Functional calculi for sectorial operators and related function theory

We construct two bounded functional calculi for sectorial operators on Banach spaces, which enhance the functional calculus for analytic Besov functions, by extending the class of functions, generalizing and sharpening estimates, and adapting the calculus to the angle of sectoriality. The calculi are based on appropriate reproducing formulas, they are compatible with standard functional calculi and they admit appropriate convergence lemmas and spectral mapping theorems. To achieve this, we develop the theory of associated function spaces in ways which are interesting and significant. As consequences of our calculi, we derive several well-known operator norm-estimates and provide generalizations of some of them.


Introduction
The theory of functional calculi forms a basis for the study of sectorial operators and semigroup generators. In particular, there were two functional calculi used extensively in the research on operator semigroups and sectorial operators through the past fifty years. One of them, the Hille-Phillips (HP) functional calculus for semigroup generators, probably stemmed from the foundational monograph [37], and it became an indispensable part of semigroup theory. The systematic approach to the other one, the holomorphic functional calculus for sectoral operators, was initiated by McIntosh and his collaborators in the 1980s. While the two calculi appeared to be very useful in applications, the operator norm-estimates within them are often problematic. The estimates within the HP-calculus are direct but rather crude, and the task of getting bounds within the holomorphic functional calculus is a priori cumbersome since the calculus is not, in general, a bounded Banach algebra homomorphism.
To circumvent those problems, a number of additional tools and methods appeared in the literature. In particular, the advanced notions and techniques related to bounded H ∞ -calculus, R-boundedness, Fourier multipliers and transference were developed in depth, and one may consult [38], [41] and [35] for many of these function-theoretical developments. Moreover, various implications of positivity of functions and their derivatives (completely monotone, Bernstein, N P + -functions) were adjusted to the operatortheoretical set-up. For clarification of the role of positivity, see [51], [31] and [6], for example.
Recently, in [7], a new functional calculus was constructed, the so-called B-calculus. First of all, the B-calculus offers a simple and efficient route to operator norm-estimates for functions of semigroup generators, thus unifying a number of estimates in the literature and leading to new ones. No supplementary arguments are required and the estimates underline the strength of the calculi. Moreover, the B-calculus possesses all attributes of classical functional calculi, see [7]. When combined properly they lead to new spectral mapping theorems and generalizations of fundamentals of semigroup theory, see [8]. To put our results into a proper context and to use some of the B-calculus properties in the sequel, we briefly recall the set-up for the B-calculus, see [7] for more details.
Let B be the algebra of holomorphic functions on the right half-plane C + such that These functions have been considered in some detail in [7] (see also [53]).
In particular, every f ∈ B belongs to H ∞ (C + ) ∩ C(C + ), and B is a Banach algebra with the norm Moreover, the algebra B modulo constants is isomorphic to the holomorphic Besov space B + ∞,1 (R), see [7,Proposition 6.2]. In the setting of power bounded operators on Hilbert spaces, the unit disc counterpart of B was employed for the study of functional calculi in [47].
Let A be a densely defined closed operator on a Banach space X such that σ(A) ⊆ C + , and for all x ∈ X and x * ∈ X * . This class of operators includes two substantial subclasses, namely the negative generators of bounded C 0 -semigroups on Hilbert spaces X and the negative generators of (sectorially) bounded holomorphic C 0 -semigroups on Banach spaces X. On the other hand, every operator in the class is the negative generator of a bounded C 0 -semigroup.
The study of functional calculus based on the algebra B was initiated in [56] for generators of bounded semigroups on Hilbert spaces and in [53] for generators of holomorphic semigroups. These works adapted and extended the approach from [47] to a more demanding and involved setting of unbounded operators. Most researchers were unaware of [56] until it became accessible a few years ago. Meanwhile the line of research put forward in [56] and [53] was continued in [36] and [52] proceeding in two different directions (additional related references can be found in [7]). In [36], by means of a new transference technique, counterparts of the results from [53] were proved in the framework of bounded semigroups on Hilbert space and certain substantial subclasses of B, while [52] offered a number of generalisations and improvements of estimates from [53]. Only [52] and [53] considered all functions in B, applied to generators of bounded holomorphic semigroups in both papers. In [7], we introduced a bounded B functional calculus for all operators satisfying (1.3), and we extended the theory in [8].
for all x ∈ X and x * ∈ X * , where f (∞) = lim Re z→∞ f (z). Using (1.3) and the definition of B (and the Closed Graph Theorem), it is easy to show that f (A) is a bounded linear mapping from X to X * * , and that the linear mapping Φ A : B → L(X, X * * ), is bounded.
It was discovered in [7] that much more is true. If A belongs to any of the classes of semigroup generators mentioned above, then the formula (1.4) defines a bounded algebra homomorphism It is natural to call the homomorphism Φ A the (B-)calculus of A. It was proved in [7] that Φ A possesses a number of useful properties. In particular, it admits the spectral inclusion (spectral mapping, in the case of bounded holomorphic semigroups) theorem and a Convergence Lemma of appropriate form. The utility of the B-calculus depends on the facts that it (strictly) extends the Hille-Phillips (HP-) calculus and it is compatible with the holomorphic functional calculi for sectorial and half-plane type operators.
Moreover, the B-calculus Φ A is the only functional calculus that one can define for A satisfying (1.3) and for functions in B. Indeed, let A be an operator on X with dense domain, and assume that σ(A) ⊆ C + . A (bounded) B-calculus for A is, by definition, a bounded algebra homomorphism Φ : B → L(X) such that Φ((z + ·) −1 ) = (z + A) −1 for all z ∈ C + . As shown in [8], if A admits a B-calculus, then the resolvent assumption (1.3) holds, and the calculus is Φ A .
While the B-calculus is optimal and unique for generators of Hilbert space semigroups, the situation is far from being so for generators of bounded holomorphic semigroups on Banach spaces (as this paper will, in particular, show). Thus using the B-calculus ideology as a guiding principle, it is natural to try to extend it beyond the Besov algebra B keeping all of its useful properties such as availability of good norm-estimates, spectral mapping theorems, Convergence Lemmas, compatibility with the other calculi, etc. Moreover, it is desirable to cover all sectorial operators regardless of their sectoriality angle.
In this paper we will construct some functional calculi encompassing wider classes of functions (including some with singularities on iR) and providing finer estimates for all negative generators of (sectorially) bounded holomorphic semigroups, and eventually for all sectorial operators. Functional calculi for generators of some classes of bounded holomorphic semigroups were constructed in [25], [26], [27], [28], and [40]. However, most of the results in those papers concern sectorial operators of angle zero, and the approaches there are based on fine estimates for the corresponding semigroups rather than fine analytic properties of resolvents.
Let A be a densely defined sectorial operator of sectorial angle θ A ∈ [0, π) on a Banach space X. It is well known that −A is the generator of a (sectorially) bounded holomorphic C 0 -semigroup on X if and only if A is sectorial and θ A < π/2 (we may write A ∈ Sect(π/2−) for this class). In this paper we address the question whether the B-calculus for A can be extended to more functions. Since the resolvent of A satisfies the estimate for all ψ ∈ (θ A , π), a direct way to define an appropriate function algebra would be to introduce a Banach space of functions f which are holomorphic on sectors Σ ψ := {z ∈ C : | arg(z)| < ψ} such that In order to apply this to all A ∈ Sect(π/2−), f should be holomorphic on C + and the assumption (1.5) should hold for all ψ ∈ (0, π/2), and in order to provide an estimate which is uniform in θ it is desirable to have sup ψ∈(0,π/2) f ψ < ∞. To our knowledge, no spaces of this type have been studied systematically in the literature, although they appear naturally in [38,Appendix H2 and Chapter 10.2], [34,Section 6] or [33,Appendix C]. This class of functions is strictly included in each of the spaces D s , s > 0 (see Proposition 4.15 and a discussion following it), which we now define.
To define a functional calculus for all A ∈ Sect(π/2−), we let D s , s > −1, be the linear space of all holomorphic functions f on C + such that If f ∈ D s , then there exists a finite limit f (∞) := lim |z|→∞, z∈Σ ψ f (z) for all ψ ∈ (0.π/2).
For every s > −1 the space D s equipped with the norm f Ds := |f (∞)| + f D s,0 , f ∈ D s , is a Banach space, but not an algebra. However, the spaces D s increase with s, and we prove in Lemma 3.21 that is an algebra.
Let f ∈ D ∞ , so f ∈ D s for some s > −1, and let A be sectorial with θ A < π/2. Define Then f Dσ (A) = f Ds (A) whenever σ > s. The following result sets out other properties of this functional calculus. The proof will be given in Section 7.
Theorem 1.1. Let A be a densely defined, closed operator on a Banach space X such that σ(A) ⊂ C + . The following are equivalent: (i) A ∈ Sect(π/2−); (ii) There is an algebra homomorphism Ψ A : D ∞ → L(X) such that and Ψ A is bounded in the sense that there exist constants C s , s > −1, such that, for every f ∈ D s , When these properties hold, Ψ A is unique, and it is defined by the formula (1.7): The homomorphism Ψ A will be called the D-calculus for A. It will be shown in Section 7 that the D-calculus is compatible with the Hille-Phillips calculus and the holomorphic calculus for sectorial operators, and a spectral mapping theorem is given in Theorem 9.3. Corollary 7.8 provides a version of this functional calculus based on the Banach algebra H ∞ (C + ) ∩ D s for a fixed value of s.
Note that H 1 (Σ π/2 ) coincides with the classical Hardy space H 1 (C + ) in the right half-plane C + . It is well-known that (H 1 (Σ ψ ), · H 1 (Σ ψ ) ) is a Banach space, and every f ∈ H 1 (Σ ψ ) has a boundary function on ∂Σ ψ . The boundary function exists as the limit of f in both an L 1 -sense and a pointwise (a.e.) sense. Moreover, the norm of f in H 1 (Σ ψ ) is attained by the L 1 -norm of its boundary function. See Section 4.1 for a succinct approach to the Hardy spaces on sectors.
Any function f ∈ H ψ has a finite limit f (∞) := lim t→∞ f (t), and moreover f ∈ H ∞ (Σ ψ ). Then H ψ becomes a Banach algebra in the norm The relationship between these spaces and the spaces D s for all s > −1 is set out in Corollary 4.3 and Lemma 4.13; in particular, for each s > −1, H π/2 is contained in D s , and D s is embedded in H ψ for ψ < π/2.
To make use of the angle of sectoriality of A, we can adjust the D-calculus to sectors as follows. If f ∈ H ψ where ψ ∈ (θ A , π), γ = π/(2ψ), and , and hence f 1/γ ∈ D 0 . This observation allows us to extend the D-calculus to the class of all sectorial operators, and makes the next definition (based on the Dcalculus) natural and plausible.
If A is sectorial and ψ ∈ (θ A , π), define One can prove (see (8.2)) that (1.11) Then (1.10) and (1.11) hold for any γ ∈ (1, π/(2θ)), and the definition of f H (A) does not depend on the choice of ψ. Now we are able to formalize our extension of the D-calculus as follows.
Theorem 1.2. Let A be a densely defined operator on a Banach space X such that σ(A) ⊂ Σ θ , where θ ∈ [0, π). The following are equivalent: The D-calculus can be given a more succinct form, by replacing (1.10) with the somewhat more transparent formula (1.12) below, inspired by results in [12]. Then where the integral converges in the uniform operator topology, and See Section 8.2 for details. The D-calculus and the H-calculus possess natural properties of functional calculi such as spectral mapping theorems and Convergence Lemmas. These properties are studied in Section 9.
The strength of the constructed calculi is illustrated by several examples showing that they lead to sharper estimates than those offered by other calculi (see Section 12 for one example). Moreover, the theory developed in this paper is successfully tested by deriving several significant estimates for functions of sectorial operators from the literature. In particular, in Section 10, we provide a proof of permanence of the class of sectorial operators under subordination and we revisit a few basic results from semigroup theory.
In developing the Dand H-calculi we prove a number of results of independent interest in function theory. Apart from the theory of the spaces D s and H ψ , their reproducing formulas, and boundedness of the associated operators elaborated in this paper, we emphasize the property (4.9) in Corollary 4.3 yielding isometric coincidence of spaces of Hardy type, Theorem 4.12 on Laplace representability of Hardy-Sobolev functions, and Theorem 5.10 on the density of rational functions in Hardy-Sobolev spaces.
Added Note: During the preparation of this paper, we have become aware of a paper by Arnold and Le Merdy [5] who consider negative generators of bounded C 0 -semigroups on Hilbert space. Inspired by ideas in [47] for the discrete case, they extend the B-calculus for those operators, to a strictly larger Banach algebra A in which B is continuously embedded. Their extension is complementary to our extensions to the Dand H-calculi for negative generators of bounded holomorphic C 0 -semigroups on Banach spaces. We are grateful to Loris Arnold for pointing out several defects in the original version of this paper.

Preliminaries
Notation. Throughout the paper, we will use the following notation: For f : C + → C, we say that f has a sectorial limit at infinity if lim |z|→∞,z∈Σ ψ f (z) exists for every ψ ∈ (0, π/2). Similarly, f has a sectorial limit at 0 if lim |z|→0,z∈Σ ψ f (z) exists for every ψ ∈ (0, π/2). We say that f has a half-plane limit at infinity if lim We say that f has a full limit at infinity or at zero if exists in C. The notation f (∞) and f (0) may denote a sectorial limit, a half-plane limit, or a full limit, according to context.
For a ∈ C + , we define functions on C by e a (z) = e −az ; r a (z) = (z + a) −1 , z = −a.
We use the following notation for spaces of functions or measures, and transforms, on R or R + : Hol(Ω) denotes the space of holomorphic functions on an open subset Ω of C, H ∞ (Ω) is the space of bounded holomorphic functions on Ω, and For a Banach space X, X * denotes the dual space of X and L(X) denotes the space of all bounded linear operators on X. The domain, spectrum and resolvent set of an (unbounded) operator A on X are denoted by D(A), σ(A) and ρ(A), respectively.
If (X , · X ) and (Y, · Y ) are normed spaces of holomorphic functions on domains Ω X and Ω Y , we will use notation as follows: Boundaries of all of the sectors appearing in this paper will be oriented from top to bottom.
Beta function. The Beta function appears in many places in the paper. It is defined for s, t > 0 by In particular, for s > −1 we will use the relations see [48, p. 386]. We note also the following limit properties: Proof conventions. We will make extensive use of the dominated convergence theorem, often for vector-valued functions. With a few exceptions, we will not give details of the relevant dominating functions, as they are usually easily identified.
We will also use the following elementary lemma on several occasions. See [21, p.21, Lemma 1] for a proof.
We will use Vitali's theorem several times, usually for holomorphic vectorvalued functions. We refer to the version given in [4,Theorem A.5].
Let X be a Banach space of holomorphic functions on a domain Ω X such that the point evaluations δ z : f → f (z), z ∈ Ω X , are continuous on X . Let (Ω, µ) be either an interval in R with length measure or an open set in C with area measure, and F : Ω → X be a continuous function such that exists as a Bochner integral in X and it can be approximated by Riemann sums. It follows that G belongs to the closed linear span of {F (t) : t ∈ Ω} in X . Now assume that F : Ω → X is locally bounded, where Ω is an open set in C, and that λ → F (λ)(z) is holomorphic on Ω for all z ∈ Ω X . We will use the fact that F : Ω → X is holomorphic in the vector-valued sense, without further comment. The result at this level of generality can be seen from [4,Corollary A.7], using the point evaluations as separating functionals. An alternative is to show that F is continuous, and then apply Morera's theorem. If the definition of F is by an integral formula, it may also be possible to apply a standard corollary of the dominated convergence theorem which leads to an integral formula for the derivative F ′ .

The Banach spaces D s and their reproducing formulas
In this section we introduce some spaces of holomorphic functions to which we will extend the B-calculus of operators in Section 7 onwards.
3.1. The spaces V s . Let s > −1, z = α + iβ, and let V s be the Banach space of (equivalence classes of) measurable functions g : C + → C such that the norm is finite, where S is the area measure on C + . Note that The following property of functions from V s is an essential element in the arguments which lead to the representations for functions in V s in Proposition 3.7 and for D s in Corollary 3.10, and eventually to the definition of a functional calculus for operators in (7.5).
Lemma 3.1. Let g ∈ V s be holomorphic, where s > −1. For every k ≥ 1 and every ψ ∈ (0, π/2), Proof. Let g ∈ V s be holomorphic, ψ ∈ (0, π/2), ψ ′ = (π/2 + ψ)/2 and b ψ = sin((π/2 − ψ)/2) = cos ψ ′ . If z ∈ Σ ψ , then Let r ∈ (0, b ψ |z|). By Cauchy's integral formula for derivatives, Multiplying by r k and integrating with respect to r over (0, b ψ |z|) gives and then 3.2. The spaces D s and the operators Q s . We now define a linear operator Q s on V s , s > −1. It will play a similar role to the operator Q on W considered in [8,Section 3], where W is the Banach space of all (equivalence classes of) measurable functions g : C + → C such that Indeed the definition of Q 1 is formally the same as the definition of Q in [8], but the domain V 1 of Q 1 is larger than W.
Indeed, both sides of this equation are holomorphic functions of λ ∈ C + , and they coincide for λ ∈ (0, ∞), so they coincide for all λ ∈ C + , by the identity theorem for holomorphic functions. Putting λ = z + α − iβ, we obtain For s > −1 let D s be the linear space of all holomorphic functions f on C + such that f ′ ∈ V s , equipped with the semi-norm If σ > s > −1, then it is immediate from (3.2) that D s ⊂ D σ . We will exhibit some functions in D s later in this section and in Section 3.3.
In the rest of this section we will obtain a reproducing formula (3.29) for functions from D s and we will describe some basic properties which will be relevant for the sequel. To this aim, we first define and study the behaviour of operators Q s on the scale of D s -spaces. Recall that in [8,Proposition 3.1] we showed that Q maps W into B. However Q s does not map the whole of V s into D s . For s > −1, a function g ∈ V s for which Q s g / ∈ D s , can be defined as follows: 1 < ρ < 2 − sin ϕ, π/4 < ϕ < π/2, and g(z) = 0 for all other points in C + . We do not give details in this paper. Instead we will show in Propositions 3.6 and 3.7 that Q s maps V s boundedly into D σ for any σ > s, and it maps holomorphic functions in V s into D s . We need the following auxiliary lemma which will be useful in a number of instances. h(t) dt (t 2 + 2t cos(ϕ + ψ) + 1) β dψ, ϕ ∈ (−π/2, π/2). Proof. Since G h,β,s (−ϕ) = G h,β,s (ϕ), we may assume that ϕ ∈ [0, π/2). Now , and we estimate these two integrals separately.
In particular, taking γ = 1 and a fixed s > 0, This estimate will play a crucial role in the proof of Theorem 1.1.
Next we consider some functions which appear frequently in the studies of holomorphic C 0 -semigroups.
We will show here that f ν ∈ D s if and only if s > ν. Moreover, if s > ν, then This estimate will be crucial for operator estimates in Section 10.
We have We use the estimates If s > ν, we obtain This establishes the claims above.
Finally we consider a function which will play an important role in our constructions of functional calculi in Section 8.
For s = 0, we have See [48, 2.5.16, (38)] for the evaluation of the integral with respect to ψ.
Proof. The first statement is immediate from the definitions of the spaces and (3.2).
For the third statement, the same estimation, but with s and σ interchanged, and using the first case of (3.19) with γ = s + 1 and h(t) = 1 + t s , shows that π 2 s (s + 1) Q s g D σ,0 ≤ K h,(s+2)/2,σ g Vs .
This establishes the third statement.
For insight on why r σ+1 λ D s,0 appears in the proof above, we refer the reader to the proof of Theorem 5.1.
Proposition 3.7. Let g ∈ V s be holomorphic, where s > −1. Then Q s g ∈ D s and (Q s g) ′ = g.
Proof. First we consider the case when g ∈ V n , where n ∈ N ∪ {0}. Then Q n g is holomorphic and (3.6) holds for s = n. It suffices to show that (Q n g) ′ = g. Let From (3.1) and Fubini's theorem, we see that ∞ 0 α n I(α) dα < ∞. Hence I(α) < ∞ for almost all α > 0 and it follows that there exists a sequence (α j ) j≥1 such that Similarly ∞ −∞ J(β) dβ < ∞, and so there exist sequences (β ± k ) k≥1 such that Let z ∈ C + be fixed. Take α > 0 with I(α) < ∞, and let j be sufficiently large that α j > 2α + Re z, and k be sufficiently large that β − k < Im z < β + k . We may apply the Cauchy integral formula around the rectangle with vertices α + iβ ± k and α j + iβ ± k , and we obtain 2π n! g (n) (2α + z) Letting k → ∞, we obtain Letting j → ∞, we obtain This holds for almost all α > 0. Substituting this into (3.6), and then integrating by parts and using Lemma 3.1, we infer that By Lemma 3.1, the integral ∞ 0 g ′ (α + z) dα converges absolutely and uniformly for z in compact subsets of C + . So, differentiating under the integral sign we get (Q n g) ′ = g. Now consider the case when s = n + δ > −1, where n ∈ N ∪ {−1, 0}, δ ∈ (0, 1), and g ∈ V n is holomorphic. Then g ∈ V n+1 , n + 1 ∈ N ∪ {0}, and (3.24) gives for z ∈ C + , t > 0 and almost all α > 0. We obtain from (3.14) and (3.13) that As in (3.25), it follows that (Q s g) ′ = g.
Proof. This is immediate from Proposition 3.7 and (3.8).
Remark 3.9. The proof of the property (Q n g) ′ = g ′ , n ≥ 0, in Proposition 3.7 uses just improper convergence of the integrals for all k ≥ 0. Indeed if g ∈ V n , then (3.26) holds for k = n by (3.24) and the definition of norm in V n . If n ≥ 1, then using Lemma 3.1, we infer that g (n−1) (α + z) = − ∞ α g (n) (s + z) ds, and hence by Fubini's theorem Repeating this argument, we conclude that (3.26) holds also for k such that 0 ≤ k < n. If k > n, then (3.26) follows directly from (3.24) and the inclusion V n ⊂ V k .
The following representation of functions in D s has appeared in [2, Corollary 4.2] (see also [3, Lemma 3.13.2] for the case s = 1).
Corollary 3.11. For every s > −1 the space D s equipped with the norm Proof. Let s > −1 be fixed and let (f k ) ∞ k=1 be a Cauchy sequence in D s . Then (3.7) and Vitali's theorem imply that (Q s f ′ k ) ∞ k=1 converges uniformly on each Σ ψ to a limit g which is holomorphic on The argument used in the proof of Corollary 3.11 also provides the following lemma of Fatou type (see also Lemma 4.11).
Remark 3.14. Corollary 3.13 implies that the point evaluation functionals δ z , z ∈ C + , are continuous on D s , s > −1. Using (3.19) and the principle set out in Section 2, we see that the function λ → r γ λ is holomorphic from C + to D s , for any s > −1, γ > 0.

3.3.
More functions in D s and their properties. In this section we give more examples of functions from D s and note some additional elementary properties which will be relevant for the sequel.
Thus B is continuously included in D s .
s > 0. We will show in Corollary 5.2 that B is dense in D s for every s > 0, and hence D s is dense in D σ for all σ > s > 0. On the other hand, we will show in Corollary 5.2 that B is not dense in D ∞ s for s > 0. For f ∈ Hol(C + ), let Lemma 3.17. Let s > −1 and t > 0. Then Proof. Note that  More generally, for The function log(1 + z)e −z is in D s , s > 0, but is also unbounded on every right half-plane.
2. Since e −z ∈ D s for s > 0, it follows from Lemma 3.17 that the functions e −t/z are in D s for all t > 0, s > 0. This shows that functions f ∈ D s may not have full limits at infinity or at zero. However, the properties (3.27) and (3.28) in Corollary 3.10 establish values for f at infinity and at zero as sectorial limits.
3. The spaces D s , s > −1, are invariant under shifts given by Indeed these operators form a bounded C 0 -semigroup on D s . See Section 11 for a proof. On the other hand, D s are not invariant under the vertical shifts when τ ∈ iR, as we see in the following example.
Since | Re arccot(z)| ≤ π/2, g ∞ = exp(π/2) = g(∞). For s > −1, we have However the boundary function of g is not continuous at z = ±i. Indeed, for a fixed ǫ > 0, Note that g(ǫ + i) does not converge as ǫ → 0+. This means that if f (z) := g(z − i), then f does not have a sectorial limit at 0 and therefore does not belong to D s for any s > −1. Thus D s is not invariant under vertical shifts.

3.4.
Bernstein functions and D s . Recall that a holomorphic function g : where a ≥ 0, b ≥ 0 and µ is a positive Borel measure on (0, ∞) such that The following properties of Bernstein functions g will be used (these properties differ slightly from those used in [6]): Here the first inequality follows from taking the real parts in (3.30), and the second inequality is shown in [7, Section 3.5, (B3)].
Further information on Bernstein functions can be found in [51].

By
s is the closure of the range of f . In particular, the spectral radius of f is f ∞ . Now we consider the linear space We will show that D ∞ is an algebra, which opens the way to an operator functional calculus on D ∞ . Lemma 3.21. For s, σ > −1, let f ∈ D s and g ∈ D σ . Then Hence, D ∞ is an algebra.
3.6. Derivatives of functions in D s . This section further clarifies the behaviour of the derivatives of functions from D s , and Lemma 3.22 is of independent interest. Corollary 3.23 will be used in Section 10.3. For m, n ∈ N, the notation z m f (n) denotes the function mapping z to z m f (n) (z). Moreover, f t is the function mapping z to f (tz) .
The argument is similar to Example 3.3 and the proof of Proposition 3.6. By Corollary 3.10, for fixed σ > s, Then similar estimates to those in Example 3.3 and Proposition 3.6 give Hence, by Fubini's theorem, since the D s+1 -norm is invariant under the change of variable z → rz (Lemma 3.17(iii)).
The following corollary is easily proved by induction.

4.
Hardy-Sobolev algebras on sectors 4.1. H p -spaces on the right half-plane and their norms. In this section and in Section 4.2 we will study the Hardy spaces H 1 (Σ ψ ) defined on sectors Σ ψ , ψ ∈ (0, π). The properties of such spaces are similar to the properties of the classical Hardy space H 1 (C + ), though their theory seems to be more involved. The Hardy spaces H p (Σ ψ ) have been studied, mostly for p > 1, but the results are scattered around various places in the literature, which is often obscure, and some proofs contain rather complicated, incomplete or vague arguments. We propose a streamlined (and probably new) approach avoiding the use of Carleson measures or log-convexity, and we obtain a new result (Corollary 4.3) on the way. The case p = 1 does not require any significant adjustments as we illustrate below. Standard references for the theory of Hardy spaces on the right half-plane are [21] and [29].
We set out the situation when ψ = π/2 in this section, and the case of general ψ in Section 4.2. Although we are mainly interested in H 1 -spaces, we present statements which are valid for H p -spaces with p ∈ [1, ∞), since the arguments are the same for all such p.
Lemmas 4.1 and 4.2 imply the next statement.
Note that the two-sided estimate (4.10) was proved in [50] and [42] in a more complicated way (see also [1], [19], [45], [54], [55]). The coincidence of norms in (4.9) seems not to have been noted before. It appears to be quite useful, as we will see in the proof of Corollary 4.5 below.
Remark 4.4. The two-sided estimate (4.10) is sharp (and cannot be improved). Indeed, let p ∈ [1, ∞) and let Then for all k, we have f k H p (C + ) = 1 and, by direct estimates, Thus, and lim In fact, for all f ∈ H p * (Σ π/2 ), the norm f H p * (Σ π/2 ) is attained at the boundary of Σ ψ/2 .

4.2.
The spaces H 1 (Σ ψ ). Now using the results of Section 4.1 for ψ = π/2, we develop basic properties of H 1 (Σ ψ ) for any ψ ∈ (0, π). Define the Hardy space H 1 (Σ ψ ) on the sector Σ ψ to be the space of all functions f ∈ Hol(Σ ψ ) such that (4.14) f We will also consider a non-symmetric version of H 1 (Σ ψ ), defined as , then the limits f (re ±iψ ) := lim ϕ→±ψ f (re iϕ ) exist a.e., and in the L 1 -sense with respect to r. Moreover Proof. The proof of (i) is clear, and (ii) is a direct verification. For ψ = π/2, the statements (iii) and (iv), excluding (4.16), were proved in Corollaries 4.3 and 4.5, and (v) is well-known. Then the general cases are reduced to the case when ψ = π/2, by means of (ii).
Remark 4.7. In addition to (4.17), it is also possible to prove that This requires additional techniques, and it is not used in this paper.

4.3.
Functions with derivatives in H 1 (Σ ψ ). For ψ ∈ (0, π), let us introduce the space In view of Corollary 4.3, and we may sometimes use the notation Such function spaces are often called Hardy-Sobolev spaces, and we will also use this terminology sporadically. Spaces more general than H ψ appear in [20]. Namely, for f ∈ Hol(Σ ψ ) it was required in [20] that the boundary values of f exist and belong (after an appropriate "rescaling") to a Besov space B s ∞,1 , s > 0. One can develop a similar approach to those spaces, but we do not see much advantage in such generality within the present context.
(i) The function f extends to a continuous bounded function on Σ ψ .
It follows from Theorem 4.8 that H ψ is an algebra for every ψ ∈ (0, π). We define a norm on H ψ by This is easily seen to be an algebra norm. Theorem 4.8(iii) shows that is an equivalent norm on H 1 (Σ ψ ).
The following lemma is simple, but crucial for our theory. The completeness of the norm is a standard fact, the scale-invariance is trivial, and the final isomorphism follows from Theorem 4.6(ii). Lemma 4.9. For every ψ ∈ (0, π), the space (H ψ , · H ψ ) is a Banach algebra. For t > 0, the map f (z) → f (tz) is an isometric algebra isomorphism on H ψ . Moreover, for any ψ 1 , ψ 2 ∈ (0, π), the map is an isometric algebra isomorphism.
We now give some examples of functions in H ψ which will play important roles in subsequent sections of this paper. The first example is of similar type to Example 3.3.
More examples can be found in Sections 4.5 and 4.6. In particular, Lemma 4.13 shows that the restriction of any function in D ∞ to Σ ψ , ψ ∈ (0, π/2), belongs to H ψ .
The following lemma is a result of Fatou type closely related to Corollary 3.12.
Proof. By Theorem 4.8, the functions {f k : k ≥ 1} are uniformly bounded on Σ ψ . By Vitali's theorem, f is holomorphic, and f ′ k (z) → f ′ (z) as k → ∞ for each z ∈ Σ ψ . By Fatou's Lemma, for |ϕ| < ψ, where (LM, · ) HP is the Hille-Phillips algebra as in Section 2. In particular, it shows that the Laplace transforms of singular measures on (0, ∞) are not in H 1,1 (C + ), which may be of interest.
and then by [49,Theorem 1.9.2] we obtain that f ∈ LM and (4.25) holds.

Now let
From the uniqueness properties of Laplace transforms it follows that Thus µ is absolutely continuous on (0, ∞), with Radon-Nikodym derivative g. Since µ is a bounded measure, g ∈ L 1 (R + ), and Hence f = f (∞) + Lg.
The spaces E ψ are studied in [38,Chapter 10]; see also [38, Appendix H2], [34, Section 6] and [33,Appendix C]. To ensure the algebra property and to relate the spaces to the H ∞ -calculus, the authors considered the algebras H ∞ (Σ ψ ) ∩ E ψ . Lemma 4.15 shows that the spaces E ψ are fully covered within the framework of the algebras D ∞ and H ψ . These algebras will be associated to the more powerful functional calculi constructed in Corollary 7.8 and Theorem 8.2. 4.6. Bernstein functions and H ψ . To illustrate the relevance of the Hardy-Sobolev spaces, we show that the "resolvent" of a Bernstein function belongs to an appropriate Hardy-Sobolev space. This observation will be used in Section 10 to provide a new proof of the permanence of subordination for holomorphic semigroups, one of the main results of [31]; see also [6] and [7].

4.7.
Representations for functions in H ψ . In this section we derive a reproducing formula for functions from H ψ and obtain certain alternative representations for its kernel.

Dense sets in D s and H ψ
In this section we establish some results concerning density and approximations in our spaces.

5.1.
Dense subsets of D s and some applications. Let R(C + ) be the linear span of {r λ : λ ∈ C + }, and R(C + ) be the sum of R(C + ) and the constant functions. Using Example 4.10(1) and Lemma 4.13, we have Theorem 5.1. The space Proof. Let R Ds (C + ) be the closure of R(C + ) in D s , and let f ∈ D s . It follows from Example 3.3 and Remark 3.14 (or a direct estimate) that the function is continuous from C + to D s , and it is Bochner integrable with respect to area measure S on C + . Since point evaluations are continuous on D s (Remark 3.14), it follows from Corollary 3.10 that as the Bochner integral of a continuous function. Hence Q s f ′ belongs to the closure in D s of the linear span of the range of the integrand, which is contained in R Ds (C + ). Now f = f (∞) + Q s f ′ which is in the closure of R(C + ) in D s .

From Proposition 3.6, we have the continuous inclusion
and from (5.1), Theorem 4.12 and Proposition 3.15, we have Here LL 1 + C is the sum of LL 1 and the constant functions, and it is a closed subspace of LM. The following density results hold.  Proof. The first three statements are immediate from Theorem 5.1.
Since any function in B extends continuously to iR, the same holds for the closure of B in D ∞ s when s > 0. The function f (z) = e −1/z ∈ D ∞ s for s > 0, (see Remarks 3.18 and Example 3.4 with ν = 0), but f is not continuous at z = 0. This establishes the final statement.
The function g(z) = exp(arccot z) considered in Example 3.19 provides another example of a function from D ∞ 0 which is discontinuous on iR, and so does not belong to the closure of B in D ∞ s for s > 0. In order to obtain operator norm-estimates for functions f (n) applied to semigroup generators (see Theorem 10.3), we will need a stronger version of Corollary 3.23 on differentiability of t → f (t·) in the D s -norm. We first prove a lemma, and we present the stronger statement in Corollary 5.5.
Then, for every s > −1, Proof. We have Hence for some C τ,λ . Since for any s > −1, the dominated convergence theorem implies (5.2).
Corollary 5.4. Let f ∈ D s , s > −1. For τ > 0, let zf ′ τ denote the function mapping z to zf ′ (τ z). Then Proof. Let τ > 0 be fixed, and Since the linear span of the functions r λ and the constants is dense in D s (see Theorem 5.1), the assertion follows.
Corollary 5.5. Let f ∈ D s , s > −1, and let Then G and F n map (0, ∞) into D s+n , G is n-times differentiable as a function from (0, ∞) to D s+n , and The proof is by induction on n. The case n = 1 is given by Corollary 5.4. Assume that G (n) = F n with values in D s+n , and let f n (z) = z n f (n) (z). Then G (n) (t)(z) = F n (t)(z) = t −n f n (tz).
By Corollary 5.4 applied to f n ∈ D s+n , G (n) is differentiable with respect to t, when considered as a function with values in D s+n+1 . Finally,

Approximations via change of variables.
Here we consider approximations of f from D s and H ψ by the functions f γ (z) = f (z γ ) as γ → 1−.

5.3.
Density of rational functions in H ψ . In Theorem 5.1 and Corollary 5.2, we established that R(C + ) and several larger spaces are dense in D s , for s > −1 or s > 0. In particular, we noted that H 1,1 (C + ) is dense in D s .
Let ψ ∈ (0, π) and R(Σ ψ ) be the linear span of {r λ : λ ∈ Σ π−ψ }. Let R H (Σ ψ ) be the closure of R(Σ ψ ) in H ψ,0 . We will prove that H ψ,0 = R H (Σ ψ ). Thus the rational functions which vanish at infinity and have simple poles outside Σ ψ are dense in H ψ modulo constants. This fact may be known, but we did not find it in the literature. Our proof involves several lemmas given below and it may be of interest as a piece of function theory. The following lemma, relating to the function spaces E ϕ from Definition (4.29), is the key step in our proof.
We have now proved the assertions (5.14) and (5.15). Thus we obtain (5.13) under the additional assumption that f (0) = 0. Now let f ∈ H ϕ,0 be arbitrary. Then consider and note that Then, by the above, f 0 ∈ R H (Σ ϕ ), and hence (5.13) holds.
By Lemma 5.9, f γ ∈ R H (Σ ψ ). Thus we obtain the following result that the rational functions with simple poles are dense in H ψ,0 .

Convergence Lemmas
In this section we formulate Convergence Lemmas for functions in D s and H ψ , composed with fractional powers. Let g ∈ D s satisfy g(0) = g(∞) = 0.
For γ ∈ (0, 1) and k ∈ N, let f k,γ and g γ be given by (6.1). Then Proof. The proof is similar to Lemma 6.2.

The D-calculus and its compatibility
Here we discuss functional calculus for sectorial operators A of angle less than π/2 and functions f ∈ D ∞ . Since f is bounded on a closed sector containing the spectrum of A (Corollary 3.10), f (A) may be considered via the extended holomorphic (sectorial) calculus. If A is injective then f (A) can be defined that way as a closed operator, but we will show that f (A) is a bounded operator when f ∈ D ∞ . Our methods provide estimates for f (A) , and we will adapt the results in Section 8 to take account of the angle of sectoriality, by using fractional powers of operators (cf. Corollary 4.14).
Recall that a densely defined operator A on a Banach space X is sectorial of angle θ ∈ [0, π/2) if σ(A) ⊂ Σ θ and, for each ϕ ∈ (θ, π], The sectorial angle θ A of A is the minimal such θ. Note that M ϕ (A) is a decreasing function of ϕ.
Let Sect(θ) stand for the class of all sectorial operators of angle θ for θ ∈ [0, π/2) on Banach spaces, and denote Sect(π/2−) := θ∈[0,π/2) Sect(θ). Then A ∈ Sect(π/2−) if and only if −A generates a (sectorially) bounded holomorphic C 0 -semigroup on X of angle (π/2) − θ A , in the sense that the semigroup has a holomorphic extension to Σ (π/2)−θ A which is bounded on each smaller subsector. Note that these semigroups are sometimes called sectorially bounded holomorphic semigroups in the literature. However, in this paper, we will adopt the convention that bounded holomorphic semigroups are bounded on sectors. We will denote the semigroup as (e −tA ) t≥0 , and e −tA then agrees with e t (A) defined in the Hille-Phillips calculus, where e t (z) = e −tz . One may consult [35] for the general theory of sectorial operators, and [4,Section 3.7] for the theory of holomorphic semigroups.
Let A be a closed, densely defined operator on a Banach space X such that Then A(z +A) −1 ≤ M A +1, z ∈ C + , and Neumann series (see [53, Lemma where θ := arccos(1/(2M A )) < π/2. So A ∈ Sect(θ) ⊂ Sect(π/2−). Conversely, if A ∈ Sect(θ) where θ ∈ [0, π/2), then (7.2) holds. Thus −A generates a bounded holomorphic semigroup if and only if (7.2) holds. The constant M A is a basic quantity associated with A, and we call it the sectoriality constant of A. Note that M tA = M A for all t > 0.
A set S of sectorial operators on the same Banach space X is uniformly sectorial of angle θ if S ⊂ Sect(θ) and, for each ϕ ∈ (θ, π), there exists C ϕ such that M ϕ (A) ≤ C ϕ for all A ∈ S. Thus S is uniformly sectorial of some angle θ < π/2 if and only if each A ∈ S satisfies (7.2) and sup A∈S M A < ∞.
In the presentation of the D-calculus that follows, we assume that the reader is familiar with the holomorphic functional calculus for sectorial operators, as in [35], and in particular with the Hille-Phillips calculus for negative generators of bounded C 0 -semigroups. We will make extensive use of fractional powers of sectorial operators in the form (A + z) −γ where γ > 0. If γ is not an integer, these operators are fractional powers which can be defined in many ways (see [44]), including using the holomorphic functional calculus (see [35,Chapter 3]). All these approaches are consistent with each other. Since D ∞ = ∞ n=0 D n , it is possible to define the D-calculus without using fractional powers, and this would simplify some proofs (for example, Lemma 7.1 becomes trivial, and the formulas (7.4) and (7.8) would not be needed). Thus we could define the D-calculus without using fractional powers, and in particular we could define the fractional powers (z + A) −γ for all γ > 0. This definition would be consistent with other definitions (see Theorem 7.4). Then we could define the D s -calculus for all s > −1 in the way described below, using fractional powers in (7.5).
The following simple lemma for fractional powers is a version of the moment inequality applied to the sectorial operator (A + z) −1 .
Let γ ∈ (0, 1). By the compatibility of our calculus with the holomorphic functional calculus for fractional powers, we may use the following standard Stiletjes formula (see [4, (3.52)], for example): Next, let z = ρe iϕ ∈ C + . Then, using Cauchy's theorem, In other cases, γ = (⌈γ⌉ − 1) + δ where δ ∈ (0, 1), and the estimate follows from the two previous cases. Now let s > −1 be fixed and let f ∈ D s . We define Note that when s = 1 and f ∈ B, (7.5) coincides with the definition of f (A) as given by the B-calculus in [7], cf. (1.4).
This definition is valid as the following simple proposition shows.
1. The map f → f Ds (A) is bounded from D s to L(X).
2. For f ∈ D s , in the operator norm topology.
Proof. Lemma 7.1 and (7.5) imply that Thus the boundedness of the map f → f Ds (A) from D s to L(X) follows.
By a standard Laplace transform representation for negative fractional powers [4, (3.56)], By the dominated convergence theorem, z → (A + z) −(s+1) is continuous (even holomorphic) on C + in the operator norm topology.
Remark 7.3. The property (7.6) can be compared with Corollary 11.2 where a direct proof is given that the shifts form a bounded holomorphic C 0semigroup on D s . To deduce (7.6), one also needs that if f ǫ (z) = f (z + ǫ), then (f ǫ ) Ds (A) = f Ds (A + ǫ). By Theorem 5.1, it suffices that this holds for f = r λ , λ ∈ C + , i.e., to show that (r λ ) Ds (A) = (λ + A) −1 . We show this in Theorem 7.6, but the argument uses (7.6).  (iii) If f ∈ LM ∩ D s , then In particular, (7.12) holds if f ∈ LM and s > 0.
Now we no longer assume that A is injective. We infer by (7.10) that for all ǫ > 0 and σ ≥ s. Letting ǫ → 0 and using Proposition 7.2, we obtain the assertion (7.11).
Finally, if f ∈ LM ∩ D s for some s > −1, then f ∈ B ∩ D s , and using (7.11), we have Recall that D s ⊂ D σ if −1 < s ≤ σ, and the space is an algebra, by Lemma 3.21. Thus it is a plausible and natural domain for a functional calculus, which we now define. Theorem 7.6. Let A ∈ Sect(π/2−). The formula (7.5) defines an algebra homomorphism: Moreover, (ii) Ψ A is bounded in the sense that for every s > −1 there exists C s (A) such that Specifically, (7.7) holds.
Moreover, Ψ A is the unique algebra homomorphism from D ∞ to L(X) which satisfies (i) and (ii).
The homomorphism Ψ A will be called the D-calculus.
We will now prove that Ψ A is a homomorphism. Let f, g ∈ D ∞ . Then f ∈ D r and g ∈ D t for some strictly positive s and t, hence f g ∈ D s+t+1 by Lemma 3.21. Since LM is dense in D s for every s > 0 by Corollary 5.2, there exist (f n ) ∞ n=1 and (g n ) ∞ n=1 from LM such that f n → f in D s and g n → g in D t , n → ∞, and then, in view of Lemma 3.21, f n g n → f g in D s+t+1 as n → ∞.
By the product rule for the HP-calculus and (7.12), n ≥ 1.
Passing to the limit when n → ∞ and using (7.13), it follows that Let Ψ : D ∞ → L(X) be an algebra homomorphism satisfying (i) and (ii). Then Ψ and Ψ A coincide on {r λ : λ ∈ C + }. Since A is densely defined, Ψ(1) = 1 (see [8, Section 6, Observation (2)], so Ψ and Ψ A coincide on the span of these functions which is dense in D s (Theorem 5.1). Since Ψ and Ψ A are both bounded on D s , it follows that they coincide on each D s and hence on D ∞ . Remark 7.7. If A is an operator for which a D-calculus exists with the properties (i) and (ii) given in Theorem 7.6, then A ∈ Sect(π/2−). This follows from (3.20) and the properties (i) and (ii). By combining this with Theorem 7.6, we obtain Theorem 1.1. Note also that, if (i) holds for some λ ∈ C + , then it holds for all λ ∈ C + , by the resolvent identity.
The Banach algebras D ∞ s , s > −1, are subalgebras of D ∞ , so we obtain the following corollary by restricting the D-calculus.
Corollary 7.8. Let A ∈ Sect(π/2−) and s > −1. Then there exists a bounded algebra homomorphism Ψ s A : D ∞ s → L(X) such that Moreover Ψ s A is the unique algebra homomorphism from D ∞ s to L(X) which satisfies (i) and (ii).
From now onwards, we will write f D (A) instead of Ψ A (f ), for f ∈ D ∞ . When f (z) = e −tz , we will continue to use the notation e −tA for f (A), since the D-calculus agrees with the HP-calculus by Theorem 7.4. So (e −tA ) t≥0 form the C 0 -semigroup generated by −A, and it extends to a bounded holomorphic semigroup.
Let g : C + → C + be a holomorphic function and assume that r λ • g ∈ D ∞ for all λ ∈ C + . Since the functions (r λ ) λ∈C + satisfy the resolvent identity, the operators ((r λ • g) D (A)) λ∈C + ⊂ L(X) also satisfy the resolvent identity, i.e., they form a pseudo-resolvent. In particular their kernels and their ranges are independent of λ, and they form the resolvent of an operator B if and only if the common kernel is {0}, and the domain of B is the common range of the pseudo-resolvent (see [57,Section VIII.4]).
Corollary 7.9. Let A, B ∈ Sect(π/2−), and let g : C + → C + be holomorphic. Assume that, for each s > −1, there exists σ > −1 such that Proof. By assumption (a), Corollary 3.13 and the Closed Graph Theorem, f → f • g is a bounded map from D s to D σ . Moreover the D-calculus for A is a bounded map from D σ to L(X). Hence the composition is a bounded map from D s to L(X), and by assumption (b) it sends r λ to (λ + B) −1 for all λ ∈ C + . Moreover the maps collectively form an algebra homomorphism from D ∞ to L(X). By the uniqueness in Theorem 7.6, this map is the D-calculus for B.
In the context of Corollary 7.9, the operator B is sometimes written as g(A), but the precise meaning depends on the specific situation.
There is also a version of Corollary 7.9 for fixed values of s and σ, using the D ∞ s -and D ∞ σ -calculi. Examples 7.10. Examples of functions g and operators B which satisfy the conditions of Corollary 7.9 include the following.
Then f • g is the functionf ∈ D s as in Lemma 3.17. Note thatf D (A) is defined as a bounded operator on X, even if A is not injective. If A 0 is the restriction of A to X 0 , the closure of the range of A, thenf D (A) acts as A −1 0 on X 0 and as the sectorial limit f (∞) on the kernel of A. If X is reflexive this determinesf D (A) on X. 2. g(z) = tz, where t > 0; σ = s, B = tA. See Lemma 3.17.
In Example 7.10(4) above, we have introduced a fractional power A γ , where γ > 0. These operators are defined in various ways, including the extended holomorphic functional calculus. To justify the example, we need the following lemma about fractional powers, which is probably known at least in simpler form. For γ ∈ (0, 1) and ν ∈ N, it follows easily from a standard result [44,Proposition 5.1.4]. We give a proof that uses the holomorphic functional calculus for fractional powers as in [35, Section 3.1].
Lemma 7.11. Let A ∈ Sect(π/2−), γ ∈ (0, 1), and ν > 0. In the operator norm topology, Proof. Let z ∈ C + be fixed, and let Then f ∈ H ∞ (C + ), and, by considering the derivative of µ → (µ + z) −ν , we see that there exists a constant C (depending on z) such that Thus f has polynomial limits at 0 and ∞, and so f belongs to the extended Riesz-Dunford class defined in [35,Lemma 2.2.3]. In other words, has polynomial decay at 0 and ∞. Moreover, there exists a constant C ′ (independent of ǫ) such that (7.14) where ψ ∈ (θ, π/2). By the dominated convergence theorem, The pointwise convergence of g ǫ − g 0 to zero is clear, and the existence of an integrable majorant follows easily from (7.14).

1.
In operator norm, Proof. The proof of (1) follows from Lemma 7.11 in essentially the same way as the last paragraph of the proof of Proposition 7.2.

The calculus on Hardy-Sobolev algebras
Given the negative generator A of a bounded holomorphic C 0 -semigroup on a Banach space X, the D-calculus allows us to extend the B-calculus to a much larger class of functions. A drawback of the D-calculus is that it does not respect the sectoriality angle of A, so the results within the D-calculus are independent of the sectoriality angle and confined to holomorphic functions on C + . To remedy that problem, we introduce in this section a version of the D-calculus adjusted to an appropriate Hardy-Sobolev algebra on a sector in the right half-plane. While the Hardy-Sobolev algebra has a "stronger" norm, it appears to be an adequate substitute for D ∞ in the setting of sectors, and it has significant applications, as we will see in Section 10.
The basic idea is a very simple change of variable in the D-calculus. If Ψ s A is the D ∞ s -calculus for a sectorial operator A, then one sets Υ A (f ) := Ψ s A γ (f 1/γ ) for appropriate values of γ, determined by the sectoriality angle θ A of A. This definition does not depend on the precise choice of γ, by Corollary 7.12. The definitions also agree for different s > −1 by Theorem 7.4(ii), and we set s = 0 for convenience. As we show below this eventually leads to a new calculus for Hardy-Sobolev algebras on sectors.
Now we can formally define the H-calculus.
The homomorphism Υ A will be called the H-calculus for A.
Proof. The boundedness of Υ A follows from either (8.13) or (8.2). The homomorphism property is implied by Theorem 7.8. Indeed, employing the functional calculus Ψ 0 A on D ∞ 0 given by Corollary 7.8, one has . The uniqueness follows from Theorem 5.10. Remark 8.3. If A is any operator for which there is an H ψ -calculus as in Theorem 8.2, then A ∈ Sect(θ) for some θ ∈ (0, ψ). This follows from (4.24), and in combination with Theorem 8.2 this yields the proof of Theorem 1.2. If Υ A (r λ ) = (λ + A) −1 for some λ ∈ Σ π−ψ then this holds for all λ ∈ Σ π−ψ , by the resolvent identity.
8.2. The operator arccot(A γ ) and the arccot-formula. In this section, we derive an alternative to the formula (8.1) for the H-calculus, in the form of an operator counterpart of Proposition 4.19 for scalar functions. In addition to its intrinsic interest, it helps us to compare our approach with the approach developed by Boyadzhiev [12], as we do at the end of this section.
If A is not injective, we have arccot int ((A + ǫ) γ ) = arccot D ((A + ǫ) γ ) from the case above. When ǫ → 0+, the left-hand side converges in operator norm to arccot int (A γ ) by applying the dominated convergence theorem in (8.5), and the right-hand side converges to arccot D (A γ ) by Proposition 7.2.
Remarks 8.7. 1. If M ψ (A) = 1 in (8.9), that is, if −A generates a holomorphic C 0 -semigroup which is contractive on Σ (π/2)−ψ , then the H-calculus is contractive. This seems to be a new feature which has not been present in constructions of other calculi in the literature.
2. An alternative to the estimate (8.9) is . This is obtained from (8.8), using the estimates (7.7) (with s = 0) and (3.23) to obtain the estimate arccot(A γ ) ≤ 3πM A γ . The constant 3 is not optimal. It is possible to provide explicit bounds for M A γ in terms of M A . However we refrain from doing so in this paper, and we refer the interested reader to [6,Propositions 5.1 and 5.2].
Note that k ψ is an even function on R \ {0}, and k ψ L 1 (R) = 1.
For any f ∈ L ∞ (R + ), let It was proved in [12,Theorem 3.1] that if A ∈ Sect(θ), A has dense range, and then A admits a bounded H ∞ (Σ ψ )-calculus given by where the integral converges absolutely (in the weak sense). Conversely, if ψ ∈ (θ, π) and ϕ ∈ (θ, ψ) are such that A has a bounded H ∞ (Σ ϕ )calculus, then (8.15) holds. (Note that in this situation A has a bounded H ∞ (Σ ψ )-calculus given by (8.16), by the uniqueness of the calculus.) The formula (8.16) is obtained in [12] by rather involved Fourier analysis, and some technical details are omitted in [12].
In [12,Proposition 5.1] it is observed that, if f ∈ H ∞ (Σ ψ ) and is holomorphic in a larger sector, and f ′ ψ ∈ L 1 (R + ), f ′ ψ (∞) = 0 (this assumption is not relevant), then one can integrate by parts and rewrite (8.16) as (8.17) f and the integral converges absolutely. This formally leads to the estimate (8.9). While our reproducing arccot-formula (8.8) was inspired by (8.17), it is not easy to put formal considerations in [12] into the theory of functional calculi considered in this paper. One can relate (8.17) to (8.8) and show that the formulas are essentially equivalent within the H-calculus. This requires a number of technicalities and we intend to communicate them elsewhere.
Here we note only that f ′ ψ ∈ L 1 (R + ) and f ∈ H ∞ (Σ ψ ) (for ψ = π/2) do not imply that f ∈ H ψ in general, as shown by an intricate example kindly communicated to us by A. Borichev. 9. Convergence Lemmas and Spectral Mapping Theorems 9.1. Convergence Lemmas. Given a negative semigroup generator A, a Convergence Lemma for the holomorphic functional calculus is a useful result allowing one to deduce the convergence of (f k (A)) ∞ k=1 to f (A) from rather weak assumptions on convergence of (f k ) ∞ k=1 to f ; see [ [7] to the current setting, we apply the change of variables method used in previous sections, and derive a variant of the Convergence Lemma for the D-calculus.
In the following result, f (A) refers to the D-calculus.
Theorem 9.1. Let A ∈ Sect(π/2−). Let s > −1 and let (f k ) k≥1 ⊂ D s be such that and there exists Let g ∈ D s satisfy g(0) = g(∞) = 0. Then In particular, if A has dense range, then for all x ∈ X.
In the following result, f (A) refers to the H-calculus. Theorem 9.2. Let A ∈ Sect(θ), and let ψ ∈ (θ, π). Let (f k ) k≥1 ⊂ H ψ be such that sup and there exists f (z) := lim In particular, if A has dense range, then for all x ∈ X.
Proof. The proof is very similar to Theorem 9.1. Corollary 3.12 is replaced by Lemma 4.11, the compatibility with fractional powers follows from the definitions and Proposition 8.1(i), Corollary 4.14 is replaced by Lemma 4.9, Lemma 6.1 is replaced by Lemma 6.2, and (7.6) is replaced by Corollary 7.12.
9.2. Spectral mapping theorems. Given a semigroup generator −A, a spectral mapping theorem for a functional calculus Ξ A signifies informally that Ξ A is associated to A in a "natural" way. However, in general the spectral "mapping" theorem states only the inclusion f (σ(A)) ⊂ σ(Ξ A (f )). Equality may fail here even for functions such as e −tz and for rather simple operators A; see [23, Section IV.3], for example. While one may expect only the spectral inclusion as above, the equality f (σ(A))∪{f (∞)} = σ(Ξ A (f ))∪ {f (∞)} sometimes holds if A inherits some properties of bounded operators such as strong resolvent estimates. Note that the spectral mapping theorem may not hold even for bounded operators if the functional calculus possesses only weak continuity properties, as discussed in [10].
The statement below shows that the D-calculus possesses the standard spectral mapping properties. It is similar to [7,Theorem 4.17], with the addition of a statement about approximate eigenvalues. Recall that for f ∈ D ∞ , its values f (∞) at infnity and f (0) at 0 are defined by (3.27) and (3.28). This convention is used below. Theorem 9.3. Let A ∈ Sect(π/2−), f ∈ D ∞ , and λ ∈ C.

One has
Proof. The statements (1) and (2) are direct corollaries of (7.5) and the reproducing formula for the D s -spaces given in Corollary 3.10.
For (3), we use the F -product of the semigroup (e −tA ) t≥0 , as introduced in [18]. Let Y be the Banach space of all bounded sequences y := (y n ) n≥1 in X such that lim t→0+ e −tA y n − y n = 0 uniformly in n, where (e −tA ) t≥0 is the bounded holomorphic C 0 -semigroup generated by −A. Let Z be the closed subspace of Y consisting of the sequences y such that lim n→∞ y n = 0, and let Y = Y /Z and Q : Y → Y be the quotient map. Then (e −tA ) t≥0 induces a bounded holomorphic C 0 -semigroup (e −tÃ ) t≥0 on Y , whose negative generator A is given by To prove the spectral mapping theorem in (4), we follow the Banach algebra method used in [7] for similar purposes and inspired by [37,Section 16.5] and [16, Section 2.2]. We may assume without loss of generality that f (∞) = 0. Let A be the bicommutant of {(z + A) −1 : −z ∈ ρ(A)} in L(X), so A is a commutative Banach algebra and the spectrum of f D (A) in A coincides with the spectrum in L(X). Observe that σ(A) ⊂ C + ∪ {0}.
Next, using the same approach via Banach algebras, we prove the analogous spectral result for the H-calculus. Theorem 9.4. Let A ∈ Sect(θ) and f ∈ H ψ for some θ < ψ < π, and let λ ∈ C.
uniform bounds on the powers of Cayley transforms and on the semigroup generated by the inverse of a semigroup generator. We then compare the results to known estimates in the literature. We also revisit the theory of holomorphic C 0 -semigroups and obtain several basic estimates along with some slight generalizations.  (i) One has If, in addition, the inverse A −1 exists and is densely defined, then A −1 generates a bounded holomorphic C 0 -semigroup (e −tA −1 ) t≥0 satisfying and, for every ν > 0, Proof. By Lemma 12.1 and Theorem 7.6, for every s > 0, Setting s = 1 we get the assertion (i).
By Proposition 3.15 or by Example 3.4 and Lemma 3.17(iii), the function e −tz ∈ D s for s > 0 and t > 0, and by (3.21) and (7.7), So the estimate (ii) follows by setting s = 1 above.
If f ν (z) := z ν e −tz , ν > 0, then f ν ∈ D s if and only if s > ν, and in that case f ν Ds = 2t −ν B((s − ν)/2, 1/2)Γ(ν + 1) (see Example 3.4). Since f ν has zero polynomial limits at zero and at infinity, (f ν ) D (A) coincides with A ν e −tA as defined by the holomorphic functional calculus (see Remark 7.5). Using (3.21), it follows that, for every s > ν, Setting s = ν + 1, the first assertion in (iii) follows. The other two estimates are consequences of Lemma 3.17(i) and the estimates for e −tA and A ν e −tA obtained above.
The results in Corollary 10.1 are not new, and it serves as an ilustration of the utility of the D-calculus. We have not aimed at finding the best possible estimates, but it seems that the H-calculus provides bounds that are fairly precise whenever it is applicable. The power-boundedness of V (A) was shown in [15] and [46], using different methods. In [7, Corollary 5.9], a weaker result was shown using the B-calculus (so all operators satisfying (1.3)). Corollary 10.1 shows how the Dand H-calculi can give a sharper estimate than the B-calculus in the case of sectorial operators. Part (ii) above is one of many estimates for the bound on a bounded holomorphic semigroup in terms of its sectorial bound, and it is clearly not sharp.
Proof. By Corollary 5.5 and the boundedness of the D s+n -calculus, (z n f (n) )(A) coincides with the derivative of order n of the function t → f (tA) evaluated at t = 1. The formula (7.5) for f (tA) ∈ D s can be differentiated repeatedly with respect to t by a standard method, and putting t = 1 then gives the formula (10.1).
Then applying the inductive hypothesis with k = n to the function f , and with k = n − 1 and k = n to the function f ′ , we obtain (z n+1 f (n+1) )(A) = A n(z n−1 f (n) ) + (z n f (n+1) ) (A) − n(z n f (n) )(A) This completes the proof of the inductive hypothesis for all n ∈ N, and hence proves (10.2) Since M A = M tA for all t > 0, it suffices to prove (10.3) for t = 1. From Lemmas 7.1 and 10.2, we obtain Remark 10.4. In Theorem 10.3, the assumption that f ∈ D ∞ and f (k) ∈ D ∞ , k = 1, 2, . . . , n, can be replaced by the assumption that f ∈ D ∞ and f (n) ∈ D ∞ , by using a result of Lyubich [43]. See Corollary 11.3.

10.2.
Norm-estimates via the H-calculus. Now we use the H-calculus to provide a new proof that holomorphy of operator semigroups generated by −A is preserved for subordinate semigroups generated by −g (A) where g is a Bernstein function. This was proved for the first time in [31].
If A is injective, then f (A) and λ + g(A) are consistently defined in the holomorphic functional calculus, and therefore f H (A) = f (A) = (λ+g(A)) −1 (see [35,Theorem 1.3

.2f)]).
When A is not injective, we follow the approach proposed in the proof of [6,Theorem 4.8].
Remarks 10.6. 1. A new feature of Theorem 10.5 is an explicit sectoriality constant for g(A), given by the right hand-side of (10.6). This could be valuable when applying the result to families of sectorial operators. Thus (10.6) offers an improvement over similar estimates in [6], [7] and [31], where the sectoriality constants for g(A) are rather implicit.
2. We take this opportunity to correct a parsing misprint in the proof [6,Theorem 4.9]. One should replace f (A) with f (A)+z in the third and fourth displays on [6, p.932] (see (10.8) and (10.9) above for similar formulas).
Finally, as an illustration, we show how the holomorphy of C 0 -semigroups generated by operators −A γ fits within the H-calculus, and how estimates of similar type to Corollary 10.1(ii) can be obtained from the representation of the H-calculus and the function arccot, as in Theorem 8.6. The following result is similar to [12,Corollary 5.2], and a generalization of the main result in [17] to non-integer γ. See also [15,Remark 2,p.83].

Appendix: Shifts on D s and H ψ
The shift semigroups on the space B had an important role in the study of the B-calculus in [7] and [8]. While the semigroups are not essential in this paper, we think they will be important for further research, and so we describe their properties on the spaces D s and H ψ . In this appendix we prove that the shifts (T (τ )) τ ∈C + given by (T (τ )f )(z) := f (z + τ ), z ∈ C + , τ ∈ C + , form a holomorphic C 0 -semigroup on D s for each s > −1. We also show that a similar statement holds for shifts on H ψ for each ψ ∈ (0, π).
The proof of the statement about the generator is almost identical to the proof for the space B in [7, Lemma 2.6].
The following corollary justifies Remark 10.4 about the assumptions in Theorem 10.3. Corollary 11.3. Let f ∈ D s , s > −1, and assume that f (n) ∈ D s for some n ∈ N. Then f (k) ∈ D s for k = 1, 2, . . . , n − 1. Now we will show that the family of shifts T also forms a bounded holomorphic C 0 -semigroup on H ψ for every ψ ∈ (0, π). If ψ > π/2, then T (τ ) are defined for τ ∈ Σ π−ψ . For this aim, we will recall the Gabriel inequality for holomorphic functions.
Clearly K ≥ 1. Moreover, it can be shown that if Γ is closed, then 2 < K < 3.7 (see [11, p.457], for example). The inequality was conjectured by J. Littlewood and first proved by Gabriel in [24,Theorem I]. It is thoroughly discussed in [11,Selected Seminars,2,4 and 5] and [32,Section 5] providing simpler proofs, more general versions and additional insights.
Theorem 11.5. Let ψ ∈ (0, π) and ψ 0 = min{ψ, π − ψ}. The family T = (T (τ )) τ ∈Σ ψ 0 is a bounded holomorphic C 0 -semigroup of angle ψ 0 on each of the spaces H ψ . The generator −A H ψ of T on H ψ is given by Proof. We will show first that the family of shifts T is uniformly bounded on H ψ for every ψ ∈ (0, π). Then the result follows quickly by a density argument.
We obtain where K > 0 is given by (11.5).
The claim about the generator −A H ψ can be justified along the lines of the proof of a similar fact for the space B in [7, Lemma 2.6].
Remark 11.6. Following a more conventional approach, one may try to prove Theorem 11.5 by reducing the estimates to the half-plane case and applying Carleson's embedding theorem for H 1 (C + ). However, the technical details become rather cumbersome, so we prefer to use Gabriel's inequality allowing for a more transparent argument.

Appendix: The D-calculus vs the HP-and the B-calculi
It is natural to compare the strength of the D-calculus with some other functional calculi, such as the recently constructed B-calculus and the wellknown HP-calculus. To show the advantages of our D-calculus with respect to the B-calculus and the HP-calculus, as an illustrative example, we consider the family of functions {f n : n ≥ 1} given by f n (z) = z − 1 z + 1 n , z ∈ C + , n ∈ N.