SEMIGROUP THEORY VIA FUNCTIONAL CALCULUS. 1. Introduction

SEMIGROUP THEORY VIA FUNCTIONAL CALCULUS MARKUS HAASE Abstract. In this survey article we present a panorama of operator classes with their associated functional calculi, relevant in semigroup theory: operators of halfplane-, strip-, sector- and parabola-type. It is shown that the basic results in the theory of C -semigroups (the Hille Yosida and the Trotter Kato theorem) follow easily from general functional calculus principles. The introduction of parabola-type operators allows to treat cosine operator functions by functional calculus methods. Dedicated to Ulf Schlotterbeck on the occasion of his 65. birthday 1. Introduction The theory of strongly continuous semigroups is now some 6 years old, with the fundamental result of Hille and Yosida dating back to 1948. Several monographs and textbooks cover the by now canonical material, each of them having its own peculiar point of view. One may focus entirely on the semigroup, and consider the generator as a derived concept (as in [3]) or one may start with the generator and view the semigroup as the pre-laplace transform of the resolvent (as in [1]). On the other side, during the last two decades the theory of functional calculus has proved to be an indispensable tool to deal with abstract evolution equations, above all in the discussion of maximal regularity. Despite their success, these methods have been mainly restriced to sectorial operators and hence to holomorphic semigroups. Eventually, C -groups were covered by a functional calculus for striptype operators [4]; but this was done because of the prominent role the groups of imaginary powers of a sectorial operator play in the parabolic world. Anyway, an approach to general semigroups via holomorphic functional calculus is missing up to now, as well as one for cosine operator functions. (In fact it was not clear for some time whether cosine functions could be treated by functional calculus methods at all.) In this article we want to close this gap. We complement the existing holomorphic functional calculi (for sectorial and strip-type operators) by two more, one on halfplanes and one on parabolas. (The first covers general semigroups, the second cosine functions.) The strength of this approach lies in the fact that we do not have to require the operator to generate a semigroup or a cosine function, but we can work just with a certain natural growth condition on the norm of the resolvent. As a conseqence we can give a short and straightfoward proof of the Hille Yosida theorem, based only on one of the cornerstones of general functional calculus theory, the so-called Convergence Lemma. A similar remark applies to the Trotter Kato theorem. Date: September 18, 26. 2 Mathematics Subject Classification. 47A6,47D6. Key words and phrases. C -semigroup, functional calculus, Hille Yosida, Trotter Kato. 1

2 MARKUS HAASE So it turns out that the theory functional calculus indeed provides a valid starting point for general semigroup theory, not only in the holomorphic case. 2. A Functional Calculus on the Half-plane Let A be the generator of a C -semigroup (T (t)) t R on a Banach space X. We would like to interpret T (t) = (e tz )(A) (t ), were e tz denotes the complex function z e tz, and this should only be a special case of a general set-up in which f(a) is defined in a reasonable way for as much functions f as possible. Our approach is based on the Cauchy integral formula; as we shall see, this to be viable requires only knowledge of the resolvent outside the spectrum of A. (In particular, the generator property is not necessary to construct the functional calculus in the first place.) As the procedure should work for all semigroup generators, we are lead to a functional calculus on (left) half-planes. Let us fix some notation. For ω [, ] define L ω := {z C Re z < ω}, R ω := {z C Re z > ω} the open left and right half-planes defined by the abscissa Re z = ω, where in the extremal cases one half-plane is understood to be empty, and the other is the whole complex plane. We say that an operator A on a Banach space X is of half-plane type ω R { }, if R ω ϱ(a) and for every α > ω. We call M α := M α (A) := sup{ R(z, A) Re z α} < s (A) := min{ω A is of half-plane type ω} [, ] the abscissa of uniform boundedness (in short: the aub) of the operator A. Since all notions are invariant under translations parallel to the real axis, we assume in the following that s (A). Fix ω (, 1) and define E(L ω ) := {f : L ω C f is holomorphic and M, s > : f(z) M z (1+s) as z }. By standard applications of Cauchy s integral theorem for f E(L ω ) the formula f(a) := 1 f(z) dz (Re a ) 2πi z a Re z=δ holds. Here δ (, ω) is arbitrary, and the direction of integration is bottom up, i.e., from δ i to δ + i. Since the resolvent R(, A) is bounded on the vertical lines (Re z = δ), δ >, the integral Φ(f) := f(a) := 1 f(z)r(z, A) dz 2πi Re z=δ converges absolutely. Proposition 2.1. The so defined mapping Φ : E(L ω ) L(X) satisfies the following properties: a) Φ is a homomorphism of algebras.

SEMIGROUP THEORY VIA FUNCTIONAL CALCULUS 3 b) If T L(X) commutes with A, i.e., T A AT, it commutes with every Φ(f), f E. c) Φ(f(z)(λ z) 1 ) = Φ(f)R(λ, A) for all Re λ > ω. d) Φ((λ z) 1 (µ z) 1 ) = R(λ, A)R(µ, A) for all Re λ, Re µ > ω. Proof. a) This follows from a combination of Fubini s theorem, the resolvent identity and Cauchy s theorem. The computation is the same as in the classical Dunford Riesz setting, see [2, VII.4.7]. b) is obvious. c) follows from the resolvent identity, since by Cauchy s theorem the integral f(z)/(λ z) dz is equal to zero. Re z=δ d) Here a standard path deformation argument is used. In the integral Re z=δ (λ z) 1 (µ z) 1 R(z, A) dz shift the path to the right, i.e., let δ grow. When passing the abscissas δ = Re λ and δ = Re µ the residue theorem yields some additive contributions which sum up to R(λ, A)R(µ, A) by the resolvent identity; if δ > Re λ, Re µ, the integral does not change any more as δ and hence it is equal to zero. Denote by M(L ω ) the field of meromorphic functions on the left half-plane Re z < ω. Then the triple (E(L ω ), M(L ω ), Φ) is a meromorphic functional calculus in the sense of [7, Section 1.3]. A meromorphic function f is called regularisable if there is a function e E such that ef E and e(a) is injective. In this case one defines f(a) := e(a) 1 (ef)(a), which is a closed operator. This definition does not depend on the chosen regulariser e E (cf. [7, Section 1.2.1]). The basic rules governing this functional calculus are the same as for any meromorphic functional calculus, see [7, Theorem 1.3.2]. The two most important of these are the laws for sums: and products f(a)g(a) (fg)(a), f(a) + g(a) (f + g)(a) D((fg)(A)) D(g(A)) = D(f(A)g(A)). In particular, one has f(a) + g(a) = (f + g)(a) and f(a)g(a) = (fg)(a) whenever g(a) L(X). Note that every bounded analytic function f H (L ω ) is regularisable, namely by the function e(z) := (1 z) 2. This is because f(z)(1 z) 2 decreases quadratially as z and e(a) = R(1, A) 2 is clearly injective. In particular, for each t the operator e ta := (e tz )(A) is defined as a closed operator and D(A 2 ) D(e ta ), t. Lemma 2.2. Let A be an operator of half-plane type, with s (A). Then for each x D(A 2 ) the function (t e ωt e ta x) : [, ) X is continuous and bounded, and its Laplace transform is e λt e ta x dt = R(λ, A)x (Re λ > ω).

4 MARKUS HAASE Proof. Write e ta x = (e tz /(1 z) 2 )(A)[(1 A) 2 x]. Then the continuity in t is clear from Lebesgue s theorem. The rest follows by Fubini s theorem. Lemma 2.3. Let A be a densely defined operator of half-plane type. Then D(A 2 ) is dense. Proof. For x D(A) and n N we write nr(n, A)x = x + R(n, A)Ax. The right hand side is bounded in n, hence R(n, A)x as n. But D(A) is dense and the operators (R(n, A)) n 1 are uniformly bounded. Hence R(n, A)x for all x X. But this implies that nr(n, A)x = x + R(n, A)Ax x whenever x D(A); so D(A) D(A 2 ) which implies that D(A 2 ) is dense in X. Proposition 2.4. Let A be an operator of half-plane type, with s (A). Then A is the generator of a C -semigroup T if and only if A is densely defined and e ta is a bounded operator for all t [, 1] satisfying sup t [,1] e ta <. In this case, T (t) = e ta for all t. Proof. Let A generate a C -semigroup (T (t)) t. Then A is densely defined. Hence D(A 2 ) is dense, by Lemma 2.3. Lemma 2.2 yields that R(, A)x is the Laplace transform of e A x for x D(A 2 ). By the uniqueness of Laplace transforms, T (t)x = e ta x, t. Since D(A 2 ) is dense and e ta is a closed operator, e ta = T (t) is a bounded operator. Conversely, suppose that A is densely defined and T (t) := e ta is a bounded operator for all t. Then T is a semigroup (by general functional calculus) and D(A 2 ) is dense, by Lemma 2.3. From the uniform boundedness sup t [,1] T (t) < one concludes easily that (T (t) t ) is uniformly bounded on compact intervals. Lemma 2.2 and the density of D(A 2 ) imply that (T (t)) is strongly continuous. Its Laplace transform coincides with the resolvent of A on D(A 2 ) (Lemma 2.2), hence on X by density. So A is the generator of T. 3. The Hille Yosida Theorem The Hille Yosida theorem is one of the most fundamental results in the elementary theory of C -semigroups. We show that it is a straightforward consequence of the following general fact of functional calculus theory. Proposition 3.1. (Convergence Lemma) Let A be a densely defined half-plane type operator on the Banach space X, with s (A). Let ω (, 1) and let (f ι ) ι be a net in H (L ω ) satisfying the following conditions: (i) sup ι f ι < ; (ii) f ι (A) L(X) for all ι, and sup ι f ι (A) < ; (iii) f(z) := lim ι f ι (z) exists for every z L ω. Then f H (L ω ), f(a) L(X), f ι (A) f(a) strongly, and f(a) lim sup ι f ι (A). Proof. The proof is analogous to the proof of [7, Proposition 5.1.4]. By Vitali s theorem, f is holomorphic and the convergence of the f ι to f is uniform on compacts. Moreover, condition (i) clearly implies that f is bounded. By Lebesgue s theorem and the definition of the elementary functional calculus (f ι (z)(1 z) 2 )(A) (f(z)(1 z) 2 )(A) in norm. Hence for x D(A 2 ), f ι (A)x = (f ι (z)(1 z) 2 )(A)(1 A) 2 x (f(z)(1 z) 2 )(A)(1 A) 2 x = f(a)x.

SEMIGROUP THEORY VIA FUNCTIONAL CALCULUS 5 Clearly f(a)x lim sup ι f ι (A). Since f(a) is a closed operator with dense domain D(f(A)) D(A 2 ), f(a) is bounded with f(a) lim sup ι f ι (A). Again by the density of D(A 2 ), f ι (A) f(a) strongly. Theorem 3.2. (Hille Yosida) Let A be a densely defined operator on the Banach space X such that (, ) ϱ(a) and M := sup n N,λ> [λr(λ, A)] n <. Then A is of half-plane type with s (A) and e ta M for all t. Proof. First we show that A is of half-plane type. Fix µ such that Re µ >. For λ > large, more precisely for λ > µ 2 /(2 Re µ), one has λ µ < λ. By the Laurent series expansion of the resolvent, R(µ, A) = (µ λ) n R(λ, A) n+1 n= and hence R(µ, A) M µ λ n /λ n+1 = M/(λ µ λ ). Now let λ to conclude R(µ, A) M/ Re µ. It follows that A is of half-plane type with s (A). Define r n,t (z) := (1 (tz)/n) n. For fixed ω (, 1) and large n N we have ( sup r n,t (z) inf Re z ω Re z ω 1 tz n ( n ) = 1 tω ) n. n Since (1 tω/n) n e tω as n, we have sup n r n,t <. Also, by hypothesis, r n,t (A) = (1 (t/n)a) n = [(n/t)r((n/t), A)] n M for all n N. Applying the Convergence Lemma yields e ta M, as desired. Remark 3.3. A more careful statement of the Convergence Lemma and an equally careful analysis of the above proof would lead to the statement that for each x X one has lim n r n,t (A)x = e ta x uniformly in t from compact subintervals of [, ). 4. The Trotter Kato Theorem While in the Convergence Lemma the function is approximated and the operator is fixed, in the following we fix the function and approximate the operator. The correct setup requires that the approximants A n are of the same type as the operator, with the relevant constants being uniformly bounded. More precisely, a family of operators (A ι ) ι is called uniformly of half-plane type ω R { } if each A ι is of half-plane type ω and sup ι M α (A ι ) < for each α > ω. Example 4.1. Let A be of half-plane type, and let A λ := λar(λ, A) for λ 1 be the Yosida approximants. Then the family (A λ ) λ 1 is uniformly of half-plane type. Indeed, a little computation shows R(µ, A λ ) = λ 2 1 R(λµ/(λ + µ), A) + (µ + λ) 2 λ + µ For the second term we have λ + µ 1 min(λ 1, (Re µ) 1 ). To estimate the first we compute ( ) µλ Re = λ2 Re µ + λ µ 2 λ + µ λ + µ 2

6 MARKUS HAASE If A happens to be the generator of a bounded C -semigroup, it satisfies an estimate R(µ, A) M(Re µ) 1 for some M 1 and all Re µ >. Given this, one obtains λ 2 M λ + µ R(µ, A λ ) µ + λ 2 λ 2 Re µ + λ µ 2 + 1 Re µ + λ M + 1 Re µ independent of λ. In the general case, fix α >, take Re µ α and define ε := α/(2α + 1). It follows that Re µ/(2 Re µ + 1) ε and this implies λ 2 (Re µ ε) 2λε Re µ, since λ 1. From this we conclude that ( ) µλ Re = λ2 Re µ + λ µ 2 λ + µ λ + µ 2 ε, and hence that R(µ, A λ ) independent of λ 1 and µ α. 2 λ 2 (λ + Re µ) 2 M 1 ε(a) + Re µ + λ M ε(a) + 1 In the previous example we clearly have lim λ R(µ, A λ ) = R(µ, A) in norm uniformly in µ from compact subsets of the open halfplane (Re z > ). Proposition 4.2. Let (A n ) n be a family of operators, uniform of half-plane, and let A be an operator such that R(µ, A n ) R(µ, A) in norm/strongly for all Re µ >. Then A is also of half-plane type. Moreover, for ω (, 1) and f E(L ω ) one has f(a n ) f(a) in norm/strongly. Suppose furthermore that A is densely defined. If f H (L ω ) and f(a n ) L(X) for all n N with C := sup n f(a n ) <, then also f(a) L(X), and f(a n ) f(a) strongly. Proof. The first two assertions are straightforward. Suppose that A is densely defined; by Lemma 2.3 this implies already that D(A 2 ) is dense in X. Take x D(A 2 ), f H (L ω ) and e(z) := f(z)(1 z) 2 E(L ω ). Then e(an )(1 A) 2 x = f(an )R(1, A n ) 2 (1 A) 2 x C R(1, An ) 2 (1 A) 2 x. Since we know already that e(a n ) e(a) and R(1, A n ) R(1, A), we conclude that f(a)x = e(a)(1 A) 2 x C R(1, A) 2 (1 A) 2 x = C x. Since D(A 2 ) is dense, it follows that f(a) L(X) with f(a) C. To prove that f(a n ) f(a) (strongly), we need only to show f(a n )x f(a)x for all x D(A 2 ). So take x D(A 2 ) and let y := (1 A) 2 x. We have seen above that f(a n )R(1, A n ) 2 y f(a)x, so we estimate the difference f(an )x f(a n )R(1, A n ) 2 y = f(an )R(1, A) 2 y f(a n )R(1, A n ) 2 y C R(1, A) 2 y R(1, A n ) 2 y by hypothesis. Remark 4.3. As in the Convergence Lemma, there is version of Proposition 4.2 that yields some uniformity: suppose that (f ι ) ι H (L ω ) is uniformly bounded and sup ι,n f ι (A n ) <. Then the convergence f ι (A n )x f ι (A)x is uniform in ι, for every x X.

SEMIGROUP THEORY VIA FUNCTIONAL CALCULUS 7 Theorem 4.4. (Trotter Kato) Suppose that, for each n N, A n is the generator of a C -semigroup, and that e ta n M for all t, n N. Suppose further that A is a densely defined operator and for some λ > one has λ ϱ(a) and R(λ, A n ) R(λ, A) strongly. Then A generates a C -semigroup and one has e tan x e ta x uniformly in t [, τ], for each x X, τ >. Proof. The theorem is a consequence of Proposition 4.2 and the remark next to it, as soon as we can show that actually {Re z > } ϱ(a) and R(µ, A n ) R(µ, A) for all Re µ >. This is done like in [3, Proposition III.4.4]. Remark 4.5. A common assumption on A implying the resolvent convergence is the following: the operator A is densely defined, λ A has dense range, and there exists a core D of A such that A n x Ax for all x D. See [3, Theorem III.4.9]. However, one might not always be given the operator A. Its existence is ensured by the following condition: R(λ, A n ) Q L(X) strongly, and Q has dense range. Indeed, by general arguments as in [7, Appendix A.5] one has Q = R(λ, A) for some possibly multi-valued operator A, which is densely defined by the range assumption on Q. It remains to show that A is in fact single-valued, i.e., Q is injective. 5. The Universal Model and the Phillips Calculus The functional calculus for operators of half-plane type provides us with a wealth of functions f where f(a) is defined. It uses the Cauchy formula and only some information of the growth of the resolvent. To obtain more information, one has to make stronger assumptions, typically of the type that certain f(a) are required to be bounded operators. An averaging over the f s yields new representation formulas different from the Cauchy formula, and so more information about the functional calculus. As an example of these fairly general considerations, let us suppose that A generates a bounded C -semigroup (T (t)) t. For a finite measure µ M[, ) one defines its Laplace transform by L(µ)(z) := e zt µ(dt) (Re z ). It is well-known that L(µ) is holomorphic on (Re z < ), and bounded and continuous on (Re z ). Moreover, µ is uniquely determined by L(µ). If f = L(µ) we define f(a)x := T (t)x µ(dt) (x X). [, ) This yields an algebra homomorphism (µ L(µ)(A)) : M[, ) L(X). By the following lemma, this is in accordance with our previous definitions. Lemma 5.1. Let ω >, and let f E(L ω ). Then there is a function g L 1 (, ) such that f(a) = g(t)t (t) dt. If f H (L ω ) is such that there is µ M[, ) with f = L(µ), then indeed f(a) = T (t) µ(dt) [, )

8 MARKUS HAASE where the left-hand side is defined by the functional calculus for A as an operator of half-plane type. Proof. To prove the first statement let δ (, ω) and define g(t) := 1 2πi Re z=δ f(z)e zt dz. Then g L 1 (, ) and Fubini s theorem does the rest. For the second assertion we assume without restriction that ω < 1. Define e(z) = (1 z) 2, so ef, e E. Find g L 1 (, ) such that L(g) = e. Then ef = L(µ g), and so (ef)(a) = T (t) (µ g)(dt) = T (t + s) µ(dt) g(s)ds = [, ) [, ) This yields the assertion. [, ) [, ) T (s) g(s)ds T (t) µ(dt) = e(a) [, ) [, ) T (t) µ(dt). Important about the Phillips calculus is the fact that it is all that one can expect in general from the functional calculus for semigroup generators. Before we can make this more precise, we introduce a very special semigroup. Example 5.2. (Shift semigroup) Let X = L 1 (, ) and let T (t)f(s) := f(t + s) for t, s. Then T is a strongly continuous contraction semigroup on X with generator A = d/dt on D(A) = W 1,1 [, ). If we think of L 1 (, ) as sitting inside L 1 (R), with all functions in L 1 (, ) being zero on (, ), then the resolvent of A can be written as R(λ, A)f = e λ f (, ), with e λ (t) = e λt 1 (,), for Re λ >. It can be shown that actually σ(a) = (Re z ). The Phillips calculus for this particular semigroup reads L(µ)(A)f = (µ f)1 (, ) where µ (B) = µ( B) for every Borel set B R. Claim: The mapping is isometric. (f L 1 (, )) (µ L(µ)(A)) : M[, ) L(L 1 (, )) Proof. Clearly the mapping is contractive. Fix µ M[, ) and denote f := L(µ). Fix ψ L 1 (, ) and ϕ C [, ) such that ψ 1, ϕ. Then t f(a)ψ, ϕ = ψ(t + s) µ(ds) ϕ(t) dt = ψ(t) ϕ(t s) µ(ds) dt Taking the supremum with respect to ψ yields t sup f(a)ψ, ϕ = sup ϕ(t s) µ(ds) ψ t By choosing suitable ϕ we see that f(a) = sup ϕ sup f(a)ψ, ϕ µ ([, t ]) ψ for any t >. Letting t concludes the proof. As a consequence one obtains the following transference type result.

SEMIGROUP THEORY VIA FUNCTIONAL CALCULUS 9 Corollary 5.3. Let A generate a bounded C -semigroup on the Banach space X. Then f(a) L(X) [sup T (t) ] f(d/dt) L(L 1 (, )) t for every f {L(µ) µ M[, )}. Proof. Let f = L(µ). Then f(a)x = T (t)x µ(dt) for all x X, whence f(a) [sup T (t) ] µ M[, ). t But µ M[, ) = f(d/dt) was shown above. We can now state the main result. Proposition 5.4. Let ω (, 1) and let f H (L ω ). Then the following statements are equivalent: (i) (ii) (iii) f(a) is a bounded operator, for each generator of a bounded semigroup T on a Banach space X. f(a) is a bounded operator, where A = d/dt generates the left shift semigroup on X = L 1 (, ). There exists µ M[, ) such that f = L(µ). Proof. The implications (iii) (i) (ii) are trivial. Suppose that f(a) is a bounded operator on X = L 1 (, ), where A = d/dt is the generator of the left shift semigroup (T (t)) t. Let f n (z) := f(z)[n/(n z)] 2. Then f n (A) f(a) strongly. By Lemma 5.1, f n = L(g n ) for certain g n L 1 (, ). By Example 5.2, g n 1 = f n (A), and this is clearly bounded independent of n. Since C [, ) is separable, the weak topology on bounded sets of M[, ) is metrizable. By compactness, there exists a subsequence (n k ) k and a measure µ M[, ) such that g nk µ weakly. For fixed Re z <, the function t e tz is in C [, ), and hence f(z)[n k /(n k z)] 2 = f nk (z) = g nk (t)e zt dt e zt µ(dt). But clearly f nk (z) f(z) also, whence f = L(µ). [, ) The proposition explains the importance of the Phillips calculus for results about approximation in norm, like the Post-Widder, the Phragmen-Doetsch and the Complex inversion formulas, see [3, Section III.5]. In fact it shows that the proofs given there are natural. 6. Sectorial and Strong Strip-Type Operators If the semigroup is a group, the spectrum of the generator is contained in a vertical strip. So the appropriate functional calculus is for functions living on that strip and not in a half plane. For reasons that will become clear later, it is desirable to consider horizontal strips instead of vertical ones. For ω > we define and St := R. St ω := {z C Im z < ω}

1 MARKUS HAASE Definition 6.1. An operator A on a Banach space X is called to be of strong strip-type ω (in short: A SStrip(ω)), if σ(a) St ω and for every ω > ω there is M ω such that M ω R(λ, A) Im λ ω ( Im λ > ω ). The least of the ω such that A SStrip(ω) is denoted by ω sst (A). For example, if ia generates a group (U(t)) t R such that U(t) Me ω t then A is of strong strip-type ω. One can weaken the hypothesis; in fact it suffices that the exponential group type θ(u) ω. Here θ(u) := inf{ω M 1 : U(t) Me ω t (t R)}. With a strong strip-type operator A SStrip(ω) there comes along a natural holomorphic functional calculus. One considers functions f holomorphic on strips St ϕ with ϕ > ω. If f has integrable decay at ±, e.g., f = O( z (1+s) ) as z for some s >, then f(a) is defined by the usual Cauchy integral, the contour being the boundary of an appropriate strip. This gives a primary functional calculus for A and is extended to a meromorphic functional calculus via the usual regularisation procedure, see [7, Section 4.2]. Of course this functional calculus extends the two half-plane calculi available for A. (Note that ±ia are both of halfplane-type and so, by Section 2, have natural functional calculi on left halfplanes.) It is clear that a result analogous to Proposition 2.4 holds. The operator ia generates a C -group if and only if A is densely defined, of strong strip-type, and (e itz )(A) is bounded for all t R. In this case, one can set up a Phillips calculus and obtains f(a)x = U(t)x µ(dt) (x X) if f(z) = µ(z) = e itz µ(dt) and µ is such that e ω t µ(dt) M(R) for some ω > θ(u). See [6] for proofs. Strong strip-type operators arise naturally as logarithms of sectorial operators [7, Proposition 3.5.1], and the fact that there are sectorial operators without bounded imaginary powers yields the existence of natural examples of strong strip-type operators that do not generate a group. For the sake of completeness, we give the definition of a sectorial operator. Let ω [, π] and define { {z arg z < ω}, (ω > ) S ω := {e z z St ω } = [, ), (ω = ). Definition 6.2. An operator A on a Banach space X is called sectorial of angle ω < π if σ(a) S ω and for every ω (ω, π] there is M ω such that λr(λ, A) M ω ( arg λ [ω, π]). The least ω such that A Sect(ω), is denoted by ω sec (A). For sectorial operators there is a natural meromorphic functional calculus set up exactly in the same way as for operators of strip- or of halfplane-type [7, Chapter 2]. (The functions live on larger sectors S ϕ, of course). The pay-off for semigroup theory lies in the fact that an operator A generates a bounded holomorphic semigroup if and only if A is sectorial of type < π/2 [7, Section 3.4]. There is strong link between sectorial and strip-type operators. It was proved essentially by Nollau that if A is an injective sectorial operator of angle ω then

SEMIGROUP THEORY VIA FUNCTIONAL CALCULUS 11 log A is of strong strip-type ω. In fact, it was proved in [4] that the strip-type of log(a) is equal to the sectoriality type of A: ω sst (log(a)) = ω sec (A). In [5] we could even show that the spectral mapping theorem σ(log(a)) = log(σ(a) \ {} holds. One can switch back and forth from A to log A, as far as the functional calculi are concerned. Indeed, there is a composition rule f(log A) = (f(log z))(a) in the sense that f(log(a)) is defined (by the f.c. for strong strip-type operators) if and only if (f(log z))(a) is defined (by the f.c. for sectorial operators). The operator i log(a) generates a C -group if and only if A has bounded imaginary powers (A is ) s R. (Since e is log z = z is and in view of the composition rule above, this does not come as a surprise.) See [7, Chapter 4] for proofs and more information. The symmetry between (injective) sectorial and strip-type operators, however, is only partial. There are strong strip-type operators that are not logarithms of sectorial ones. (An example is d/dt on L 1 (R)). Here, the best result up to now is by Monniaux [8]; it states that if ia generates a C -group U of type θ(u) < π, and if the space X is UMD, then A is the logarithm of a sectorial operator. See [6] for a recent new proof. Let us note that as in the semigroup case, there are Convergence Lemmas and Trotter-Kato type results for sectorial and strip-type operators, cp. [7, Sections 2.6 and 5.1]. 7. Cosine Functions In this section we turn to the treatment by functional calculus methods of cosine operator functions. As a guiding intuition, we think of a generator A of a cosine function as A = ( ib) 2 where ib generates a group. It is then pretty natural to consider a functional calculus on the parabola Π ω = {z 2 z St ω }. To define an operator of parabola-type ω we need to specify a resolvent estimate that should hold outside every larger parabola. A natural way to find such a condition is to look at the negative generator of a cosine function. So let A generate a cosine function Cos on the Banach space X, and assume that Cos(t) Me ω t, t R. By definition, λr(λ 2, A) = Taking norms and estimating yields e λt Cos t dt R( λ 2, A) M λ (Re λ ω) (Re λ > ω). (Re λ > ω). (1) The function (z z 2 ) : (Im z > ω) C \ Π ω is biholomorphic, its inverse being on a branch of the square root. Writing µ = λ 2 in 1 yields R(µ, A) M µ ( Im µ ω ) (µ / Π ω ). This expression is actually independent of the branch of the square root we take. It yields the canonical resolvent estimate for a parabola-type operator.

12 MARKUS HAASE Definition 7.1. Let ω and define Π ω := {z 2 z St ω }. An operator A on a Banach space X is called to be of parabola-type ω (in short: A Para(ω)) if σ(a) Π ω and for each ω > ω there exists M ω such that R(µ, A) M ω µ ( Im µ ω ) (µ / Π ω ). The least of the ω such that A Para(ω) is denoted by ω par (A). We have seen above that if A generates a cosine function of exponential growth type ω then A is of parabola-type ω. Here is another example. Lemma 7.2. Let B SStrip(ω). Then A := B 2 Para(ω). Proof. Fix ω > ω and µ C \ Π ω. Since B is of strong strip-type, we find M ω such that R(λ, B) M ω ( Im λ ω ) 1 ( Im λ > ω ). Taking λ := µ (either choice) yields R(µ, A) = (λ 2 B 2 ) 1 = R(λ, B)R( λ, B) = 1 (R(λ, B) R( λ, B)) 2λ by the resolvent identity. Estimating the norm yields the assertion. And yet another one. Lemma 7.3. Let ω and let A be an operator on a Banach space X such that σ(a) Π ω and R(µ, A) M dist(µ, Π ω ) 1 (µ / Π ω ) for some M 1. Then A Para(ω). More precisely, R(µ, A) M µ ( Im µ ω ) (µ / Π ω ). Proof. Let Im λ > ω. It suffices to show that λ ( Im λ ω) dist(λ 2, Π ω ), i.e. λ 2 z 2 = λ z λ + z λ ( Im λ ω) for all z such that Im z ω. But z λ ( Im λ ω) and z + λ ( Im λ ω). And at least one of the factors λ z, λ + z is λ, for trivial geometrical reasons. Corollary 7.4. Let X = H be a Hilbert space and let A be such that its numerical range and spectrum is contained in Π ω. Then A satisfies the condition of Lemma 7.3 with M = 1, and so is of parabola-type. We set up a functional calculus for a parabola-type operator A Para(ω) in the obvious fashion. Let ϕ > ω and let { E(Π ϕ ) := f O(Π ϕ ) f(z) = O ( z ε ) } as z, for some ε > 1/2 Then f(w) = 1 dz f(z) 2πi Π ω z w by Cauchy s theorem, where ω (ω, ϕ). We define f(a) := 1 2πi Π ω f(z)r(z, A) dz = 1 πi iω + iω (w Π ω ) f(z 2 )zr(z 2, A) dz

SEMIGROUP THEORY VIA FUNCTIONAL CALCULUS 13 and as usual this does not depend on the choice of ω. In the usual fashion one shows that this defines an algebra homomorphism and that it respects resolvents: Φ := (f f(a)) : E(Π ϕ ) L(X), (λ z) 1 (A) = R(λ, A) (λ / Π ϕ ). Therefore, a meromorphic functional calculus in the sense of [7, Section 1.3] is defined, and there is a canonical definition of f(a) for meromorphic functions f on Π ϕ that are regularisable by elements of E(Π ϕ ). Of course one obtains a corresponding Convergence Lemma in the case that A is densely defined. If the parabola-type operator A arises as a square A = B 2 of a strong strip-type operator B, then there is an obvious composition rule: Proposition 7.5. Let B SStrip(ω) and A := B 2. Let ϕ > ω and f M(Π ϕ ) such that f(a) is defined. Then [f(z 2 )](B) is defined and f(z 2 )(B) = f(a). Proof. By [7, Proposition 1.3.6] one may suppose without loss of generality that f E(Π ϕ ). Then one may perform a computation similar to [7, p.43,p.96/97]. But it is even simpler: f(a) = 1 2πi Π ω f(z)r(z, A) dz = 1 f(z 2 ) 2z R(z 2, B 2 ) dz 2πi Im z=ω = 1 f(z 2 ) 2z R(z 2, B 2 ) dz 2πi Im z=ω = 1 f(z 2 ) [R(z, B) R( z, B)) dz 2πi Im z=ω = 1 f(z 2 )R(z, B) dz = [f(z 2 )](B) 2πi Im z =ω with the appropriate orientations of the contours. If we shift a parabola-type operator far enough to the right, we obtain a sectorial operator (of arbitrary small angle). Proposition 7.6. Let ω and let A be an operator on the Banach space X such that M R(µ, A) ( ) (µ / Π ω ). µ Im µ ω Let B θ := A + (ω/ cos θ) 2. Then B θ is sectorial of angle π/2 θ. For θ = one has even R(µ, B ) 2M (Re µ < ), Re µ whence B is actually of half-plane type. Proof. We first prove the assertion for B. Let Re µ < and find λ = x + iy such that λ 2 = µ ω 2 and y > ω. By assumption on µ, x 2 < y 2 ω 2. Then R(µ, B ) = R(λ 2, A) But Re µ = Re λ 2 + ω 2 = (y 2 ω 2 x 2 ), and hence M x2 + y 2 (y ω). Re µ x2 + y 2 (y ω) = y2 ω 2 x 2 x2 + y 2 (y ω) y2 ω 2 y(y ω) = y + ω = 1 + y y ω 2.

14 MARKUS HAASE Fix θ (, π/2). We will establish the estimate R(µ, e iθ B θ ) 2M (cos θ) Re µ (Re µ < ) and this implies what we stated in the proposition. Let Re µ < and find λ = x+iy such that y > ω and λ 2 = e iθ µ (ω/ cos θ) 2. Then µ = e iθ (λ 2 + (ω/ cos θ) 2 ) and so Re µ cos θ = y2 (ω/ cos θ) 2 x 2 + (tan θ)2xy = (y 2 ω 2 )(1 + tan 2 θ) (x y tan θ) 2 and this is > by hypothesis. All in all we can estimate R(µ, e iθ B θ ) = R(λ 2, A) = M cos θ Re µ M cos θ Re µ [ M λ (Im λ ω) = M cos θ Re µ (y 2 ω 2 )(1 + tan 2 θ) (x y tan θ) 2 x2 + y 2 (y ω) [ (y + ω)(1 + tan 2 θ) x2 + y 2 ] ( Re µ/ cos θ) x2 + y 2 (y ω) ] 2M (cos θ) Re µ. This concludes the proof. One can easily show that a composition rule f(b θ ) = [f(z + (ω/ cos θ) 2 )](A) holds when f(b θ ) is defined by the functional calculus for sectorial operators (or also, in the case θ =, by the one for halfplane-type operators). By [7, Proposition 1.3.6] one needs to know it only for functions f belonging to a generating set for the primary calculus for B θ. Here the assertions follows from Cauchy s theorem. How can we access cosine operator functions by the functional calculus? Let A be of parabola-type ω. Consider the functions f t (z) := cos(t z) (t R) which are bounded holomorphic functions on every parabola Π ϕ, ϕ >. So Cos A (t) := f t (A) (t R) (2) is a well-defined family of closed operators, and it is no surprise that the analogue of Proposition 2.4 holds. Proposition 7.7. Let A be an operator of parabola-type ω on the Banach space X. Then A generates a cosine function (Cos(t)) t R if and only if A is densely defined and Cos A (t) (defined by (2)) is a bounded operator for each t R, satisfying sup t [,2] Cos A (t) <. In this case Cos(t) = Cos A (t), t R. Proof. Suppose that A generates a cosine function Cos, fix µ > ω 2, and let x D(A). Then x D(Cos A (t)) for every t. One has Cos A (t)x = [f t /(µ + z)](a)(µ + A)x

SEMIGROUP THEORY VIA FUNCTIONAL CALCULUS 15 and t Cos A (t)x is continuous, by Lebesgue s theorem. Taking Laplace transforms and applying Fubini s theorem yields (with y := (µ + A)x) e λt Cos A (t)x dt = 1 ( e λt cos(t ) 1 z) dt R(z, A) dz y 2πi µ + z = 1 2πi Π ω Π ω λ (λ 2 R(z, A) dz y + z)(µ + z) = λ(λ 2 + A) 1 (µ + A) 1 y = λr(λ 2, A)x for λ > ω. Uniqueness of Laplace transforms yields Cos A (t)x = Cos(t)x, t. Since D(A) is dense and Cos A (t) is a closed operator, Cos A (t) = Cos(t) is a bounded operator for every t. Conversely, suppose that Cos A (t) is bounded operator for every t and that sup t [,2] Cos A (t) <. By general functional calculus, Cos A satisfies the cosine law [1, (3.88)], and so [1, Lemma 3.14.3] shows that Cos A is exponentially bounded. As seen above, Cos A (t)x is continuous for all x D(A), hence by the density of D(A), this is true even for all x X. Hence, Cos A is cosine function. But the computation above (together with density of D(A)) shows that the Laplace transform of Cos A is λr(λ 2, A), whence A is the generator of Cos A. Using a Convergence Lemma for the functional calculus on parbolas, one can aim for rational approximation results for cosine functions. Similarly, a Trotter-Kato type result holds. References [1] Wolfgang Arendt, Charles J.K. Batty, Matthias Hieber, and Frank Neubrander. Vector- Valued Laplace Transforms and Cauchy Problems. Monographs in Mathematics. 96. Basel: Birkhäuser. xi, 523 p., 21. [2] John B. Conway. A Course in Functional Analysis. 2nd ed. Graduate Texts in Mathematics, 96. New York etc.: Springer-Verlag. xvi, 399 p., 199. [3] Klaus-Jochen Engel and Rainer Nagel. One-Parameter Semigroups for Linear Evolution Equations. Graduate Texts in Mathematics. 194. Berlin: Springer. xxi, 586 p., 2. [4] Markus Haase. Spectral properties of operator logarithms. Math. Z., 245(4):761 779, 23. [5] Markus Haase. Spectral mapping theorems for holomorphic functional calculi. J. London Math. Soc. (2), 71(3):723 739, 25. [6] Markus Haase. Functional calculus for groups and applications to evolution equations. Submitted, 26. [7] Markus Haase. The Functional Calculus for Sectorial Operators. Number 169 in Operator Theory: Advances and Applications. Birkhäuser-Verlag, Basel, 26. [8] Sylvie Monniaux. A new approach to the Dore-Venni theorem. Math. Nachr., 24:163 183, 1999. Department of Pure Mathematics, University of Leeds, Leeds LS2 9JT, United Kingdom E-mail address: haase@maths.leeds.ac.uk