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 )) = fµb j µ 2 M(R )g: This algebra, which is called the Fourier{Stieltjes algebra1 of Rd, is en- dowed with the norm d kµbkFS := kµkM (µ 2 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 2 FSS(R )); (6:2) where, as always, M = sup kT k. T t2S t Example 6.1. Let Ω = (Ω; Σ; ν) be a measure space and a : Ω ! Rd a measurable function. Fix p 2 [1; 1) and abbreviate X := Lp(Ω). For t 2 Rd let Tt 2 L(X) be defined by −it·a(·) Ttx := e x (x 2 X): Then (T ) d is a bounded and strongly continuous group and, by Exercise t t2R 5.8, Tµx = (µb ◦ a) x (x 2 X) for all µ 2 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 2 FS(R ) one has −1 −1 1 d Ψτ (f)x = µ ∗ x = F (µb · xb) = F (f · xb)(x 2 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 2 L(X) is called doubly power-bounded if T is invertible and M := sup kT nk < 1. Such operators correspond in a one-to-one T n2Z 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 = fz 2 C : jzj = 1g. ∼ 1 By Exercise 5.9, M(Z) = ` (Z) and FSZ(R) consists of all functions X −int 1 f = αne (α = (αn)n 2 ` ): (6:3) n2Z These functions form a subalgebra of C2p(R), the algebra of 2p-periodic functions on R. By the Fourier{Stieltjes calculus, the function f as in (6.3) is mapped to X n ΨT (f) = αnT : n2Z Hence, this calculus is basically the same as a Laurent series calculus X n X n ΦT : αnz 7−! αnT (6:4) n2Z n2Z 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 j α 2 ` (Z) ⊆ C(T) n2Z 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) ! C2p(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 2 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 1 X −int 1 f = αne (α = (αn)n 2 ` ): (6:5) n=0 These functions form a subalgebra of C2p(R). Under the Fourier{Stieltjes calculus, the function f as in (6.5) is mapped to 1 X n ΨT (f) = αnT : n=0 This calculus is isomorphic to the power-series calculus 1 1 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 P1 n phism f 7! f(e ). (Observe that for α 2 ` 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 2 M(R+) is the function Z −zt Lµ : C+ ! C; (Lµ)(z) = e µ(dt): R+ Here, C+ := fz 2 C : Re z > 0g is the open right half-plane. By some standard arguments, kLµk1 ≤ kµkM and Lµ 2 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 2 R (6:7) and since the Fourier transform is injective, so is the Laplace transform. Let us call its range LS(C+) := fLµ : µ 2 M(R+)g the Laplace-Stieltjes algebra and endow it with the norm kLµkLS := kµkM (µ 2 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 2 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 2 LS(C+) can be interpreted as a function on C+, on C+, on iR, or on (0; 1) 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 func- tional 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.