative, ΦT (f) commutes with the resolvent of A, hence with A. The operator A of Theorem 6.4 is called the generator of the Hille– Phillips calculus ΦT and one often writes f(A) in place of ΦT (f). With this convention we have

−tz −tA Tt = ΦT (Lδt) = ΦT (e ) = e (t ≥ 0).

A little unconveniently, it is the operator −A (and not A) which is called the generator of the semigroup T . One writes −A ∼ (Tt)t≥0 for this. We shall see in Theorem 6.6 below that the semigroup T is uniquely determined by its generator.

6.3 General C0-Semigroups and C0-Groups

Suppose now that T = (Tt)t≥0 is a C0-semigroup, but not necessarily bounded. Then, by the uniform boundedness principle, T is still operator norm bounded on compact intervals. This implies that T is exponentially bounded, i.e., there is M ≥ 1 and ω ∈ R such that

ωt kTtk ≤ Me (t ≥ 0) (6.9)

(see Exercise 6.1). One says that T is of type (M, ω) if (6.9) holds. The number

ω0(T ) := inf{ω ∈ R | there is M ≥ 1 such that (6.9) holds} is called the (exponential) growth bound of T . If ω0(T ) < 0, the semigroup is called exponentially stable. For each ω ∈ C one can consider the rescaled semigroup T ω, deﬁned by

ω −ωt T (t) := e Tt (t ≥ 0), which is again strongly continuous. Since T is exponentially bounded, if Re ω is large enough, the rescaled semigroup T ω is bounded and hence has a generator −Aω, say. The following tells in particular that the operator A := Aω −ω is independent of ω.

Theorem 6.5. Let T = (Tt)t≥0 be a C0-semigroup on a Banach space X and let λ, ω ∈ C such that T λ and T ω are bounded semigroups with generators −Aω and −Aλ, respectively. Then

A := Aω − ω = Aλ − λ. (6.10) 6.3 General C0-Semigroups and C0-Groups 93

Furthermore, the following assertions hold: a) A is densely deﬁned.

b) TtA ⊆ ATt for all t ≥ 0. R t R t c) 0 Ts dsA ⊆ A 0 Ts ds = I − Tt for all t ≥ 0. d) For x, y ∈ X the following assertions are equivalent: (i) −Ax = y; 1 (ii) y = lim (Ttx − x); t&0 t d (iii) T (·)x ∈ C1( ; X) and T x = T y on . R+ dt t t R+ Proof. Without loss of generality we may suppose that Re λ ≥ Re ω. Then Z ∞ Z ∞ −1 −t −λt −t −(λ−ω)t −ωt (1 + Aλ) = e e Tt dt = e e e Tt dt 0 0 −1 = (1 + λ − ω + Aω) .

This establishes the ﬁrst claim. Assertion a) is clear and b) holds true since, by construction, each Tt commutes with the resolvent of A. Assertion c) follows directly from d) and e) of Theorem 6.4. For the proof of d) we note that the implication (iii) ⇒ (ii) is trivial. (i) ⇒ (iii): If −Ax = y then (A + λ)x = −y + λx =: z and hence Z ∞ Z ∞ Z ∞ −λs −λs λt −λs Ttx = Tt e Tsz ds = e Tt+sz ds = e e Tsz ds. 0 0 t By the fundamental theorem of calculus (Theorem A.3) and the product rule, the orbit T (·)x is diﬀerentiable with derivative

d T x = λT x − eλte−λtT z = T (λx − z) = T y dt t t t t t as claimed. (ii) ⇒ (i): This is left as Exercise 6.2.

If A is as in (6.10), the operator −A is called the generator of the semigroup T . By construction, Z ∞ −1 −λt (λ + A) = e Tt dt 0 for all suﬃciently large Re λ.

Theorem 6.6. A C0-semigroup is uniquely determined by its generator. 94 6 Integral Transform Functional Calculi

Proof. Suppose that B is the generator of the C0-semigroups S and T on the Banach space X. Fix x ∈ dom(B) and t > 0, and consider the mapping

f : [0, t] → X, f(s) := T (t − s)S(s)x.

Then by Lemma A.5, f 0(s) = −BT (t − s)S(s)x + T (t − s)BS(s)x = −T (t − s)BS(s)x + T (t − s)BS(s)x = 0 for all s ∈ [0, t]. Hence, f is constant and therefore T (t) = f(0) = f(t) = S(t).

Let −A be the generator of a C0-semigroup T = (Tt)t≥0 of type (M, ω). Then the operator −(A + ω) generates the bounded semigroup T ω and hence A + ω generates the associated Hille–Phillips calculus ΦT ω . It is therefore reasonable to deﬁne a functional calculus ΦT for A by ΦT (f) := ΦT ω f(z − ω) (6.11) for f belonging to the Laplace–Stieltjes algebra

LS(C+−ω) := {f | f(z−ω) ∈ LS(C+)}.

This calculus is called the Hille–Phillips calculus for T (on C+−ω). Note the boundedness property

kΦ (f)k ≤ Mkfk (f ∈ LS( −ω)) T LS(C+−ω) C+ where kfk := kf(z−ω)k . LS(C+−ω) LS(C+) Remark 6.7. Since the type of a semigroup is not unique, the above terminology could be ambiguous. To wit, if T is of type (M, ω), it is also of type ω (M, α) for each α > ω. Accordingly, one has the Hille–Phillips calculi ΦT on α C+−ω and ΦT on C+−α for A. However, these calculi are compatible in the sense that (by restriction) LS(C+−α) ⊆ LS(C+−ω) and

α ω ΦT (f) = ΦT (f|C+−ω)(f ∈ LS(C+−α)) (Exercise 6.4). We see that a smaller growth bound results in a larger calculus.

C0-Groups

A C0-group on a Banach space X is just a strongly continuous representation U = (Us)s∈R of R on X. From such a C0-group, two C0-semigroups can be derived, the forward semigroup (Ut)t≥0 and the backward semi- −1 group (U−t)t≥0. Obviously, each determines the other, as U−t = Ut for all t ≥ 0. The generator of the group U is deﬁned as the generator of the corresponding forward semigroup.

Theorem 6.8. Let B be the generator of a C0-semigroup T = (Tt)t≥0. Then the following assertions are equivalent. 6.3 General C0-Semigroups and C0-Groups 95

(i) T extends to a strongly continuous group.

(ii) −B is the generator of a C0-semigroup.

(iii) Tt is invertible for some t > 0. −1 In this case −B generates the corresponding backward semigroup (Tt )t≥0.

Proof. (ii) ⇒ (i): Let −B ∼ (St)t≥0. Then, for all x ∈ dom(B) d S(t)T (t)x = (−B)S(t)T (t)x + S(t)BT (t)x dt = −S(t)BT (t)x + S(t)BT (t)x = 0 by Lemma A.5. Since dom(B) is dense, it follows that S(t)T (t) = S(0)T (0) = I for all t ≥ 0. Interchanging the roles of S and T yields T (t)S(t) = I as well, hence each T (t) is invertible with T (t)−1 = S(t). It is now routine to check that the extension of T to R given by T (s) := S(−s) for s ≤ 0 is a C0-group. (i) ⇒ (iii) is clear.

(iii) ⇒ (ii): Fix t0 > 0 such that T (t0) is invertible. For general t > 0 we can find n ∈ N and r > 0 such that t + r = nt0. Hence, T (t)T (r) = T (r)T (t) = n −1 T (t0) is invertible, and so must be T (t). Define S(t) := T (t) for t ≥ 0. Then S is a semigroup, and strongly continuous because for fixed τ > 0

S(t) = S(τ)T (τ)T (t)−1 = S(τ)T (τ − t)T (t)T (t)−1 = S(τ)T (τ − t) for 0 ≤ t ≤ τ. Let C be the generator of S and x ∈ dom(B). Then for 0 < t < τ,

S(t)x − x T (τ − t)x − T (τ)x = S(τ) → −S(τ)T (τ)Bx = −Bx t t as t & 0. Hence, −B ⊆ C. By symmetry, it follows that C = −B.

A C0-group U = (Us)s∈R is said to be of type (M, ω) for some M ≥ 1 and ω ≥ 0 if ω|s| kUsk ≤ Me (s ∈ R).

By the results from above, each C0-group is of some type (M, ω). The quantity

θ(U) := inf{ω ≥ 0 | ∃ M ≥ 1 : U is of type (M, ω)} is called the group type of U.

Let B be the generator of a bounded group U and let ΨU be the associated Fourier–Stieltjes calculus. Deﬁne A := iB, so that B = −iA. Then A is the generator of ΨU as

−isz ΨU (e ) = Uδs = Us (s ∈ R) and, for Im λ > 0, 96 6 Integral Transform Functional Calculi 1 Z ∞ (λ − z)−1 = eiλse−isz ds i 0 as functions on R. If U is not bounded, one can still deﬁne a functional calculus based on the Fourier transform. However, one has to restrict to a certain subalgebra of measures/functions. See Exercise 6.7.

6.4 Supplement: Continuity Properties and Uniqueness

In this supplementary section we shall present a uniqueness statement for the Fourier-Stieltjes calculus of a bounded strongly continuous representation of d d S in the cases S = R and S = R+. These statements involve a certain continuity property of the calculus, interesting in its own right. We shall make use of the results b)-d) of Exercise 6.8. We start with observing that in certain cases the inequality (6.2) is an identity. Lemma 6.9. For each µ ∈ M(Rd) one has

kµkM = kτµk 1 d = kτµk d . L(L (R )) L(C0(R ))

1 d d In other words: For X = L (R ) and X = C0(R ) the regular representation d τ : M(R ) → L(X), µ 7→ τµ, is isometric.

Proof. The inequality kτµk ≤ kµk (both cases) is (6.2). For the converse, let d f ∈ C0(R ). Then Z (τµf)(0) = hτµf, δ0i = hτtf, δ0i µ(dt) = hSf, µi . d R Hence, by replacing f by Sf,

|hf, µi| ≤ kτµSfk∞ ≤ kτµkkSfk∞ = kτµkkfk∞,

d which implies that kµk ≤ kτµk. This settles the case X = C0(R ). For the case X = L1(Rd) we employ duality and compute

kτµkL(L1) = sup sup |hµ ∗ f, gi| = sup sup |hf, Sµ ∗ gi| f g g f = kτ k = kSµk = kµk , Sµ L(C0) M M

d where the suprema are taken over all g in the unit ball of C0(R ) and all f in the unit ball of L1(Rd). d Deﬁnition 6.10. A sequence (µn)n in M(R ) converges strongly to µ ∈ M(Rd) if 6.4 Supplement: Continuity Properties and Uniqueness 97

1 1 d µn ∗ f → µ ∗ f in L -norm for all f ∈ L (R ). 1 d In other words: µn → µ strongly if τµn → τµ strongly in L(L (R )). By Lemma 6.9 and the uniform boundedness principle, a strongly convergent sequence is uniformly norm bounded. From this it follows easily that the convolution product is simultaneously continuous with respect to strong convergence of sequences. Strong convergence implies weak∗ convergence (un- d ∼ d 0 der the identiﬁcation M(R ) = C0(R ) ), see Exercise 6.10. Note also that if d tn → t in R then δtn → δt strongly. 1 d A sequence (ϕn)n in L (R ) is called an approximation of the identity 4 if ϕnλ → δ0 strongly, and a Dirac sequence if Z fϕn dλ → f(0) (n → ∞) d R d for each f ∈ Cb(R ; X) and any Banach space X. We say that (ϕn)n is a Dirac sequence on a closed subset E ⊆ Rd if it is a Dirac sequence and supp(ϕn) ⊆ E for all n ∈ N. Each Dirac sequence is an approximation of the identity. Observe that Dirac sequences are easy to construct (Exercise 6.9) and that we have already used a special Dirac sequence on R+ in the proof of Theorem 6.4. The following result underlines the importance of our notion of “strong convergence”.

d d Theorem 6.11. Let S ∈ {R , R+} and let T : S → L(X) be a bounded, strongly continuous representation on a Banach space X. Then the associated calculus M(S) → L(X) has the following continuity property: If (µn)n is a sequence in M(S) and µn → µ strongly, then µ ∈ M(S) and Tµn → Tµ strongly in L(X).

Proof. As already mentioned, strong convergence implies weak∗-convergence. Hence supp(µ) ⊆ S, i.e., µ ∈ M(S).

Since the µn are uniformly norm bounded, so are the Tµn . Hence, it suﬃces to check strong convergence in L(X) only on a dense set of vectors. Let (ϕm)m be a Dirac sequence on E = S. For each x ∈ X and m ∈ N one has

Tµn Tϕm x = Tµn∗ϕm x → Tµ∗ϕm x = TµTϕm x

as n → ∞. But Tϕm x → x as m → ∞, and we are done. Let us call a sequence f = µ ∈ FS ( d) strongly convergent to f = µ ∈ n cn S R b d FSS(R ), if µn → µ strongly. And let us call a functional calculus

d Ψ : FSS(R ) → L(X)

4 That is our terminology. Diﬀerent deﬁnitions exist in the literature. 98 6 Integral Transform Functional Calculi

d strongly continuous if whenever fn → f strongly in FSS(R ) then Ψ(fn) → Ψ(f) strongly in L(X). With this terminology, Theorem 6.11 simply tells that in the case of a bounded and strongly continuous representation of S = Rd d or S = R+, the associated Fourier–Stieltjes calculus is strongly continuous. The following is the uniqueness result we annouced.

d d d Theorem 6.12. Let S = R or S = R+ and let Ψ : FSS(R ) → L(X) be a strongly continuous calculus. Then T : S → L(X), deﬁned by Tt := Ψ(et) for t ∈ S, is a bounded and strongly continuous representation, and Ψ coincides with the associated Fourier–Stieltjes calculus.

Proof. The continuity assumption on Ψ implies that Ψ is norm bounded (via the closed graph theorem) and that the representation T is strongly continuous. By the norm boundedness, T is also bounded. d Let E := {f ∈ FSS(R ) | Ψ(f) = ΨT (f)}. Then E is a strongly closed sub- d algebra of FSS(R ) containing all the functions et, t ∈ S. Hence, the claim follows from the next lemma.

d d Lemma 6.13. Let S = R+ or S = R and let M ⊆ M(S) be a convolution subalgebra closed under strong convergence of sequences. Then M = M(S) in each of the following cases:

1) M contains δt for each t ∈ S. 2) M contains some dense subset of L1(S). Proof. As M is strongly closed, it is norm closed. Suppose that 2) holds. Then L1(S) ⊆ M. As L1(S) contains an approximation of the identity and M is strongly closed, M(S) ⊆ M. Suppose that 1) holds and consider ﬁrst the case d = 1. Fix ϕ ∈ Cc(R) such that supp(ϕ) ⊆ [0, 1]. Then the sequence of measures

n 1 X k µn := ϕ δ k n n n k=1

∗ converges to ϕλ in the weak -sense (as functionals on C0(R)). As the sup- ports of the µn are all contained in a fixed compact set, Exercise 6.11 yields that µn → ϕλ strongly. Obviously, with slightly more notational effort this 1 argument can be carried out for each ϕ ∈ Cc(S). As Cc(S) is dense in L (S), we obtain condition 2) and are done. For arbitrary dimension d ∈ N one can employ a similar argument (with but even more notational effort). 6.4 Supplement: Continuity Properties and Uniqueness 99 Exercises

6.1 (Growth Bound). Let T = (Tt)t≥0 be a C0-semigroup on a Banach space X. a) Show that

log kTtk log kTtk ω0(T ) := inf = lim ∈ R ∪ {−∞}. (6.12) t>0 t t→∞ t

b) Show that to each ω > ω0(T ) there is M ≥ 1 such that

ωt kTtk ≤ Me (t ≥ 0).

Then show that ω0(T ) is actually the inﬁmum of all ω ∈ R with this property.

The number ω0(T ) is called the growth bound of the semigroup T .

6.2. Let −A be the generator of a C0-semigroup T = (Tt)t≥0 on a Banach 1 space X and suppose that x, y ∈ X are such that y = limt&0 t (Ttx − x). a) Show that T (·)x is differentiable on R+ and its derivative is T (·)y. [Hint: Show first that the orbit is right differentiable, cf. [3, Lemma II.1.1].] b) Show that −Ax = y. [Hint: For λ > 0 sufficiently large, compute d −λt dt e Ttx and integrate over R+.] 6.3 (Hille–Yosida Estimates). Let for λ ∈ C and n ∈ N the function fn : R → C be defined by tn−1 f := e−λt1 . n (n−1)! R+

Prove the following assertions. 1 a) If λ ∈ C+ then fn ∈ L (R+) and fn = f1 ∗ · · · ∗ f1 (n times).

b) Let B be the generator of a C0-semigroup T = (Tt)t≥0 of type (M, ω) on a Banach space X. Then, for Re λ > ω and n ≥ 0, Z ∞ n tn−1 −λt R(λ, B) = (n−1)! e Tt dt and 0 M kR(λ, B)nk ≤ . (Re λ − ω)n

[Hint: Reduce to the case ω = 0 by rescaling and then employ a).]

6.4. Let −A be the generator of a C0-semigroup T of type (M, ω) and let α > ω. Prove the following assertions: ωt 1 R ∞ −zt a) If f : R+ → C is such that e f ∈ L (R+), then g := 0 e f(t) dt ∈ LS(C+−ω) and 100 6 Integral Transform Functional Calculi Z ∞ g(A) = f(t)Tt dt. 0

b) Via restriction, the algebra LS(C+−α) can be considered to be a subalgebra of LS(C+−ω), and the Hille-Phillips calculus for A on the larger algebra restricts to the Hille–Phillips calculus for A on the smaller one.

6.5. a) Let A be a closed operator on a Banach space X such that ρ(A) 6= ∅. Show that each operator An, n ∈ N, is closed. b) Let −A be the generator of a C0-semigroup T = (Tt)t≥0, let n ∈ N, and let D ⊆ dom(An) be a subspace which dense in X and invariant under the semigroup T . Show that D is a core for An.

c) Let j ∈ {1, . . . , d} and consider the shift semigroup (τtej )t≥0 in the p d d direction ej on X = L (R ), 1 ≤ p < ∞, or X = C0(R ). Show that its generator B is the closure (as an operator on X) of the operator

∂ B0 = − ∂xj

∞ d defined on the space D = Cc (R ) of smooth functions with compact support. [Hint: For b) observe first that if y ∈ D and λ is sufficiently large, then n −n xλ,y := λ (λ + A) y is contained in the k · kAn closure of D, and second that one can find elements of the form xλ,y arbitrarily k · kAn -close to any given x ∈ dom(An), see [3, Proposition II.1.7]. For c) use b).]

6.6 (Right Shift Semigroup on a Finite Interval). Let τ = (τt)t≥0 be the right shift semigroup on X = Lp(0, 1), where 1 ≤ p < ∞. This can be described as follows: τtx := τtxe |(0,1) (x ∈ X, t ≥ 0), where xe is the extension by 0 to R of x and τ on the right hand side is the just the regular representation of R on X. It is easy to see that this yields a C0-semigroup on X. Let −A be its generator.

a) Show that τt = 0 for t ≥ 1. Conclude that σ(A) = ∅ and A has a Hille–Phillips calculus for LS(C++ω) for each ω ∈ R (however large). b) Show that A−1 = V , the Volterra operator on X (see Section 1.4), and that σ(V ) = {0}.

c) Let r > 0 and (αn)n be a sequence of complex numbers such that −n P∞ −n M := supn≥0 |αn| r < ∞. Show that the function f := n=0 αnz is contained in LS(C++ω) for each ω > r. Show further that

∞ X n f(A) = αnV n=0 6.4 Supplement: Continuity Properties and Uniqueness 101

where f(A) is defined via the Hille–Phillips calculus for A. d) Find a function f such that f(A) is defined via the Hille–Phillips calculus for A, but g := f(z−1) is not holomorphic at 0, so g(V ) is not defined via the Dunford–Riesz calculus for V . [Hint: for the first part of b) observe that point evaluations are not continuous on Lp(0, 1); for the second cf. Exercise 2.4; for c) cf. Exercise 6.3 and show that the series defining f converges in FS(C++ω).] d Remark: A closer look would reveal that A = dt is the weak derivative operator with domain

1,p 1,p dom(A) = W0 (0, 1) := {u ∈ W (0, 1) | u(0) = 0}. For the case p = 2 this can be found in [4, Section 10.2].

6.7 (Fourier–Stieltjes Calculus for Unbounded C0-Groups). Let −iA be the generator of a C0-group U = (Us)s∈R of type (M, ω). Deﬁne Z ω|s| Mω(R) := {µ ∈ M(R) | kµkMω := e |µ| (ds) < ∞}. R Show the following assertions:

a)M ω(R) is unital subalgebra of M(R) and a Banach algebra with respect

to the norm k · kMω . b) The mapping Z Mω(R) → L(X), µ 7→ Uµ := Us µ(ds) R

is a unital algebra homomorphism with kUµk ≤ MkµkMω . c) The Fourier transform µb of µ ∈ Mω(R) has a unique extension to a function continuous on the strip Stω = {z ∈ C : |Im z| ≤ ω} and holomorphic in its interior. d) The spectrum σ(A) of A is contained in the said strip and one has M kR(λ, A)k ≤ (|Im λ| > ω). |Im λ| − ω

[Hint for b): show and use that Mc(R) is dense in Mω(R) with respect to the norm k · kMω .] The mapping ΨU deﬁned on {µb : µ ∈ Mω(R)} by Z ΨU (µb) = Us µ(ds)(µ ∈ Mω(R)) R is called the Fourier–Stieltjes calculus associated with U. 102 6 Integral Transform Functional Calculi

6.8 (Multiparameter C0-Semigroups). A (strongly continuous) represen- d tation T : R+ → L(X) is called a d-parameter semigroup (C0-semigroup). Such d-parameter semigroups T are in one-to-one correspondence with d- tuples (T 1,...,T d) of pairwise commuting 1-parameter semigroups via

1 d T (t1e1 + ··· + tded) = T (t1) ··· T (td)(t1, . . . , td ∈ R+), cf. Remark 5.1. a) Show that a d-parameter semigroup T is strongly continuous if and only if each T j, j = 1, . . . , d, is strongly continuous. j Let T be a d-parameter C0-semigroup. Then each of the semigroups T has a generator −Aj, say.

b) Show that T is uniquely determined by the tuple (A1,...,Ad).

c) Show that dom(A1) ∩ · · · ∩ dom(Ad) is dense in X.

d) Let −A be the generator of the C0-semigroup S, deﬁned by

1 d S(t) := T (te1 + ··· + ted) = T (t) ··· T (t)(t ≥ 0).

Show that A = A1 + ··· + Ad. [Hint: use d) and Exercise 6.5.b).] If, in addition, T is bounded, we can consider the associated Fourier–Stieltjes calculus. However, as in the case d = 1, one rather often works with the Hille–Phillips calculus which is based on the “d-dimensional” Laplace transform. d e) Try to give a deﬁnition of the Laplace transform of measures in M(R+) and built on it a construction of the “Hille–Phillips calculus” for bounded d-parameter C0-semigroups. What is the connection between this calculus and the invidual calculi for the semigroups T j?

6.9 (Dirac Sequences). Let 0 ∈ E ⊆ Rd be closed. A Dirac sequence on 1 d E is a sequence (ϕn)n in L (R ) such that supp(ϕn) ⊆ E for all n ∈ N and Z fϕn dλ → f(0) (n → ∞) d R whenever f ∈ Cb(E; X) and X is a Banach space. 1 d a) Let (ϕn)n be a sequence in L (R ) with the following properties: 1) supp(ϕn) ⊆ E for all n ∈ N.

2) supn kϕnk1 < ∞. R 3) limn→∞ d ϕn = 1. R R 4) limn→∞ |x|≥ε |ϕn(x)| dx → 0 for all ε > 0.

Show that (ϕn)n is a Dirac sequence on E. 6.4 Supplement: Continuity Properties and Uniqueness 103

1 d R d b) Let ϕ ∈ L ( ) such that d ϕ = 1. Deﬁne ϕn(x) := n ϕ(nx) for R R d x ∈ R . Show that (ϕn)n is a Dirac sequence on E = R+ supp(ϕ).

c) Let T : E → L(X) be bounded and strongly continuous, and let (ϕn)n

be a Dirac sequence on E. Show that Tϕn → I strongly.

Supplementary Exercises

d ∗ 6.10. Show that if µn → 0 strongly in M(R ) then µn → 0 weakly (under d ∼ d 0 the identiﬁcation M(R ) = C0(R ) ) and µcn → 0 uniformly on compacts. d d ∗ 6.11. Let (µn)n be a sequence in M(R ) such that µn → µ ∈ M(R ) weakly d as functionals on C0(R ). Show that µn → µ strongly if, in addition, there is d compact set K ⊆ R such that supp(µn) ⊆ K for all n ∈ N.

6.12. Let H be a Hilbert space and let U : Rd → L(Rd) be a strongly continuous representation of Rd by unitary operators on H. Show that the associated d Fourier–Stieltjes calculus ΨU : FS(R ) → L(H) is a ∗-homomorphism. [Hint: recall Exercise 5.10.]

References

[1] E. Hille and R. S. Phillips. Functional Analysis and Semi-Groups. Vol. 31. Colloquium Publications. Providence, RI: American Mathemat- ical Society, 1974, pp. xii+808. [2] K.-J. Engel and R. Nagel. A Short Course on Operator Semigroups. Universitext. New York: Springer, 2006, pp. x+247. [3] K.-J. Engel and R. Nagel. One-Parameter Semigroups for Linear Evo- lution Equations. Vol. 194. Graduate Texts in Mathematics. Berlin: Springer-Verlag, 2000, pp. xxi+586. [4] M. Haase. Functional analysis. Vol. 156. Graduate Studies in Mathe- matics. An elementary introduction. American Mathematical Society, Providence, RI, 2014, pp. xviii+372. 104 6 Integral Transform Functional Calculi