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 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 6.12 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 6.5 The Heat Semigroup on R d The Heat Semigroup on R d In this section we shall apply our knowledge of (semi)groups and the corresponding functional calculi in order to become familiar with a special example: the heat semigroup on R d. We shall first treat the case d = 1. In the following we use the symbol s for the real coordinate function and z for the coordinate function in Fourier/Laplace domain. Let g t := 1 2πt e s2 /4t (t > ). The family (g t ) t> is called the heat kernel on R. Here are its most important properties. Lemma The following assertions hold: a) g t = 1 t g 1 ( s t ) for all t >. b) g t 1 = R g t = 1 for all t >, and (g t ) t> is a (generalized) Dirac sequence as t. c) F(g t ) = e tz2 for all t >. d) g t g s = g s+t for all s, t >. e) The map (t g t ) : (, ) L 1 (R) is continuous. Proof. a) is immediate and c) is well known. b) Since g t, g t 1 = R g t. By a) and substitution, R g t = R g 1 = 1 2π R e s2 /4 ds = 1, which is also well known. d) follows from b) since the Fourier transform is injective and turns convolutions into products. e) Given < a < b < one has g t 1 2πa e s2 /4b (a t b). Since obviously t g t (s) is continuous for each fixed s R, the claim follows from Lebesgue s theorem. Let ia be the generator of a bounded C -group (U s ) s R on a Banach space X, with associated Fourier Stieltjes calculus Ψ U. Further, let G = (G t ) t be defined by G t := Ψ U (e tz2 ) = { R g t(s) U s ds for t >, I for t =.

16 1 6 Integral Transform Functional Calculi Then (G t ) t is a bounded C -semigroup on X. (The strong continuity follows from Lemma 6.14.b) and Exercise 6.9.c).) This semigroup is called the heat semigroup or the Gauss Weierstrass semigroup associated with the group (U s ) s R. Let us denote its generator by B and the associated Hille Phillips calculus by Φ G. For the following result one should recall that elements f LS(C + ) can be interpreted as functions on R +. Theorem Let (U s ) s R be a bounded C -group on a Banach space X and let (G t ) t be its associated Gauss Weierstrass semigroup. If f LS(C + ) then f(z 2 ) FS(R) and Φ G (f) = Ψ U (f(z 2 )). Proof. Given a measure µ M(R + ) write µ = αδ + ν where α C and ν M(, ). Then Lµ = α1 + Lν and it suffices to prove the claim for f = Lν. By Lemma 6.14.e) and the definition of the norm on FS(R), the mapping is bounded and continuous. Hence, (, ) FS(R), t e tz2 Lν(z 2 ) = e tz2 ν(dt) FS(R) since the integral converges in FS(R). (Note that point evaluations are continuous on FS(R).) Finally, Φ G (f) = as claimed. G t ν(dt) = = Ψ U (f(z 2 )) Ψ U (e tz2 ) ν(dt) = Ψ U ( Theorem 6.15 helps to identify the generator B of (G t ) t. Corollary In the situation from above we have B = A 2. Proof. Applying the theorem with f = 1 1+z yields (I + B) 1 ( 1 ) ( 1 ) = Ψ U = ΨU 1 + z 2 (1 + iz)(1 iz) ) e tz2 ν(dt) = (I + ia) 1 (I ia) 1 = ( (I ia)(i + ia) ) 1 = (I + A 2 ) 1 (see Theorem A.2) from which the claim follows.

17 6.5 The Heat Semigroup on R d 11 Note that B = A 2 = ( ia) 2, so the generator of G is simply the square of the generator of U. Examples We apply these results to various shift groups, in which case one simply speaks of the heat or Gauss Weierstrass semigroup on the respective space. 1) Let X = UC b (R) and U = τ the right shift group. The associated heat semigroup on X is given by G t f = g t f = 1 2πt f(s)e (x s)2 /4t ds (t > ). (6.12) Its generator is d2 dx 2 R with domain UC 2 b(r) = {f C 2 (R) f, f, f UC b (R)}. This follows from the simple-to-prove fact that the generator of τ is d dx with domain UC 1 b(r) = {f C 1 (R) f, f UC b (R)}. An analogous result holds for the heat semigroup on X = C (R). 2) Let X = L p (R) for 1 p < and U = τ the right shift group. Then its associated heat semigroup is again given by (6.12) (recall Exercise 5.7). The generator of U is the closure of d dx defined on C c (R), see Exercise 6.5.b). By Exercise 6.5.a), its square which is the generator d of G is the closure of 2 dx on C 2 c (R). (Its domain is W 2,p (R), but we do not prove this here.) 3) Fix 1 j d and let U be the shift group in the direction of e j on X = UC b (R d ) or X = C (R d ), i.e., U s = τ sej for all s R. Its generator is x j with domain { f X f x j } exists everywhere and yields a function in X. So the associated heat semigroup has generator 2 x 2 j with domain consisting of those f X such that f x j and 2 f x 2 j exist and are in X. 4) Fix 1 j d and let U be the shift group in the direction of e j on X = L p (R d ) for 1 p <. Its generator D j, say, is the closure of the operator x j defined originally on C c (R d ). (It is true that D j f = g x j f = g in the weak sense

18 12 6 Integral Transform Functional Calculi for f, g L p (R d ), but we do not prove this here.) It follows that the generator of the associated heat semigroup is the closure of 2 (defined x 2 j on C c (R d )). The multidimensional case Let us turn to the multidimensional situation. The d-dimensional heat kernel is the family (g d,t ) t> given by 1 g d,t = g t... g t = (2πt) d 2 e s 2 /4t (t > ). (The modulus here denotes the Euclidean norm on R d.) It is easily seen that Lemma 6.14 holds mutatis mutandis: (g d,t ) t> is a generalized Dirac sequence and continuous in t > and F(g d,t ) = e t z 2 for all t >. If (U s ) s R d is a bounded, strongly continuous representation of R d on a Banach space X with corresponding Fourier Stieltjes calculus Ψ U the associated heat semigroup (also: Gauss Weierstrass semigroup) is (G t ) t, defined by G t := Ψ U (e t z 2 ) (t ). This means that G t = R d g d,t (s)u s ds whenever t >. Then, with pretty much the same proof, we obtain the following analogue of Theorem Theorem Let (U s ) s R d be a bounded and strongly continuous group on a Banach space X with associated Fourier Stieltjes calculus Ψ U. Let, furthermore, (G t ) t be the associated Gauss Weierstrass semigroup and Φ G its Hille Phillips calculus. If f LS(C + ) then f( z 2 ) FS(R d ) and Φ G (f) = Ψ U (f( z 2 )). In the situation of Theorem 6.18, let ia j be the generator of the bounded C -group U j, defined by U j s = U sej for s R and 1 j d. Then for g FS(R d ) one can think of Ψ U (g) as Ψ U (g) = g(a 1,..., A d ) similar to the one-dimensional case. Now if B denotes the generator of (G t ) t then, as in the proof of Corollary 6.16, ( (I + B) 1 1 ) ( 1 ) = Ψ U 1 + z 2 = 1 + z (A 1,..., A d ). z2 d Hence, we would like to conclude

19 6.6 Supplement: Subordinate Semigroups 13 B = (z z 2 d)(a 1,..., A d )? = A A 2 d. (6.13) The first identity can be justified (e.g. by results of the next chapter), but the second one fails in general. The best one can say here is the following. Theorem In the situation just described, B = A A2 d. Proof. By Corollary 6.16, the operator A 2 j is the generator of Gj, the heat semigroup associated with the group U j. Since the groups U j are pairwise commuting, so are the semigroups G j. Now observe (by a little computation) that G(t) = G 1 (t) G d (t) (t ). Hence, Exercise 6.8.d) yields the claim. Examples 6.2. Consider the Gauss Weierstrass semigroup associated with the right shift group on X = C (R d ) or X = L p (R d ), 1 p <. Then it follows from Example ) and Theorem 6.19 that its generator X, say, is the closure in X of the operator d := 2 x 2 j=1 j (6.14) on C c (R d ). It can be shown but we do not do this here that X is a restriction of the distributional Laplacian to X. We shall see later that, due to the boundedness of the so-called Riesz transforms, for 1 < p < the domain of L p is W 2,p (R d ). 6.6 Supplement: Subordinate Semigroups In this section we review our findings from Section 6.5 on the Gauss Weierstrass semigroups from a more abstract point of view. A family of measures (µ t ) t in M(R d ) is called a convolution semigroup if µ = δ and µ t µ s = µ s+t whenever s, t. It is called strongly continuous if µ t δ strongly (as defined in Section 6.4 above) as t. In the following we suppose that S = R d or S = R d + and (µ t ) t is a convolution semigroup in M(S). Then to each bounded and strongly continuous representation S : S L(X) a semigroup (T t ) t on X is given by T t := Ψ S ( µ t ) = S s µ t (ds) L(X) (t ). S The semigroup T is called subordinate to the representation S, and the family (µ t ) t is the so-called subordinator. One has the following lemma.

20 14 6 Integral Transform Functional Calculi Lemma If the convolution semigroup (µ t ) t is strongly continuous then so is the semigroup (T t ) t. Proof. It follows from Theorem 6.11 that T is strongly continuous at t =. By Exercise 6.14, T is strongly continuous on the whole of R +. For strongly continuous convolution semigroups, the following result is of fundamental importance. Theorem Let (µ t ) t be any strongly continuous and uniformly bounded convolution semigroup in M(S). Then there is a uniquely determined continuous function a : R d C + such that µ t = e ta (t ). Moreover, for each f LS(C + ) one has f a FS S (R d ) and Φ T (f) = Ψ S (f a) whenever S : S L(X) is a strongly continuous and bounded representation with Fourier Stieltjes calculus Ψ S, T is the semigroup subordinate to S with respect to the subordinator (µ t ) t, and Φ T is its Hille Phillips calculus. For the proof, we need the following auxiliary result, interesting in its own right. Theorem Let T be a bounded linear operator on L 1 (R d ). Then the following assertions are equivalent: (i) T commutes with all translations. (ii) T (ν ψ) = ν T ψ for all ν M(R d ) and ψ L 1 (R d ). (iii) T (ϕ ψ) = ϕ T ψ for all ϕ, ψ L 1 (R d ). (iv) There is a function a : R d C such that F(T ψ) = a ψ for all ψ L 1 (R d ). (v) There is µ M(R d ) such that T ψ = µ ψ for all ψ L 1 (R d ). In this case a = µ and µ is uniquely determined by (v). Proof. (v) (iv): take a = µ. (iv) (ii): For ν M(R d ) and ψ L 1 (R d ): F(T (ν ψ)) = af(ν ψ) = a ν ψ = νa ψ = νf(t ψ) = F(ν T ψ) and hence, by the injectivity of the Fourier transform, T (ν ψ) = ν T ψ. (ii) (i),(iii): For (i) take ν = ϕλ and for (iii) take ν = δ t, t R d. (i) (ii): This follows from Lemma 5.3 applied to the translation group.

21 6.6 Supplement: Subordinate Semigroups 15 (iii) (v): Let (ϕ n ) n N be any Dirac sequence in L 1 (R d ). Then by the continuity of T, the commutativity of convolution, and hypothesis (iii) T ψ = lim n T (ϕ n ψ) = lim n T ϕ n ψ as a limit in L 1 (R d ) for each ψ L 1 (R d ). The sequence (T ϕ n ) n N is bounded and can be regarded as a sequence in M(R d ). By the identification M(R d ) = C (R d ) and the Banach Alaoglu theorem, passing to a subsequence we may suppose that there is µ M(R d ) such that ϕ n µ weakly. Then, for each ψ L 1 (R d ) and f C (R d ) f, T ψ = lim n f, T ϕ n ψ = lim n Sψ f, T ϕ n = Sψ f, µ = f, µ ψ. It follows that T ψ = µ ψ, as claimed. Uniqueness follows from Lemma 6.9. Remark Suppose that in Theorem 6.23 there is a closed subset E R d such that L 1 (E) contains a Dirac sequence and is invariant under T. Then supp(µ) E. Indeed, the proof shows that µ is a weak -limit of functions T ϕ n, where (ϕ n ) n is a certain subsequence of an arbitrary Dirac sequence in L 1 (R d ). In particular, one can suppose that supp(ϕ n ) E for all n N. Hence, if T leaves L 1 (E) invariant, then supp(µ) E as well. We can now prove Theorem Proof of Theorem First, consider the multiplication operator semigroup (T t ) t on C (R d ) defined by T t f = µ t f (f C (R d ), t ). is the bounded strongly con- By Example 5.6.1), T t = S µt, where (S s ) s R d tinuous group on C (R d ) given by S s f = e s f (f C (R d )). Hence, by Lemma 6.21, (T t ) t is strongly continuous. By Exercise 6.15 there is a unique continuous function a : R d C + such that µ t = e ta for all t. For the second part we first deal with the case S = R d. Let ν M(R + ) and f = Lν. By Theorem 6.23 there is a unique measure µ M(R d ) such that µ t ψ ν(dt) = µ ψ for all ψ L 1 (R d ). Taking Fourier transforms and inserting x R d we see that

22 16 6 Integral Transform Functional Calculi e ta(x) ψ(x) ν(dt) = µ(x) ψ(x). This yields f a = µ FS(R d ). Finally, with S and T as in the hypotheses of the theorem and ψ L 1 (R d ), Φ T (f)s ψ = T t ν(dt) S ψ = S µt S ψ ν(dt) = = S µt ψ ν(dt) = S µ ψ = S µ S ψ = Ψ S (f a)s ψ. S µt ψ ν(dt) As ψ L 1 (R d ) is arbitrary, it follows that Φ T (f) = Ψ S (f a) as claimed. Now consider the case S = R d +. Then supp(µ t ) R d + for all t. By Remark 6.24, µ M(R d +). The remaining parts of the proof carry over unchanged, save that one has to take ψ L 1 (R d +) in the final argument. Example 6.25 (Heat Semigroups). Of course, the heat semigroups of Section 6.5 are examples of subordinate semigroups for the special choice µ t = g d,t λ for t >. Remark Strong continuity of a convolution semigroup (µ t ) t is not easy to check. However, if all µ t are probability measures, the strong continuity is equivalent with the weak convergence of µ t to δ as t (Exercise 6.16). So it should not come as a surprise that subordinate semigroups were first studied by Bochner and Feller in the context of probability theory. 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 { }. (6.15) 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).

23 6.6 Supplement: Subordinate Semigroups 17 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 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) (Nelson s Lemma) 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 is dense in X and invariant under the semigroup T. Show that D is a core for A n. (Note that by Theorem A.2 each operator A n is closed.) b) (Coordinate Shifts) 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.

24 18 6 Integral Transform Functional Calculi [Hint: For a) 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 b) use a).] 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= 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) <. R

25 6.6 Supplement: Subordinate Semigroups 19 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 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. 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).]

26 11 6 Integral Transform Functional Calculi 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. 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. 6.1 (Self-adjoint Semigroups). Show that for an operator A on a Hilbert space H the following assertions are equivalent: (i) (ii) A is a positive self-adjoint operator. A generates a C -semigroup T = (T t ) t of self-adjoint operators. Show further that in this case the Borel calculus for A coincides with the Hille Phillips calculus for A on LS(C + ). Supplementary Exercises 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 a compact set K R d such that supp(µ n ) K for all n N.

27 6.6 Supplement: Subordinate Semigroups 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.] Let T = (T t ) t be an operator semigroup on a Banach space X such that lim t T t = I strongly. Show that T is strongly continuous on R +. [Hint: Show first that there is δ > such that sup t δ T t <. See also [3, Prop.I.5.3].] Let Ω be a locally compact metric space and let (e t ) t be a family of functions in C b (Ω) such that e t+s = e t e s for all t, s and e = 1, and such that the operator family (T t ) t defined by T t = M et for all t is a bounded strongly continuous semigroup on X = C (Ω). a) For x Ω and t define ϕ x (t) := e t (x). Show that there is a unique a(x) C such that ϕ x (t) = e ta(x) for all t. b) Prove that a(ω) C + and that a is continuous. [Hint: For a) note first that ϕ is continuous; then the result is classical, see [3, Prop. I.1.4]; a different proof proceeds by showing that the Laplace transform of ϕ x is (λ + a(x)) 1 for some a(x) C and all Re λ >. For b) consider the operator e t T t dt.] Let (µ n ) n be a sequence of Borel probability measures on R d such that µ n δ weakly as n. Show that R d f dµ n f() (n ) whenever f C b (R d ; X) and X is any Banach space. Conclude that µ n δ strongly. 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.

28 112 6 Integral Transform Functional Calculi