Chapter 6 Integral Transform Functional Calculi

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1 Chapter 6 Integral Transform Functional Calculi In this chapter we continue our investigations from the previous one and encounter functional calculi associated with various semigroup representations. 6.1 The Fourier Stieltjes Calculus Recall from the previous chapter that the Fourier transform F : M(R d ) UC b (R d ) is a contractive and injective unital algebra homomorphism. Hence, it is an isomorphism onto its image FS(R d ) := F(M(R d )) = { µ µ M(R d )}. This algebra, which is called the Fourier Stieltjes algebra 1 of R d, is endowed with the norm µ FS := µ M (µ M(R d )), which turns it into a Banach algebra and the Fourier transform F : M(R d ) FS(R d ) into an isometric isomorphism. (The Fourier algebra of R d is the closed ideal(!) A(R d ) := F(L 1 (R d )) of Fourier transforms of L 1 -functions.) Let S R d be a closed subsemigroup. Then M(S) is a Banach subalgebra of M(R d ) and FS S (R d ) := 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(R d ) for it. 85

2 86 6 Integral Transform Functional Calculi called its associated Fourier Stieltjes algebra, is a Banach subalgebra of FS(R d ). Let T : S L(X) be a strongly continuous and bounded representation with associated algebra representation M(S) L(X), µ T µ = T t µ(dt). (6.1) Since the Fourier transform F : M(S) FS S (R d ) is an isomorphism, we can compose its inverse with the representation (6.1). In this way a functional calculus Ψ T : FS S (R d ) L(X), Ψ T ( µ) := T µ, is obtained, which we call the Fourier Stieltjes calculus for T. Note that by the definition of the norm on FS(R d ) and by (5.6) we have Ψ T (f) M T f FS (f FS S (R d )), (6.2) where, as always, M T = sup t S T t. Example 6.1. Let Ω = (Ω, Σ, ν) be a measure space and a : Ω R d a measurable function. Fix p [1, ) and abbreviate X := L p (Ω). For t R d let T t L(X) be defined by S T t x := e it a( ) x (x X). Then (T t ) t R d is a bounded and strongly continuous group and, by Exercise 5.8, T µ x = ( µ a) x (x X) for all µ M(R d ). 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(R d ). Example 6.2. Let T = τ be the regular (right shift) representation of R d on X = L 1 (R d ). Then, for f = µ FS(R d ) one has Ψ τ (f)x = µ x = F 1 ( µ x) = F 1 (f x) (x L 1 (R d )). That is, the operator Ψ τ (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 +.

3 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 T := sup n Z T n <. Such operators correspond in a one-to-one 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}. By Exercise 5.9, M(Z) = l 1 (Z) and FS Z (R) consists of all functions f = n Z α n e int (α = (α n ) n l 1 ). (6.3) These functions form a subalgebra of C 2π (R), the algebra of 2π-periodic functions on R. By the Fourier Stieltjes calculus, the function f as in (6.3) is mapped to Ψ T (f) = n Z α n T n. Hence, this calculus is basically the same as a Laurent series calculus Φ T : n Z α n z n n Z α n T n (6.4) 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 A(T) := { α n z n α l 1 (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) C(T) C 2π (R), f f(e it ). This restricts to an isomorphism η : A(T) FS Z (R) by virtue of which the Fourier Stieltjes calculus is isomorphic to the Wiener calculus.

4 88 6 Integral Transform Functional Calculi Power-bounded operators Power-bounded operators T L(X) correspond in a one-to-one fashion to Z + -representations on X. By Exercise 5.9, M(Z + ) = l 1 (Z + ) and FS Z+ (R) consists of all functions f = α n e int (α = (α n ) n l 1 ). (6.5) n= These functions form a subalgebra of C 2π (R). Under the Fourier Stieltjes calculus, the function f as in (6.5) is mapped to Ψ T (f) = α n T n. n= This calculus is isomorphic to the power-series calculus Φ T : A 1 +(D) L(X), ( Φ T α n z n) = α n T n. (6.6) n= n= The isomorphism of the two calculi is again given by the algebra homomorphism f f(e it ). (Observe that for α l 1 the function f = n= α nz n can be viewed as a function on D, or on D or on T or on (, 1) and in either interpretation α is determined by f.) 6.2 Bounded C -Semigroups and the Hille Phillips Calculus We now turn to the case S = R +. A strongly continuous representation of R + on a Banach space is often called a C -semigroup 2. 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 = (T t ) t be a bounded C -semigroup on a Banach space X. Then we have the Fourier Stieltjes calculus Ψ T : FS R+ (R) L(X), Ψ T ( µ) = T t µ(dt). R + 2 The name goes back to the important monograph [1, Chap. 1.1] of Hille and Phillips, where different continuity notions for semigroup representations are considered.

5 6.2 Bounded C -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 Lµ : C + C, (Lµ)(z) = e zt µ(dt). R + Here, C + := {z C : Re z > } is the open right half-plane. By some standard arguments, Lµ µ M and Lµ UC b (C + ) Hol(C + ), and it is easy to see that the Laplace transform L : M(R + ) UC b (C + ), µ 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 Then the mapping Lµ LS := µ M (µ M(R + )). LS(C + ) FS R+ (R), f f(is) is an isometric isomorphism of unital Banach algebras. Given a C -semigroup (T t ) t one can compose its Fourier Stieltjes calculus Ψ T with the inverse of this isomorphism to obtain the calculus Φ T : LS(C + ) L(X), Φ T (Lµ) := T t µ(dt). R + This calculus is called the Hille Phillips calculus 3 for T. It satisfies the norm estimate Φ T (f) M T µ M (f = Lµ, µ M(R + )). (6.8) 3 One would expect the name Laplace Stieltjes calculus but we prefer sticking to the common nomenclature.

6 9 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 (, ) 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 C -Semigroup Suppose as before that T = (T t ) t is a bounded C -semigroup on a Banach space X. For Re λ, Re z > one has 1 λ + z = e λt e zt dt. In other words, the function 1 λ+z (defined on C +) is the Laplace transform of the L 1 -function e λt 1 R+. As such, (λ + z) 1 LS(C + ) for each λ C +. Theorem 6.4. Let T = (T t ) t be a bounded C -semigroup on a Banach space X with associated Hille Phillips calculus Φ T. 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: a) [ Re z < ] ρ(a) and R( λ, A) = Φ T ((λ + z) 1 ) for all Re λ >. b) dom(a) is dense in X. c) λ(λ + A) 1 I strongly as < λ. d) For all w C, t > and x X: t e ws T s x ds dom(a) and (A w) e) Φ T (f)a AΦ T (f) for all f LS(C + ). Proof. The operator family ( 1 ) ( R(w) := Φ T = Φ T w z t 1 ( w) + z e sw T s x ds = x e tw T t x. ), Re w <, 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 < (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 <. As ker(r(w))

7 6.2 Bounded C -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 >, x X, and w, λ C. Then a little computation yields (z + λ) t e ws e sz ds = (z w) = 1 e wt e tz + (w + λ) t t That means that f = Lµ LS(C + ) with e ws e sz ds + (w + λ) t e ws e sz ds e ws e sz ds =: f(z) (Re z > ). µ = δ e wt δ t + (w + λ)1 [,t] e ws ds. For each Re λ > one can divide by z + λ and then apply the Hille Phillips calculus to obtain t e ws T s x ds = Φ T ( t ) e sw e sz ds x = R( λ, A)Φ T (f)x dom(a). For w = we hence obtain t T sx ds dom(a), and since 1 t t T s x ds x as t by the strong continuity of T, we arrive at x dom(a). c) It follows from the norm estimate (6.8) that (λ + A) 1 M T e Re λt dt M T Re λ for all Re λ >. In particular, sup λ> λ(λ + A) 1 <. Hence, for fixed λ > the resolvent identity yields λ(λ + A) 1 (λ + A) 1 = λ λ λ ( (λ + A) 1 (λ + A) 1) (λ + A) 1 in operator norm as λ. Since dom(a) = ran((λ + A) 1 ) is dense, assertion c) follows. d) Let V := t ews T s ds. In b) we have seen that V = (λ + A) 1 Φ T (f), hence (λ + A)V = Φ T (f) = I e wt T t + (w + λ)v. By adding scalar multiples of V we obtain the identity (λ + A)V = I e wt T t + (w + λ)v

8 92 6 Integral Transform Functional Calculi for all λ C, in particular for λ = w. e) As LS(C + ) is commutative, Φ T (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 Φ T and one often writes f(a) in place of Φ T (f). With this convention we have T t = Φ T (Lδ t ) = Φ T (e tz ) = e ta (t ). A little unconveniently, it is the operator A (and not A) which is called the generator of the semigroup T. One writes A (T t ) t for this. We shall see in Theorem 6.6 below that the semigroup T is uniquely determined by its generator. 6.3 General C -Semigroups and C -Groups Suppose now that T = (T t ) t is a C -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 T t Me ωt (t ) (6.9) (see Exercise 6.1). One says that T is of type (M, ω) if (6.9) holds. The number ω (T ) := inf{ω R there is M 1 such that (6.9) holds} is called the (exponential) growth bound of T. If ω (T ) <, the semigroup is called exponentially stable. For each ω C one can consider the rescaled semigroup T ω, defined by T ω (t) := e ωt T t (t ), 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 A ω, say. The following tells in particular that the operator A := A ω ω is independent of ω. Theorem 6.5. Let T = (T t ) t be a C -semigroup on a Banach space X and let λ, ω C such that T λ and T ω are bounded semigroups with generators A ω and A λ, respectively. Then A := A ω ω = A λ λ. (6.1)

9 6.3 General C -Semigroups and C -Groups 93 Furthermore, the following assertions hold: a) A is densely defined. b) T t A AT t for all t. c) t T s dsa A t T s ds = I T t for all t. d) For x, y X the following assertions are equivalent: (i) Ax = y; (ii) 1 y = lim t t (T tx x); (iii) T ( )x C 1 (R + ; X) and d dt T tx = T t y on R +. Proof. Without loss of generality we may suppose that Re λ Re ω. Then (1 + A λ ) 1 = e t e λt T t dt = = (1 + λ ω + A ω ) 1. e t e (λ ω)t e ωt T t dt This establishes the first claim. Assertion a) is clear and b) holds true since, by construction, each T t 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 T t x = T t e λs T s z ds = e λs T t+s z ds = e λt e λs T s z ds. By the fundamental theorem of calculus (Theorem A.3) and the product rule, the orbit T ( )x is differentiable with derivative as claimed. d dt T tx = λt t x e λt e λt T t z = T t (λx z) = T t y (ii) (i): This is left as Exercise 6.2. If A is as in (6.1), the operator A is called the generator of the semigroup T. By construction, for all sufficiently large Re λ. (λ + A) 1 = e λt T t dt Theorem 6.6. A C -semigroup is uniquely determined by its generator. t

10 94 6 Integral Transform Functional Calculi Proof. Suppose that B is the generator of the C -semigroups S and T on the Banach space X. Fix x dom(b) and t >, and consider the mapping f : [, t] X, f(s) := T (t s)s(s)x. Then by Lemma A.5, f (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 = for all s [, t]. Hence, f is constant and therefore T (t) = f() = f(t) = S(t). Let A be the generator of a C -semigroup T = (T t ) t 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 Φ T ω. It is therefore reasonable to define a functional calculus Φ T for A by ( ) Φ T (f) := Φ T 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 Φ T (f) M f LS(C+ ω) (f LS(C + ω)) where f LS(C+ ω) := f(z 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 (M, α) for each α > ω. Accordingly, one has the Hille Phillips calculi Φ ω T on C + ω and Φ α T on C + α for A. However, these calculi are compatible in the sense that (by restriction) LS(C + α) LS(C + ω) and Φ α T (f) = Φ ω T (f C+ ω) (f LS(C + α)) (Exercise 6.4). We see that a smaller growth bound results in a larger calculus. C -Groups A C -group on a Banach space X is just a strongly continuous representation U = (U s ) s R of R on X. From such a C -group, two C -semigroups can be derived, the forward semigroup (U t ) t and the backward semigroup (U t ) t. Obviously, each determines the other, as U t = Ut 1 for all t. 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 C -semigroup T = (T t ) t. Then the following assertions are equivalent.

11 6.3 General C -Semigroups and C -Groups 95 (i) (ii) T extends to a strongly continuous group. B is the generator of a C -semigroup. (iii) T t is invertible for some t >. In this case B generates the corresponding backward semigroup (T 1 t ) t. Proof. (ii) (i): Let B (S t ) t. 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 = by Lemma A.5. Since dom(b) is dense, it follows that S(t)T (t) = S()T () = I for all t. 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 is a C -group. (i) (iii) is clear. (iii) (ii): Fix t > such that T (t ) is invertible. For general t > we can find n N and r > such that t + r = nt. Hence, T (t)t (r) = T (r)t (t) = T (t ) n is invertible, and so must be T (t). Define S(t) := T (t) 1 for t. Then S is a semigroup, and strongly continuous because for fixed τ > 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 t τ. Let C be the generator of S and x dom(b). Then for < t < τ, S(t)x x t T (τ t)x T (τ)x = S(τ) S(τ)T (τ)bx = Bx t as t. Hence, B C. By symmetry, it follows that C = B. A C -group U = (U s ) s R is said to be of type (M, ω) for some M 1 and ω if U s Me ω s (s R). By the results from above, each C -group is of some type (M, ω). The quantity θ(u) := inf{ω 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 Ψ U be the associated Fourier Stieltjes calculus. Define A := ib, so that B = ia. Then A is the generator of Ψ U as and, for Im λ >, Ψ U (e isz ) = U δs = U s (s R)

12 96 6 Integral Transform Functional Calculi (λ z) 1 = 1 e iλs e isz ds i 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 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 S in the cases S = R d and S = R d +. 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(R d ) one has µ M = τ µ L(L1 (R d )) = τ µ L(C (R d )). In other words: For X = L 1 (R d ) and X = C (R d ) the regular representation τ : M(R d ) L(X), µ τ µ, is isometric. Proof. The inequality τ µ µ (both cases) is (6.2). For the converse, let f C (R d ). Then (τ µ f)() = τ µ f, δ = τ t f, δ µ(dt) = Sf, µ. R d Hence, by replacing f by Sf, f, µ τ µ Sf τ µ Sf = τ µ f, which implies that µ τ µ. This settles the case X = C (R d ). For the case X = L 1 (R d ) we employ duality and compute τ µ L(L 1 ) = sup f sup g µ f, g = sup g = τ Sµ L(C ) = Sµ M = µ M, sup f, Sµ g f where the suprema are taken over all g in the unit ball of C (R d ) and all f in the unit ball of L 1 (R d ). Definition 6.1. A sequence (µ n ) n in M(R d ) converges strongly to µ M(R d ) if

13 6.4 Supplement: Continuity Properties and Uniqueness 97 µ n f µ f in L 1 -norm for all f L 1 (R d ). In other words: µ n µ strongly if τ µn τ µ strongly in L(L 1 (R d )). 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 (under the identification M(R d ) = C (R d ) ), see Exercise 6.1. Note also that if t n t in R d then δ tn δ t strongly. A sequence (ϕ n ) n in L 1 (R d ) is called an approximation of the identity if ϕ n λ δ strongly, and a Dirac sequence 4 if R d fϕ n dλ f() (n ) for each f C b (R d ; X) and any Banach space X. We say that (ϕ n ) n is a Dirac sequence on a closed subset E R d if it is a Dirac sequence and supp(ϕ n ) 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. Theorem Let S {R d, R d +} 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 (µ n ) n is a sequence in M(S) and µ n µ strongly, then µ M(S) and T µn T µ strongly in L(X). Proof. As already mentioned, strong convergence implies weak -convergence. Hence supp(µ) S, i.e., µ M(S). Since the µ n are uniformly norm bounded, so are the T µn. Hence, it suffices to check strong convergence in L(X) only on a dense set of vectors. Let (ϕ m ) m be a Dirac sequence on E = S. For each x X and m N one has T µn T ϕm x = T µn ϕ m x T µ ϕm x = T µ T ϕm x as n. But T ϕm x x as m, and we are done. Let us call a sequence f n = µ n FS S (R d ) strongly convergent to f = µ FS S (R d ), if µ n µ strongly. And let us call a functional calculus Ψ : FS S (R d ) L(X) 4 That is our terminology. Different definitions exist in the literature.

14 98 6 Integral Transform Functional Calculi strongly continuous if whenever f n f strongly in FS S (R d ) then Ψ(f n ) 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 = R d or S = R d +, the associated Fourier Stieltjes calculus is strongly continuous. The following is the uniqueness result we annouced. Theorem Let S = R d or S = R d + and let Ψ : FS S (R d ) L(X) be a strongly continuous calculus. Then T : S L(X), defined by T t := Ψ(e t ) 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. Let E := {f FS S (R d ) Ψ(f) = Ψ T (f)}. Then E is a strongly closed subalgebra of FS S (R d ) containing all the functions e t, t S. Hence, the claim follows from the next lemma. Lemma Let S = R d + or S = R d 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 δ t for each t S. 2) M contains some dense subset of L 1 (S). Proof. As M is strongly closed, it is norm closed. Suppose that 2) holds. Then L 1 (S) M. As L 1 (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 C c (R) such that supp(ϕ) [, 1]. Then the sequence of measures µ n := 1 n n k=1 ϕ ( k n) δ k n converges to ϕλ in the weak -sense (as functionals on C (R)). As the supports of the µ n are all contained in a fixed compact set, Exercise 6.11 yields that µ n ϕλ strongly. Obviously, with slightly more notational effort this argument can be carried out for each ϕ C c (S). As C c (S) is dense in L 1 (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).

15 6.4 Supplement: Continuity Properties and Uniqueness 99 Exercises 6.1 (Growth Bound). Let T = (T t ) t be a C -semigroup on a Banach space X. a) Show that log T t log T t ω (T ) := inf = lim R { }. (6.12) t> t t t b) Show that to each ω > ω (T ) there is M 1 such that T t Me ωt (t ). Then show that ω (T ) is actually the infimum of all ω R with this property. The number ω (T ) is called the growth bound of the semigroup T Let A be the generator of a C -semigroup T = (T t ) t on a Banach space X and suppose that x, y X are such that y = lim t 1 t (T tx 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 > sufficiently large, compute d dt e λt T t x and integrate over R +.] 6.3 (Hille Yosida Estimates). Let for λ C and n N the function f n : R C be defined by f n := Prove the following assertions. tn 1 (n 1)! e λt 1 R+. a) If λ C + then f n L 1 (R + ) and f n = f 1 f 1 (n times). b) Let B be the generator of a C -semigroup T = (T t ) t of type (M, ω) on a Banach space X. Then, for Re λ > ω and n, R(λ, B) n = R(λ, B) n t n 1 (n 1)! e λt T t dt M (Re λ ω) n. and [Hint: Reduce to the case ω = by rescaling and then employ a).] 6.4. Let A be the generator of a C -semigroup T of type (M, ω) and let α > ω. Prove the following assertions: a) If f : R + C is such that e ωt f L 1 (R + ), then g := e zt f(t) dt LS(C + ω) and

16 1 6 Integral Transform Functional Calculi g(a) = f(t)t t dt. 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 a) Let A be a closed operator on a Banach space X such that ρ(a). Show that each operator A n, n N, is closed. b) Let A be the generator of a C -semigroup T = (T t ) t, let n N, and let D dom(a n ) be a subspace which dense in X and invariant under the semigroup T. Show that D is a core for A n. c) Let j {1,..., d} and consider the shift semigroup (τ tej ) t in the direction e j on X = L p (R d ), 1 p <, or X = C (R d ). Show that its generator B is the closure (as an operator on X) of the operator B = x j defined on the space D = C c (R d ) of smooth functions with compact support. [Hint: For b) observe first that if y D and λ is sufficiently large, then x λ,y := λ n (λ + A) n y is contained in the A n closure of D, and second that one can find elements of the form x λ,y arbitrarily A n-close to any given x dom(a n ), see [3, Proposition II.1.7]. For c) use b).] 6.6 (Right Shift Semigroup on a Finite Interval). Let τ = (τ t ) t be the right shift semigroup on X = L p (, 1), where 1 p <. This can be described as follows: τ t x := ( τ t x ) (,1) (x X, t ), where x is the extension by 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 C -semigroup on X. Let A be its generator. a) Show that τ t = 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 ) = {}. c) Let r > and (α n ) n be a sequence of complex numbers such that M := sup n α n r n <. Show that the function f := n= α nz n is contained in LS(C + +ω) for each ω > r. Show further that f(a) = α n V n n=

17 6.4 Supplement: Continuity Properties and Uniqueness 11 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, 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 L p (, 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).] Remark: A closer look would reveal that A = d dt is the weak derivative operator with domain dom(a) = W 1,p (, 1) := {u W1,p (, 1) u() = }. For the case p = 2 this can be found in [4, Section 1.2]. 6.7 (Fourier Stieltjes Calculus for Unbounded C -Groups). Let ia be the generator of a C -group U = (U s ) s R of type (M, ω). Define M ω (R) := {µ M(R) µ Mω := e ω s µ (ds) < }. Show the following assertions: a) M ω (R) is unital subalgebra of M(R) and a Banach algebra with respect to the norm Mω. b) The mapping M ω (R) L(X), µ U µ := R R U s µ(ds) is a unital algebra homomorphism with U µ M µ Mω. c) The Fourier transform µ of µ M ω (R) has a unique extension to a function continuous on the strip St ω = {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 R(λ, A) M Im λ ω ( Im λ > ω). [Hint for b): show and use that M c (R) is dense in M ω (R) with respect to the norm Mω.] The mapping Ψ U defined on { µ : µ M ω (R)} by Ψ U ( µ) = U s µ(ds) (µ M ω (R)) R is called the Fourier Stieltjes calculus associated with U.

18 12 6 Integral Transform Functional Calculi 6.8 (Multiparameter C -Semigroups). A (strongly continuous) representation T : R d + L(X) is called a d-parameter semigroup (C -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 T (t 1 e t d e d ) = T 1 (t 1 ) T d (t d ) (t 1,..., t d 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. Let T be a d-parameter C -semigroup. Then each of the semigroups T j has a generator A j, say. b) Show that T is uniquely determined by the tuple (A 1,..., A d ). c) Show that dom(a 1 ) dom(a d ) is dense in X. d) Let A be the generator of the C -semigroup S, defined by S(t) := T (te te d ) = T 1 (t) T d (t) (t ). Show that A = A A d. [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. e) Try to give a definition of the Laplace transform of measures in M(R d +) and built on it a construction of the Hille Phillips calculus for bounded d-parameter C -semigroups. What is the connection between this calculus and the invidual calculi for the semigroups T j? 6.9 (Dirac Sequences). Let E R d be closed. A Dirac sequence on E is a sequence (ϕ n ) n in L 1 (R d ) such that supp(ϕ n ) E for all n N and R d fϕ n dλ f() (n ) whenever f C b (E; X) and X is a Banach space. a) Let (ϕ n ) n be a sequence in L 1 (R d ) with the following properties: 1) supp(ϕ n ) E for all n N. 2) sup n ϕ n 1 <. 3) lim n R d ϕ n = 1. 4) lim n x ε ϕ n(x) dx for all ε >. Show that (ϕ n ) n is a Dirac sequence on E.

19 6.4 Supplement: Continuity Properties and Uniqueness 13 b) Let ϕ L 1 (R d ) such that R d ϕ = 1. Define ϕ n (x) := n d ϕ(nx) for x R d. Show that (ϕ n ) n is a Dirac sequence on E = R + supp(ϕ). c) Let T : E L(X) be bounded and strongly continuous, and let (ϕ n ) n be a Dirac sequence on E. Show that T ϕn I strongly. Supplementary Exercises 6.1. Show that if µ n strongly in M(R d ) then µ n weakly (under the identification M(R d ) = C (R d ) ) and µ n uniformly on compacts Let (µ n ) n be a sequence in M(R d ) such that µ n µ M(R d ) weakly as functionals on C (R d ). Show that µ n µ strongly if, in addition, there is compact set K R d such that supp(µ n ) K for all n N Let H be a Hilbert space and let U : R d L(R d ) be a strongly continuous representation of R d by unitary operators on H. Show that the associated Fourier Stieltjes calculus Ψ U : FS(R d ) L(H) is a -homomorphism. [Hint: recall Exercise 5.1.] References [1] E. Hille and R. S. Phillips. Functional Analysis and Semi-Groups. Vol. 31. Colloquium Publications. Providence, RI: American Mathematical Society, 1974, pp. xii+88. [2] K.-J. Engel and R. Nagel. A Short Course on Operator Semigroups. Universitext. New York: Springer, 26, pp. x+247. [3] K.-J. Engel and R. Nagel. One-Parameter Semigroups for Linear Evolution Equations. Vol Graduate Texts in Mathematics. Berlin: Springer-Verlag, 2, pp. xxi+586. [4] M. Haase. Functional analysis. Vol Graduate Studies in Mathematics. An elementary introduction. American Mathematical Society, Providence, RI, 214, pp. xviii+372.

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