A Besov algebra calculus for generators of operator semigroups and related norm-estimates

We construct a new bounded functional calculus for the generators of bounded semigroups on Hilbert spaces and generators of bounded holomorphic semigroups on Banach spaces. The calculus is a natural (and strict) extension of the classical Hille-Phillips functional calculus, and it is compatible with the other well-known functional calculi. It satisfies the standard properties of functional calculi, provides a unified and direct approach to a number of norm-estimates in the literature, and allows improvements of some of them.


Introduction
Given a linear operator A on a Banach space X, a fundamental matter in operator theory is to define a functional calculus for A and to get reasonable norm-estimates for functions of A. A rich enough functional calculus for A yields various spectral decomposition properties and leads to a detailed spectral theory. One well-known instance of that is the functional calculus for normal operators on Hilbert spaces.
A common procedure to create a functional calculus for an unbounded operator A is to define a function algebra A on the spectrum σ(A) of A and a homomorphism from A into the space of bounded linear operators on X with suitable continuity properties and norm-estimates. One instance of this is the classical Riesz-Dunford calculus given by Cauchy's reproducing formula (1.1) where A can be chosen to be the algebra Hol (O) of functions holomorphic in a neighbourhood O of σ(A) ∪ {∞}, and Γ ⊂ O is an appropriate contour. Unfortunately, for unbounded A (1.1) produces bounded operators f (A) only in very restricted settings, and thus (1.1) is only a starting point for the construction of an extended functional calculus, for example, for sectorial or similar classes of operators A. Here sharp norm-estimates are hardly available.
The major problem in defining a functional calculus as in (1.1) (or similar contexts) is that one has to start from appropriate norm bounds for the resolvent of A, thus having some information about at least one function of A in advance. A very natural and useful class of unbounded operators leading to a holomorphic functional calculus via (1.1) is the class of sectorial operators A on Banach spaces having their spectrum in Σ θ ∪ {0} where θ ∈ (0, π) and Σ θ := {λ ∈ C\{0} : | arg(λ)| < ω}, and allowing a linear resolvent estimate (λ − A) −1 ≤ C/|λ| for λ ∈ C \ Σ θ ∪ {0}. A homomorphism from the algebra A σ of holomorphic functions on Σ σ , σ > ω, decaying sufficiently fast at zero and at infinity, (1.2) A σ := f : Σ σ → C holomorphic and |f (λ)| ≤ C |λ| ε 1 + |λ| 2ε for some ε, C > 0 depending on f , is only the starting point here. The holomorphic functional calculus for A is constructed as a mapping taking values in the closed operators on X and extending the homomorphism from (1.1) in a canonical way. The so-called holomorphic (or sometimes McIntosh) functional calculus for sectorial operators became an indispensable tool in applications of operator theory to PDEs and harmonic analysis. For more information, we refer to [55], [43] and [27].
If −A generates a C 0 -semigroup (e −tA ) t≥0 on X, then different reproducing formulas with the semigroup can be used. One of them leads to the classical Hille-Phillips (or HP-) functional calculus from the 1950s, see e.g. [48,Chapter XV]. The calculus is given by a bounded homomorphism from the space of bounded measures M (R + ) into L(X) : Despite its very direct nature, the HP-calculus has proved to be crucial in many areas of analysis, including probability theory, approximation theory, theory of Banach algebras, spectral theory, etc. It is a base for so-called transference principles and their applications in harmonic analysis and operator theory. While classical aspects of the transference techniques can be found in [20], its modern treatment and applications to semigroup theory are contained in [44] (see also [57]).
In most approaches to functional calculi one starts with a reproducing formula for sufficiently regular functions, such as (1.2), and then extends it via a regularization procedure to include more singular functions of interest, for example z α and log z. The reproducing formula determines the function algebra for the extended functional calculus, and thus it is basic for the calculus construction. Using the regularization idea, the HP-calculus was extended by Balakrishnan [7], while the holomorphic functional calculus for unbounded sectorial operators has been developed by McIntosh (preceded by pioneering work of Bade), and then further developed and extended to other geometries (strip-type, half-plane type, etc) by a number of mathematicians.
Other functional calculi include the Hirsch calculus where a reproducing formula arises as the Stieltjes-type representation formula for complete Bernstein functions; the Bochner-Phillips calculus, where following Bochner's representation formula for Bernstein functions, a measure µ in (1.3) is replaced by a convolution semigroup of measures (µ t ) t≥0 , and a Bernstein function f (A) of A arises as the generator of strongly continuous semigroup ( ∞ 0 e −sA dµ t (s)) t≥0 ; and the Davies-Dyn'kin (or Helffer-Sjöstrand) calculus based on the Cauchy-Green reproducing formula and thus even allowing non-holomorphic functions. See [66,Chapter 13], and also [38] and [8], for more on the Hirsch and Bochner-Phillips calculi, and [24] and [3] for the third calculus.
Having defined f (A) in any calculus, it is natural and useful to look for sharp norm-estimates for f (A) in terms of f . Note that the HP-calculus produces a norm-estimate in terms of the representing measure µ rather than Lµ, and this can hardly be optimal apart from a few very special cases. The question for which A the holomorphic calculus is bounded in the sense that f (A) ≤ C A f ∞ for all f ∈ A σ , with C A depending only on A, is of major importance, since the uniform estimate allows one to avoid a drawback of "a priori" estimates in reproducing formulas. The boundedness of H ∞ -calculus is linked to square function estimates on Hilbert and Banach spaces, and thus eventually to hard problems in harmonic analysis involving boundedness of singular integrals. We refer to [27], [44] and [55] for more details and pertinent comments.
While the boundedness of H ∞ -calculus is very useful, for example in the study of maximal regularity properties in PDE theory, it imposes stringent conditions on A that are hard to verify in the abstract context. (Nevertheless it is well known that many concrete differential operators do have that property.) This remark extends to the other calculi, including the HP-calculus where the uniform estimate Lµ(A) ≤ C A µ allows one to deal with norm bounds for functions of A as if A generates a unitary C 0 -group. An example of such a situation is provided by the generators of C 0 -contraction semigroups, where the uniform estimate for Lµ(A) is provided by a unitary dilation. This suggests an important task of identifying classes of functions (or even single functions), and classes of semigroup generators, such that the corresponding functional calculus satisfies operator norm-estimates which are weaker than those given by the H ∞ -norm, but as close as possible to the H ∞ -norm. The problem of obtaining sharp norm-estimates for functions of semigroup generators has been addressed in a series of recent papers [44], [45], [67], [76], and [82].
Ideally, the functional calculus should be defined for substantial classes of operators and functions, take values in the algebra of bounded operators, and provide sharp enough estimates for the operator norms. This paper offers a new functional calculus that fits this description to a large extent. More precisely, let −A be the generator of a bounded C 0 -semigroup on a Hilbert space X or the generator of a bounded holomorphic semigroup on a Banach space X. Then for all x ∈ X and x * ∈ X * its weak resolvent (· + A) −1 x, x * belongs to the space E given by (1.4) E := g ∈ Hol(C + ) : Note that · E 0 is a seminorm on E vanishing on the constant functions. As observed in [76], a Banach algebra B can be defined as with the norm f B := f ∞ + f B 0 . Then E can be paired with B via a duality ·, · B given by The Laplace transforms of measures µ ∈ M (R + ) belong to B, and one has a reproducing formula of Cauchy type Consequently, for A as above, we define a mapping Φ A : B → L(X, X * * ), Φ A (f ) = f (A), as a w * -integral: for all x ∈ X and x * ∈ X * . However it is not immediately clear that it yields a functional calculus for A, since L(M (R + )) is not dense in B. The duality ·, · B is only partial, since the spaces E and B have rather complicated structures; see Section 2 for details of the spaces. To make our observations rigorous and to establish functional calculus properties of Φ A , we prove several intermediate statements of independent interest leading to the following theorem which summarises the main results of this paper (see Section 4). Theorem 1.1 can be applied to C 0 -semigroups which are not bounded, since any C 0 -semigroup can be made bounded by rescaling, and most normestimates are of interest even for semigroups which decay exponentially.
The use of Besov functions for functional calculus goes back to Peller's foundational paper [64] in the discrete case. Peller defined and explored the functional calculus for power bounded operators on a Hilbert space X, based on the space B 0 ∞,1 (D) which is the analogue of B for the unit disc. He proved a counterpart of (1.9) for power bounded operators on X, and also obtained several generalizations of (1.9) using several specific algebras related to B 0 ∞,1 (D). This line of research was continued, in particular, in [44], [73], [74], and [75], where similar classes of operators have been considered. The polynomials are dense in B 0 ∞,1 (D), and this simple but important fact greatly simplifies the construction in [64]. Unfortunately, in the setting of C 0 -semigroups, there is only a partial replacement for polynomials provided by those entire functions of exponential type which are bounded on C + . Consequently we have to use a different approach via a duality. It leads to the reproducing formula (1.8) which is apparently new and crucial for the calculus bound (1.9).
There was a substantial contribution to this topic in the PhD thesis of S. White [78] from 1989. He adapted a large part of Peller's estimates [64] to the more demanding context of semigroup generators. Unfortunately, the results were not published in journals, and thus were overlooked by the mathematical community until very recently. Employing the ideas from [64] in the semigroup setting, a calculus for the generators of bounded holomorphic semigroups on a Banach space X was constructed by Vitse in [76].
Vitse's results were put in a wider context of transference methods by Haase in [44]. In [44], functional calculus estimates were reduced to estimates of the Fourier multiplier norms, and the semigroup was not assumed to be holomorphic. Instead there were additional geometric assumptions on X, and so the most complete results were for Hilbert spaces. The aim of constructing a full Besov functional calculus for generators of bounded semigroups on Hilbert space and its compatibility to other calculi was posed explicitly in [44, p.2992]. Other contributions were made in [44], [45], [67], [78] and [82], where operator norm-estimates were obtained for particular classes of Besov functions for A as in Theorem 1.1. The emphasis in those papers was on functions of the form Lµ for µ ∈ M (R + ) treated by either the HP-calculus or the holomorphic functional calculus, and the arguments there relied on the Littlewood-Paley decomposition of Besov functions (see Section 6 for more on that). Thus the generality of Theorem 1.1 was out of reach. Related functional calculi for generators of C 0 -groups of polynomial growth (thus having their spectrum on iR) and for sectorial operators of zero angle were studied in [21], [52] and [53], again by means of Fourier analysis.
Once the B-calculus has been established, it leads to a number of sharp norm-estimates. The estimates are direct consequences of (1.9) and certain elementary (but not straightforward) estimates for norms of functions in B. As a sample result, we formulate the next statement which generalises results in [45, Theorem 1.1(c)] (see Corollary 5.6).
where h(s) := ess sup |t|≥s |f (it)| and f (i·) is the boundary value of f. If −A is the generator of an exponentially stable C 0 -semigroup on a Hilbert space X, then f (A) ∈ L(X) and Here C A is a constant which depends on A, but not on f .
For generators of bounded semigroups on Hilbert space, other applications include sharp norm-estimates for functions which have holomorphic extensions to larger half-planes, for the function e −1/z and its regularizations, and for powers of Cayley transforms (z − 1) n /(z + 1) n , n ∈ N. In the context of generators of bounded holomorphic semigroups, we give sharp constants for the norms of B-functions of the generators of bounded holomorphic semigroups, and provide sharp estimates for resolvents of Bernstein functions for the same class of semigroup generators. All of these applications have substantial motivations, and they recover and/or improve notable known results, in particular from [44], [45], [67], [76], [80], and [82]. We refer to Section 5 for a fuller discussion of the norm-estimates provided by the B-calculus. However, we emphasize that the most attractive feature of the B-calculus is not the estimates as such, but the fact that all of them can be obtained in a single, technically simple, manner.
Although we use the term Besov algebra in the title of the paper, we use only the definition of B in (1.5), together with techniques from complex function theory, Fourier analysis, and operator theory. In particular, we do not use the Littlewood-Paley decompositions that frequently appear in connection with Besov spaces. In an appendix (Section 6), we show how B (or more precisely, a subspace of codimension 1), is a realization of the analytic part of a conventional Besov space on R defined via Littlewood-Paley dyadic decomposition. In [76], the algebra B arises as a half-plane realization of the analytic Besov algebra B 0 ∞,1 (C + ), and the space E is shown there to contain a continuously embedded analytic Besov space B 0 1,∞ (C + ).
Notation. Throughout the paper, we shall use the following notation: supp(f ) denotes the support of a function or distribution f on R, For a ∈ C + , we define functions on C + by e a (z) = e −az , r a (z) = (z + a) −1 .
We use the following notation for spaces of functions or measures, and transforms, on R or R + : We shall also consider F applied to measures and distributions on R, and F −1 will be the inverse Fourier transform on R.
For a Banach space X, 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.

The Besov algebra, related spaces and a duality
In this section we present material which we shall need for what follows. Some of this material is quite standard, but we think it will be helpful to readers with various backgrounds if the material is collected together.
2.1. The algebra H ∞ (C + ). Let H ∞ (C + ) be the Banach algebra of bounded holomorphic functions on C + , equipped with the supremum norm Each function f ∈ H ∞ (C + ) has a boundary function on iR, given by We need the Poisson kernel and Poisson semigroup , P (t)g = P t * g, t > 0, y ∈ R, g ∈ L ∞ (R).
We will use the following standard facts.
The proof of (3) via Cauchy's integral formula around a circular contour shows that C n can be taken to be n!. However any f ∈ H ∞ (C + ) satisfies the Cauchy integral representation as a principal value integral. This may be seen by applying Cauchy's integral formula around semi-circles and letting the radius tend to infinity. Similarly, or by differentiating the formula as an absolutely convergent integral. This formula shows that C 1 can be taken to be 1/2, and repeated differentiation of the formula shows that C n can be n! 2π R (t 2 + 1) −(n+1)/2 dt = n!Γ(n/2) 2 √ πΓ((n+1)/2) . We shall use only the cases n = 1 and n = 2.
Then the mapping f → f b is an isometric isomorphism from H ∞ (C + ) onto H ∞ (R) and its inverse is given in Lemma 2.1(1) [46, Section II. 1.5].
If f ∈ H ∞ (C + ) and f b ∈ L 1 (R) then for every z ∈ C + , the Cauchy representation becomes (see [46, p.170]). Thus the Fourier-Laplace transform on the right hand side of (2.2) provides the analytic extension of f b to C + .
2.2. The Banach algebra B. We define B to be the space of those holomorphic functions f on C + such that We note the following elementary properties of functions f ∈ B.
The extended function f is uniformly continuous on C + . 5. If U is an open set containing the range of f , and h is a bounded holomorphic function with bounded derivative on U , then h • f ∈ B. 6. If f is bounded away from 0, then 1/f ∈ B. 7. The spectrum of f in the Banach algebra B is the closure of the range of f . 8. Assume that the range of f is contained in Σ π . If β > 1, then f β (z) := f (z) β ∈ B. If f is bounded away from 0, then f β ∈ B for all β ∈ R.
Proof. We prove only the first two statements. Note first that (2.3) implies that f ′ ∈ L 1 (R + ), and hence for any x > 0. Now let z = x + iy, and assume, without loss of generality, that y > 0. Let ε > 0. Integrating f ′ along the line segment from z to x + ε −1 y, and then the horizontal half-line [x + ε −1 y, ∞) gives Hence Letting ε → 0+ gives The first two statements follow immediately. The remaining statements are straightforward.
It is easily seen that B is a Banach algebra in the norm The proof of Proposition 2.2 shows that · B is equivalent to each norm of the form |f (a)| + f B 0 (a ∈ C + ∪ {∞}). It is convenient to use the algebra norm for the theory developed in the rest of Section 2 and in Section 4, although that norm is not optimal for the norm-estimates in Section 5.
The space B has been denoted in a different way in some papers, as if it were a Besov space, but this is questionable; see the Appendix for further discussion of this point.
There is a superficial resemblance between (2.3) and the definition of the Hardy space H 1 (C + ) on the half-plane C + , which consists of the holomorphic functions f on C + such that See [29,Chapter 11] or [34,Chapter II] for details of the Hardy space, noting that those references consider the spaces on the upper half-plane. It is well known that if f ′ ∈ H 1 (C + ) then f ′ ∈ L 1 (R + ). For example this can be shown by applying the Carleson embedding theorem [34,Theorem II.3.9] to Lebesgue measure on R + ; see also [29, p.198]. Here we show the stronger result that f ∈ B.
In Section 2.4 and Section 3 we shall consider some other classes of functions which are in B.

Spectral decompositions. For any closed subset
The sense in which these subspaces are "spectral" will become clear later in this section.
The subspaces H ∞ (I) for closed intervals I will play a special role in what follows. Some of them allow a very simple description. For example, by the Phragmen-Lindelöf theorem (see e.g. [46, p.175, F]), one has where e z (λ) = e −zλ , reflecting the fact that shifts of the distributional Fourier transform correspond to multiplication by exponential functions. So Using the distributional Paley-Wiener-Schwartz theorem ([50, Section 7.4]; see also [18,Theorem 3.5]), an analytic extension to C + for f ∈ H ∞ (I), where I ⊂ R + is compact, can be written in a similar way via the distributional Fourier-Laplace transforms: where e z (λ) = e −zλ and the right-hand side is well-defined since the support of the distribution F −1 f b is compact, so it can be applied to C ∞ -functions without ambiguity. It follows that the Fourier transforms of restrictions of f ∈ H ∞ (I) to vertical lines are essentially given by the Fourier transform of f b , i.e., denoting f x (y) := f (x + iy), x > 0, we have Alternatively, one may observe that f (x + iy) = (P x * f b )(y) and use that (FP x )(s) = e −x|s| , x > 0, and that F −1 f b has compact support in R + . The support of The Paley-Wiener theorem shows that, if g ∈ L ∞ (R) and σ > 0, then supp(F −1 g) ⊂ [−σ, σ] if and only if g extends to an entire function of exponential type not exceeding σ [46,Chapter II.5.7]. This can be easily transformed into a characterization of H ∞ (I) for arbitrary I by multiplication with an exponential function. The conclusion is that the smallest closed interval I such that an entire function f of exponential type belongs to H ∞ (I) has [18,Theorem 3.5]).
Recall that if g ∈ L 2 (iR) and F −1 (g(−i·)) has support in [0, σ], then F −1 (g(−i·)) ∈ L 2 (0, σ) by Plancherel's theorem, and g extends to C + with the representation This classical Paley-Wiener theorem will be used in Lemma 2.12 and then in Lemma 2.19 which plays an important role in Section 4.
We recall Bernstein's inequality (see [ We note also Bohr's inequality (see [58,Theorem 2]), which gives the following for f ∈ H ∞ [ε, ∞): We now show that H ∞ [ε, σ] ⊂ B, with a continuous embedding, when 0 < ε < σ < ∞, directly from our definition of B in Section 2.2. Using an approach via dyadic decompositions (se the Appendix of this paper), a similar result was obtained in [78, Lemma 5.5.10] and related ideas appear in [76] and [44].
If f ∈ H ∞ (I) and g ∈ H ∞ (J), then f g ∈ H ∞ (I + J) [51,Lemma VI.4.7]. Let Then G is a subalgebra of B by Lemma 2.4, and all functions f ∈ G extend to entire functions of exponential type on C, they are bounded on iR, and they decay exponentially as x → ∞. We shall show in Proposition 2.9 that the closure of G in B is So any function f ∈ B is the sum of a function in G and a constant function.
Our proof of Proposition 2.9 will use some techniques from general operator theory, applied to the shift operators, to the right and vertically, on B. There is a less abstract but much more technical proof of Proposition 2.9.
The following lemma is quite simple but it plays a crucial role here and in the development of the functional calculus in Section 4.
1. For each f ∈ B,

The generator of the
Since f is uniformly continuous on C + , T B (a)f − f ∞ → 0 as a → 0. For x > 0 and y ∈ R, integrating f ′′ along a line-segment and applying Lemma 2.1(3) gives By the maximum principle for f ′ on {z : Re z ≥ x}, and this function is integrable over R + . By the dominated convergence theorem, This formula holds pointwise, and the integral is a B-valued integral with continuous integrand. Letting t → 0+ gives f ∈ D(A B ).
3. This follows immediately from [4, Section 3.9] or from direct calculations. (2) and (3) that σ(A B ) ⊂ R + . On the other hand, for any a ∈ R + , e a is an eigenvector of A B with eigenvalue a.

It follows from
5. This is simple to check.
We shall show in Proposition 4.6 that (T B (a)) a∈C + is a holomorphic C 0semigroup.
Next we recall the notion of spectral subspaces introduced by Arveson in the context of bounded representations of locally compact abelian groups on Banach spaces, particularly operator algebras. We need only the special case of C 0 -groups which is described in [79] and [23,Chapter 8]. To our knowledge this theory has not previously been applied to the study of holomorphic function spaces.
Let (S(t)) t∈R be the C 0 -group of shifts on BUC(R): A simple calculation shows that Remark 2.6. The notion of spectral subspaces also applies to the duals of C 0 -groups, and in particular to the shifts on L ∞ (R) regarded as the dual of a C 0 -group on L 1 (R). Since (2.14) is also valid for f ∈ L ∞ (R), the space H ∞ (R) is the spectral subspace of L ∞ (R) corresponding to I = R + . We will not use that fact, but we will use Lemma 2.8 below which is related to it.
The following abstract result is a consequence of [79, Corollary 3.5] or [62, Corollary 8.1.8], but those results are set in more general contexts and they rely on different definitions of the spectral subspaces. We give a simple direct proof.
Proposition 2.7. Let (T (t)) t≥0 be a bounded C 0 -group on a Banach space X, with generator A, and assume that the range of A is dense in X.

The set {X
Proof. Let f ∈ S(R) be such that Ff has compact support and is constantly 1 in a neighbourhood of 0. Define {g a : a > 0} ⊂ L 1 (R) by Then Fg a is a compact subset of R \ {0}. Let x = Ay, where y ∈ D(A), and set .
Since R f (t) dt = 1 and (T (t)) t≥0 is strongly continuous, Moreover, using integration by parts we obtain Thus we infer that lim a→∞ x a = x. Now (2.15) and the density of the range of A imply both claims.
Consider the C 0 -semigroup (T B (t)) t≥0 and the C 0 -group of isometries on B, with generators −A B and iA B respectively, as in Lemma 2.5.
Proof. Let (S(t)) t∈R be the C 0 -group of shifts on BUC(R), as in (2.13). Let K : B → BUC(R) be the isometric injection given by Kf = f b . Then and hence, for f ∈ B, This implies that B G (I) ⊂ H ∞ (I) for every closed subset I of R + .
On the other hand, if I is a compact subset of (0, ∞), then H ∞ (I) ⊂ B by Lemma 2.4, and then (2.16) implies that H ∞ (I) ⊂ B G (I).
Proof. We consider the restrictions of G(t) to B 0 denoted by the same symbol, and note that this does not change the spectral subspaces B G [ε, δ].
for every x > 0, and Then, by the bounded (or monotone) convergence theorem, This shows that the range of A B is dense in B 0 . It now suffices to apply Proposition 2.7 to (G(t)) t∈R on B 0 , and then apply Lemma 2.8.
Remark 2.10. We can now give a quantified version of Lemma 2.5(1), which estimates T B (t)f − f B in terms of the "K-functional" K(f ; t) which is a basic tool in approximation theory. A number of regularity properties of f can be described in terms of K (and similar quantities).
Let f ∈ B, and define Note that K(f ; t) → 0 as t → 0+, by Proposition 2.9.
Since the Laplace transform µ → Lµ is a bounded algebra homomorphism from M (R + ) to B, LM is a subalgebra of B. The convolution-exponential exp * (µ) of µ in M (R + ) may be calculated in LM, and its Laplace transform is the function z → e m(z) . Examples 2.11. 1. For a ∈ R + , let e a (z) = e −az . Then e a is the Laplace transform of the Dirac delta measure δ a , e a B = 2 if a > 0, and e a − e b B = 4 whenever a, b are distinct and non-zero. 2. For a = x + iy ∈ C + , let r a (z) = (z + a) −1 . Then r a is the Laplace transform of the function e a ∈ L 1 (R + ) and r a B = 2/x. Hence e ±tra ∈ LM and e ±tra B ≤ e 2t/x . We shall give a sharper estimate in Lemma 3.4. 3. Let χ = 1 − 2r 1 , so Then χ is the Laplace transform of the measure δ 0 − 2e −t , and χ B = 3. 4. Let Then η is the Laplace transform of Lebesgue measure on [0, 1], and η B = 2.
The following lemma is not new; for example, a similar argument is given in [76, p.250]. It plays an important role in this subject, and we give a precise statement and proof here. Proof.
We will now consider some topological properties of B and LM. Lemma 2.13. Consider B with its norm-topology.
Proof. 1. The first statement is an immediate consequence of Example 2.11(1).
2. If LM were closed in B, then · HP and · B would be equivalent norms on LM. We shall show in Sections 3.4 and 5.5 that this is not so.  [13,Theorem 12.11.1]). Hence there exist an entire function G of exponential type and δ > 0 such that, for all µ ∈ M (R), where G R is the restriction of G to R. Let σ be greater than the exponential type of G, so that the support of Proof. 1. Since U is contained in the closed unit ball of H ∞ (C + ), which is compact in the topology of uniform convergence on compact sets, by Montel's Theorem, it suffices to consider a sequence (f n ) in U which converges to a holomorphic function f uniformly on compact sets and to show that f ∈ U .
From this and Fatou's Lemma, it follows that as required.
2.5. Duality and weak approximation. Let E be the space of holomorphic functions g on C + such that The constant functions are in E, and · E 0 is a seminorm which vanishes on the constant functions.
Functions in E have the following properties.
Proof. 1. By the definition of E, the function g ′ (z + 1) belongs to the Hardy space and hence lim Re z→∞ g(z) = g(∞).
We can now define a norm · E on E by Then E becomes a Banach space. However we shall work mainly with the seminorm · E 0 .
The proof of Proposition 2.15 has shown a relation between E and H 1 (C + ), and the following proposition gives another relation between them (cf. Proposition 2.3).
Then g ∈ H ∞ (C + ), and for x > 0, Since the choice of ϕ with ϕ ∞ = 1 was arbitrary, we infer that It follows that f ∈ E and Examples 2.17. 1. The functions r a (z) := (z + a) −1 (a ∈ C + ) are in E and r a E 0 = π. Moreover, the function a → r a is continuous from C + to E. Hence for any bounded measure µ on C + , the function There is a (partial) duality between E and B given by The duality is bounded in the sense that However the duality is only partial in the sense that B and E are not the dual or predual of each other with respect to this duality, and the constant functions in each space are annihilated in the duality. This duality appeared in [76, p.266] (where it is presented in slightly different form, but (2.21) can be converted into the formula in [76] by putting g(z) = G(z)). It was also noted in [76] that Green's formula on C + transforms (2.21) into for "good functions" (the argument in [76] requires a correction: In the notation of [76], one should set u 1 = x, u 2 = F · G). See [70,Lemma 17 and remark] for a precise statement and a different proof. The approach of extending pairings of functions on R into C + via Green's formula has been used systematically in [33]. We shall use (2.22) only in cases where f and g have holomorphic extensions across iR and they and their derivatives decay at infinity.
Then the following hold: 2. Let f ∈ B and T B (a)f be as in Lemma 2.5. Then Proof. 1. This is very simple.
2. For a = is ∈ iR, the statement follows from a simple change of variable. So we may assume that a > 0.
Since f ∈ B and g ∈ E, the integrals of h along iR and a + iR are absolutely convergent, and By applying Cauchy's theorem to the integral of h around the rectangles with vertices at ±y n and a ± iy n , for suitable y n → ∞, we conclude that the integrals of h along iR and a + iR coincide, so that Multiplying by x and integrating with respect to x over R + gives (2.23).

CHARLES BATTY, ALEXANDER GOMILKO, AND YURI TOMILOV
Thus, In other words, R + ∋ a → T E (a)r 0 is not continuous at 0.
4. This is very simple.
Next we show that we can approximate functions in B weakly with respect to our duality by multiplying them by a suitably chosen approximate identity. This will play a crucial role at several places in the construction of the functional calculus.
Then, for all g ∈ E, lim Proof. We may assume that f (∞) = 0, so f ∈ G by (2.12). Since {m δ : 0 < δ < 1} is a bounded set in B (Lemma 2.5(5)), it suffices to consider the case when f ∈ H ∞ [ε, σ] for some 0 < ε < σ. Using Bohr's inequality (2.6) we obtain From the assumptions, γ δ (z) is bounded uniformly in z and δ, and From these properties, the absolute convergence of the repeated integral in (2.21) and the dominated convergence theorem, we obtain that 2.6. Representation of Besov functions. Here we show that any function in B can be represented in terms of the duality as in (2.21). When g = r z where r z (λ) = (z + λ) −1 for some z ∈ C + and (2.22) holds, one obtains the Cauchy integral formula: where the contour integral on iR is in the downward direction. We make this more precise in the following.
Proof. First assume that f = Lµ ∈ LM. Then where we have used that y → (x − iy + z) 2 is the inverse Fourier transform of t → 2πte −xt e −zt on R + (extended to R by 0).
Next we consider f ∈ G, so f (∞) = 0. Let δ > 0 and η δ (z) = η(δz), where η is as in (2.18). By Lemma 2.12, f δ ∈ LM. By applying the case above to f δ and using Lemma 2.19, The formula extends by continuity to f ∈ B 0 , and then to f ∈ B by adding constants.
Remark 2.21. If z, a ∈ C + , we now have two formulas for f (z + a): Since r z+a = T E (a)r z , this agrees with (2.23).

2.7.
Dual Banach spaces. Our partial duality induces a contractive map Ψ B : E → B 0 * , where B 0 * can be identified in the natural way with the space of functionals in B * which annihilate the constant functions. It follows from Proposition 2.20 that the range of Ψ B is weak*-dense in B 0 * . However it is not norm-dense.
For a ∈ R + , the function e a of Example 1.8(1) belongs to B, and its boundary function is e b a (y) = e −iay which belongs to the Banach algebra AP(R) of almost periodic functions on R.
Take any χ in the Bohr compactification of R which is not in R, so χ is a character of AP(R) which is not evaluation at any point in R. Extend χ to a bounded linear functionalχ on BUC(R) and define ψ ∈ B 0 * by would be a measurable semigroup homomorphism from R + to the unit circle T, and so there would exist s ∈ R such that χ(e b a ) = exp(−ias) for all a ∈ R + and then for all a ∈ R. By linearity and continuity of χ, χ(g) = g(s) for all g ∈ AP(R). This contradicts the choice of χ.
For any g ∈ E, a → g, e a B is measurable, and so a → ϕ(e a ) is measurable for any ϕ which is in the norm-closure of the range of Ψ B , and also for the We now give an alternative proof of Proposition 2.22. Suppose that the range of Ψ is norm-dense in B 0 * (for a contradiction). Let f ∈ G. Then Lemma 2.19 would imply that f δ tends to f weakly in B 0 , and Lemma 2.12 shows that f δ ∈ LM. Hence LM would be weakly dense in G and then in B 0 . By Mazur's Theorem, LM would be norm-dense in B. This would contradict Lemma 2.13. Thus we conclude that the range of Ψ B is not norm-dense in B 0 * .
* with the space of linear functionals in E * which annihilate the constant functions. Our duality then provides a contractive map Ψ E from B to E 0 * .
Proof. We argue by contradiction. Suppose that the range of Ψ E is normdense in E 0 * . Let (T B (t)) t≥0 and (T E (t)) t≥0 be the shift semigroups defined in Lemmas 2.5 and 2.18. Let g ∈ E. By (2.23) and Lemma 2.5, as t → 0+. It follows from our hypothetical assumption and a simple approximation that lim t→0+ ψ(T E (t)g) = ψ(g) for all ψ ∈ E 0 * and g ∈ E 0 . By [23,Proposition 1.23], this implies that lim t→0+ T E (t)g − g E = 0 for all g ∈ E 0 . This contradicts Lemma 2.18(4).

Norm-estimates for some subclasses of B
In this section we obtain estimates of the B-norms of some more specific classes of functions in B. We give explicit forms for most of the estimates, showing how they depend on any parameters and giving explicit (but not necessarily optimal) values for any absolute constants. The estimates will be applied to operators in Sections 4 and 5.
3.1. Holomorphic extensions to the left. In this subsection we consider some functions which have holomorphic extensions to a strip in the left halfplane. For fixed ω > 0, we let Proof. Applying Lemma 2.1(3) to g(z) := f (z − ω) shows that, for x + iy ∈ C + , This implies that f ′ ∈ B and Let g ∈ H ∞ ω for some ω > 0. Then f g ∈ B and Proof. Applying the estimate (3.1) for g we obtain that as required.
by an inequality for the (modified) integral exponential function [1, 5.1.20]. Moreover g(z) = e τ z f (z) and By (1), we obtain that f ∈ B and

3.2.
Functions with decay on iR. In the following result, the assumption (3.2) and monotonicity of the integrand implies that f (is) → 0 as |s| → ∞.
This implies that f (t) → 0 as t → ∞, from Poisson's integral formula.
and assume that Let ω > 0 and g(z) = f (z + ω). Then g ∈ B and Proof. Let z = x + iy ∈ C + . By (2.1), For fixed x > 0 we estimate sup y∈R |f ′ (x+ω+iy)|. Without loss of generality we may assume that y ≥ 0. We consider two cases. ds.
Combining the two estimates above, we infer that and hence

Exponentials of inverses.
Let f (z) = exp(−1/z). Then f ∈ H ∞ (C + ), but it does not have a continuous extension to C + , and hence f / ∈ B. This can also be seen by observing that |f ′ (x + iy)| = (ex) −1 when x ∈ (0, 1) and As observed in Example 2.11 (2), the function e −tr 1 (z) = exp(−t/(z+1)) ∈ LM for t ∈ R. In Lemma 3.4 we estimate the B-norm of these functions for t > 0, and then Lemma 3.5 gives a stronger result with a more complicated proof. In Section 5.4 we shall show that Lemma 3.4 leads easily to a known result concerning the inverse generator problem for operator semigroups (Corollary 5.7), while Lemma 3.5 leads to a more general result on the same problem (Corollary 5.8).
Thenf t ∈ LM for every t > 0 and It was shown in the proof of [26,Theorem 3.3] that the functions g t (z) = z(z + 1) −1 e −t/z are in LM. In the the next lemma, we consider the function f t := g 2 t/2 ∈ LM. We estimate f t B directly, as this gives a sharp estimate in (3.4) in a fairly simple way, while estimating the L 1 -norm of the inverse Laplace transform of f t does not appear to lead to such an estimate. The estimate in Lemma 3.4 can be deduced (up to a multiplicative constant) from the estimate (3.4), since Then there is a constant C such that Proof. Let z = x + iy ∈ C + . Note first that f t ∞ = 1 and we obtain It is already apparent from this that f t ∈ B and To obtain a logarithmic estimate, we shall compare g t (x + iy) with ϕ t (x) := t + 2 (1 + x) 2 + tx for x > 0 and y ∈ R. There are three cases, as follows.
Case 3: y 2 < tx and (1 + x) 2 < tx. Then Combining these three cases, we obtain that g t (x + iy) ≤ 2ϕ t (x) for all x + iy ∈ C + . Let which is asymptotically equivalent to 4 log t for large t. Now the estimates (3.5) for small t, and (3.6) for large t, imply (3.4).
Remark 3.6. The estimate in Lemma 3.5 is sharp because for some c > 0. For t ≥ 5, the upper bound in (3.6) is sharper than the upper bound t + 3 in (3.5). We believe that (3.4) is valid for all t when C = 5.
3.4. Cayley transforms. As observed in Example 2.11(3), the function χ = 1 − 2r 1 : z → (z − 1)/(z + 1) belongs to LM and therefore so do its powers. Estimating χ n HP is quite complicated, as it involves the asymptotics of Laguerre polynomials, but it is known that they grow like n 1/2 (see Section 5.5). The B-norms of these functions grow only logarithmically, as shown in the following lemma. Then f n ∈ B and f n B ≤ 3 + 2 log(2n), for each n ∈ N.

Bernstein to Besov. A Bernstein function is a holomorphic function
where a ≥ 0, b ≥ 0 and µ is a positive measure on (0, ∞) such that 1 − Re e −zs dµ(s) For further information on Bernstein functions, see [66].
Remark 3.9. Note the constant C in Proposition 3.8 is independent of the function f , provided that f is Bernstein.
Although we have stated Proposition 3.8 for Bernstein functions, its proof uses only the properties (B1)-(B3). Slightly more general properties would suffice, although the constant C might then depend on the conditions. For results in this direction, see [9].

Functional calculus for B
In this section X is a complex Banach space, X * is its Banach dual, and the duality between X and X * is written as x, x * for x ∈ X, x * ∈ X * . We shall write z ∈ C + as z = α + iβ in this section, as we shall be using x to denote vectors in X.

Definition. Let
A be a closed operator on a Banach space X, with dense domain D(A). We assume that the spectrum σ(A) is contained in C + and for all x ∈ X and x * ∈ X * . By the Closed Graph Theorem, there is a constant c such that for all α > 0, x ∈ X and x * ∈ X * . Note that (4.1) says precisely that the function belongs to E. We let γ A be the smallest value of c such that (4.2) holds, so It was shown in [35] and [69] that if A satisfies the assumptions above then −A is the generator of a bounded C 0 -semigroup (T (t)) t≥0 . Moreover, the following holds for all x ∈ X, x * ∈ X * and α > 0: This representation is not explicit in the papers, but it is at the core of the theory. To see that (4.5) is true, multiply the equation by t, and then take Laplace transforms of each side with respect to t. The resulting functions of z are both (z + A) −2 x, x * , so uniqueness of Laplace transforms implies (4.5) (see [19, p.505] for the full argument).
2. If −A generates a (sectorially) bounded holomorphic C 0 -semigroup, then a very simple argument shows that (4.2) holds. We will discuss this further and give an estimate of γ A in Section 4.2.
Assume that (4.1) holds, so that the functions g x,x * in (4.3) are in E, and let f ∈ B. We aim to define f (A) by replacing z by A and r z by (z + A) −1 in the representation formula in Proposition 2.20. To ensure that the corresponding integrals are convergent, we need to work in the weak operator topology, and we use the duality between B and E considered in Section 2.5.
Let f ∈ B and A be as above. Define for all x ∈ X and x * ∈ X * . It is easily seen that this defines a bounded linear mapping f (A) : X → X * * , and that the linear mapping is bounded. Indeed, We will show that f (A) ∈ L(X) and Φ A is an algebra homomorphism. First we establish consistency of the definition in (4.10) with the Hille-Phillips (or HP-) calculus, by following the argument in Proposition 2.20 for f ∈ LM. Proof. Since f ∈ LM, f = Lµ for some µ ∈ M (R + ). First assume that f (∞) = 0. Then where we have used that β → (α − iβ + A) −2 x, x * is the inverse Fourier transform of t → 2πte −αt T (t)x, x * on R + (extended to R by 0), and both functions are in L 1 (R). We now have the required result when f (∞) = 0. The general case follows by applying the above case to f − f (∞).
In particular, Lemma 4.2 implies that (λ + A) −1 = r λ (A), where λ ∈ C + , and T (t) = e t (A), where e t (z) = e −tz . Consequently, from here on we will write e −tA in place of T (t).
By Lemma 2.19, for x ∈ X and x * ∈ X * , we now have Now we present the main result showing that the map Φ A has the essential properties of a bounded functional calculus. We shall subsequently refer to Φ A as the B-calculus for A. Proof. The mapping Φ A : f → f (A) is bounded from B into L(X, X * * ). If f ∈ G, then f η δ ∈ LM by Lemma 2.12, so Lemma 4.3 applies to all such f . In particular, Φ A maps G into L(X). Since G is norm-dense in B 0 , it follows that Φ A (f ) ∈ L(X) whenever f (∞) = 0. Since Φ A (1) = I, it follows that Φ A maps B into L(X).
To show that Φ A is multiplicative, it suffices to consider f, g ∈ G. Then f g ∈ G. Take δ, δ ′ > 0. Since f η δ , gη δ ′ ∈ LM, Letting δ ′ → 0+ with δ fixed and using Lemma 4.   [44], and a smaller class in [78]). We obtain the estimate for all functions in B. Applications to bounded C 0 -semigroups on Hilbert spaces are given in Section 5.
We note two other simple properties of the B-calculus. Proof. The first statement follows from (4.10) and an application of (2.23) with g = g x,x * . The second statement is a simple change of variables in (4.10): t = aα, s = aβ.
We shall consider some more general properties of the B-calculus in Sections 4.3 (compatibility with other calculi), 4.4 (convergence lemmas) and 4.5 (spectral inclusion).
Let A be an operator and assume that σ(A) ⊂ C + and (4.13) It follows from (4.13) and Neumann series (see [76, then (4.13) holds, with M A equal to C π/2 in (4.12). Thus −A generates a bounded holomorphic semigroup if and only if σ(A) ⊂ C + and (4.13) holds.
In that case, M A is a basic quantity associated with A, which we call the sectoriality constant of A.
Before discussing bounded holomorphic semigroups in general, we show that the semigroup (T B (a)) a∈C + of shifts is a bounded holomorphic semigroup on B, as mentioned after Lemma 2.4. A short proof of holomorphy appeals to a theorem of Arendt and Nikolski (see [4,Theorem A.7]). Since the functionals f → f (z) for z ∈ C + form a separating subspace of B * and since the functions a → (T B (a)f )(z) = f (z + a) are holomorphic on C + , and T B (a) is a contraction, it follows from that theorem that T B is holomorphic on C + . Another proof proceeds by applying Proposition 2.5(2),(3) and general semigroup theory (see [4,Section 3.9]).
We now give a more explicit proof of holomorphy exhibiting some estimates which may be useful for other purposes. Proof. We will show directly that T B : C + → L(B) is holomorphic. Let f ∈ B and a ∈ C + . Since T B (a)f ∈ H ∞ ω for ω = Re a, it follows from Lemma 3.1 that (T B (a)f ) ′ ∈ B. It remains to show that Since the operators T B (is) are invertible isometries on B, we may assume that a > 0.
We use the representation and Using (4.14), we have We estimate these integrals separately.
Some applications of the B-calculus for operators in Sect(π/2−) are discussed in Section 5.

Compatibility with other calculi.
It is important to compare the B-calculus with the other calculi in the literature. Here the rule of thumb is that all calculi are compatible whenever they are well-defined, and the Bcalculus does not deviate from that general principle, as we will see below.
We first remark that according to Theorem 4.4 the B-calculus is a strict extension of the HP-calculus. Thus all of the compatibility results for the HP-calculus are valid in the setting of the B-calculus restricted to the subalgebra LM of B. Compatibility results for the HP-calculus can be found in [43,Proposition 3.3.2], [55] and [8,Sections 4,5], for example.
In this section we will show that the B-calculus is compatible with two other functional calculi, namely the classical sectorial holomorphic functional calculus (see [43] and [55]) and the half-plane holomorphic functional calculus (see [10]). Their constructions involve a primary functional calculus defined by contour integration, followed by an extension procedure, and the resulting operators may themselves be unbounded. We refer to [43,Chapter 1] for the general background of this theory of functional calculi for unbounded operators, in particular, the notions of primary and extended functional calculus, and the use of regularisers in the extension procedure. For the definitions of the primary calculus and the properties of the extended sectorial and half-plane calculi, we refer to [43,Chapter 2,etc] and [10] respectively.
Let A ∈ Sect(π/2−), and let f → Ψ A (f ) stand for the (extended) sectorial holomorphic functional calculus whenever it is defined as a closed operator. If f ∈ B then Ψ A (f ) is defined in that calculus. The following proposition shows that the B-calculus and the sectorial holomorphic functional calculus are compatible. Variants of this result were shown by Vitse in [76], where the construction of the calculus for Besov functions was based on the sectorial calculus. It is possible to establish compatibility of our B-calculus and Vitse's calculus, and hence with sectorial calculus, initially on G and then by approximation arguments on B. Here we give a more direct argument.
a) First, assume that 0 ∈ σ(A), and let f ∈ B ∩ A π/2 , where A π/2 is given by (1.2). Then, for any θ ′ ∈ (θ, π/2), It follows from the definition of Ψ A (f ), Lemma 2.20 and Fubini's theorem that From the primary functional calculus for invertible sectorial operators, we obtain and then, from Corollary 4.8, b) Now we assume that A is invertible, and consider arbitrary f ∈ B. Applying the case a) with f replaced by (1 − r 1 )f ∈ B ∩ A π/2 , we obtain c) Finally we assume that 0 ∈ σ(A), and we consider f ∈ B. The case b) above yields Φ A+ε (f ) = Ψ A+ε (f ) for all ε > 0. Then, using continuity properties for Φ A and Ψ A given in Lemma 2.5 and [43, Lemma 2.6.7] respectively (or by a direct estimate), we have where the limits are in operator norm. The proposition follows.
In particular, Proposition 4.9 implies that the Composition Rule [43, Theorem 2.4.2] for the sectorial functional calculus can be used within the Bcalculus for A ∈ Sect(π/2−).
We now consider compatibility of the B-calculus with the half-plane holomorphic calculus, and we use the notation Ψ * A (f ) for the (extended) halfplane calculus whenever it can be applied for A and f .
Then f (A) is defined in each calculus, Φ A (f ) is a bounded operator, and Ψ * A (f ) is a closed operator. We now show that Ψ * A (f ) = Φ A (f ). Proof. First, we assume that (C1) holds. Then the proof is very similar to Proposition 4.9, with the following changes.
In part b), the regularizer 1 − r 1 is replaced by r 2 1 . This completes the proof when (C1) holds.
In each case, one lets η → 0+, to obtain the equality for f and A, using the Convergence Lemma for the half-plane calculus.
2. Proposition 4.10 suggests an alternative way to define the B-calculus. For f ∈ B, one could define f (A + ε) by the half-plane functional calculus, and then show that f (A + ε) satisfies (4.10). This implies that f (A + ε) ∈ L(X) and f (A + ε) ≤ C f B for f ∈ B. Then one can conclude from Lemma 2.5 that lim exists in the operator norm, and this could become the definition of f (A). This process reverses some of the steps that we have taken.
Another compatibility result between the B-calculus and the sectorial calculus is the following. Here the assumptions on A and f are such that both the B-calculus and the sectorial calculus can be applied, and we denote the corresponding operators by Φ A (f ) and Ψ A (f ), respectively. Proposition 4.12. Assume that A satisfies (4.1). Let f ∈ H ∞ (Σ θ ) for some θ ∈ (π/2, π) and assume that f Proof. Let ε > 0, and f ε (z) = f (z + ε). Then Ψ A (f ε ) = Ψ A+ε (f ) by a very routine argument (or a very special case of the Composition Rule). Moreover It is possible to weaken the assumption in Proposition 4.12 that f is bounded on Σ θ to polynomial boundedness in a similar way to case (C2) of Proposition 4.10.

4.4.
A Convergence Lemma. The Convergence Lemma in the holomorphic functional calculus of sectorial operators [43, Proposition 5.1.4] is a result of a tauberian character. Assume, for simplicity, that A ∈ Sect(π/2−) has dense range, and A admits bounded H ∞ (C + )-calculus (so f (A) ∈ L(X) for all f ∈ H ∞ (C + )). Then pointwise convergence of (f n ) n≥1 ⊂ H ∞ (C + ) to f on C + , together with sup n≥1 f n ∞ < ∞, implies strong convergence of f n (A) to f (A). Moreover, f n (A) → f (A) in the operator norm if f n → f in H ∞ (C + ). The Convergence Lemma allows one to replace convergence of (f n ) n≥1 in H ∞ (C + ) by convergence of (f n ) n≥1 in a weaker topology at the price of getting only strong convergence of (f n (A)) n≥1 . However, such convergence often suffices in various applications. See [43] and [55] for fuller discussions of that, and also for variants of the Convergence Lemma for other types of operators, including the situation when the H ∞ (C + )-calculus for A is unbounded. Several instances of applications of the Convergence Lemma for sectorial or half-plane operators can be found in Section 4.3 of this paper.
Since the B-calculus is compatible with the sectorial and half-plane calculi the Convergence Lemmas for those calculi can be applied to the B-calculus, when the assumptions of Proposition 4.9, Proposition 4.10 or Proposition 4.12 hold uniformly for the approximating sequence. In particular all the functions must be holomorphic on the same open sector or half-plane.
The statement below is a Convergence Lemma which is specific to the B-calculus. It applies to operators whose spectrum may be as large as C + and functions which may not extend beyond C + , and pointwise convergence of (f n ) n≥1 is assumed only on C + . This is compensated by adding an assumption on the behaviour of (f ′ n ) n≥1 near the imaginary axis. Then f 0 ∈ B and, for every x ∈ X, Proof. The assumption (4.22) implies that that f n (iβ) = lim δ→0+ f n (δ + iβ) uniformly in n, and hence (4.21) holds also for z ∈ iR.
By Lemma 2.14, f 0 ∈ B. Replacing f n by f n − f 0 , we may assume that f 0 = 0. Then considering (f n −f n (0)) n≥1 we may also assume that f n (0) = 0 for each n ∈ N.
Let x ∈ D(A 2 ), n ∈ N and x * ∈ X * with x * = 1. For r ≥ 1, let D r = {z ∈ C + : |z| < r}. We apply the definition (4.10) and the reproducing formula in Proposition 2.20 for z = 0. Writing z = α + iβ and dS(z) for area measure on C + , we obtain and S 2 (r, n, x * ) = We will treat the terms S 1 and S 2 separately.
We consider S 2 first. Note that From this, (4.7) and the inequality 2|z| 2 ≥ (β 2 + r 2 ), |z| ≥ r ≥ 1, we have Let ε > 0. From the estimate above, we may take r ≥ 1 so large that |S 2 (r, n, x * )| ≤ ε for all n ∈ N and all x * ∈ X * with x * = 1. Now, we have By the assumption (4.22), there exists δ > 0 such that the first integral in the last line of the display is less than ε for all n ∈ N. By Vitali's theorem, (f ′ n ) n≥1 converges uniformly to zero on compact subsets of C + . Thus, for fixed r and δ, the second integral converges to zero as n → ∞. Let N = N (δ, r) be such that the second integral is less than ε for all n ≥ N . Then, summing the estimates one has Even if (4.21) holds for z ∈ C + (but (4.22) does not hold), the conclusion of Theorem 4.13 may fail. Let µ be a Rajchman measure on R + , that is, Fµ vanishes at ±∞. If f = Lµ and f (0) = 0, then f n (z) → 0, in the closed half-plane C + . However (f n (A)) n≥1 does not converge strongly to zero for the generator A of any unitary C 0 -group on a Hilbert space X.
In general, conditions which guarantee the strong convergence of (e −nA ) n≥1 are rather involved and depend on the spectral properties of A.
If A ∈ Sect(π/2−) and the range of A is dense in X, then e −tA → 0 as t → ∞, in the strong operator topology due to specific properties of such A (e.g. sup t>0 tAe −tA < ∞).

Spectral inclusion and mapping.
With a few exceptions, given a semigroup generator −A, the spectral "mapping" theorem for a functional calculus Υ A usually takes the form of the spectral inclusion f (σ(A)) ⊂ σ(Υ A (f )). In general, equality fails dramatically, even for the very natural Proof. 1. From (4.10) and Proposition 2.20, for all x * ∈ X * , we have 2. This is similar to (1).
3. We shall follow the Banach algebra method used in [48,Section 16.5] and [23,Section 2.2]. We may assume without loss of generality that f (∞) = 0. Let A be the bicommutant of {(λ − A) −1 : λ ∈ ρ(A)} in L(X), so the spectrum of f (A) in A coincides with the spectrum in L(X).
First, assume that the resolvent of A is bounded on the left half-plane. By the resolvent identity, (z + A) −2 (1 + A) −2 ≤ C(1 + |z − 1|) −2 for z ∈ C + . Then we may obtain the following formula in which the integral is absolutely convergent in the operator norm: (4.23) Applying (4.10) with x replaced by (1 + A) −2 x gives a weak form of (4.23), and then this strong form follows.
Proof. This follows from Lemma 3.1. To obtain the constant 4πK 2 A instead of 5πK 2 A , note that f ′ (∞) = 0 and apply (4.11).
Similar results to Corollaries 5.3, 5.4 and 5.5 could be deduced in the same way for A ∈ Sect(π/2−). However those results can be proved directly by defining f (A), f (A)(λ + A) −α and f ′ (A) respectively by absolutely convergent integrals as used in the functional calculus for invertible sectorial operators [43,Section 2.5.1]. The relevant functions f (z), f (z)(λ + z) −α and f ′ (z) respectively all decay at least at a polynomial rate along rays with arguments in (−π/2, π/2) and the contour can pass to the left of 0. Then straightforward estimates produce results of this type. 5.3. Exponentially stable semigroups. Corollaries 5.3, 5.4 and 5.5 can all be adapted to the case when −A generates an exponentially bounded C 0semigroup on a Hilbert space, so e −tA ≤ M e −ωt , t ≥ 0. In this situation, one may apply the corollaries above with A replaced by A − ω and f ∈ H ∞ (C + ) replaced by f (· + ω) ∈ H ∞ ω . For example, the conclusion of this version of Corollary 5.4 becomes that where the function f (A) may be defined by the half-plane calculus.
Instead of giving full details, we now give another result which we formulate for exponentially stable semigroups on Hilbert space and we give a proof which uses this technique in order to apply Lemma 3.3. It extends a result of Schwenninger and Zwart [68] who considered the case when |f (is)| ≤ (log(|s| + e) −α for some α > 1.

5.4.
Inverse generator problem. Let −A be the generator of a bounded C 0 -semigroup, and assume that A has dense range. Then A is injective, and we may consider the operator A −1 whose domain is the range of A. The longstanding inverse generator problem asks whether −A −1 also generates a C 0 -semigroup. The problem was raised by de Laubenfels [25], and he pointed out the simple positive solution in the case of a bounded holomorphic C 0semigroup on a Banach space. Then A is sectorial of angle θ ∈ [0, π/2) and A −1 is sectorial of the same angle. A negative answer to the problem was first given in [80]. Moreover, essentially any growth of a semigroup (e −tA ) t≥0 rules out a positive answer, and the answer is also negative for bounded semigroups on L p -spaces when p = 2; see [40], or [32] for a somewhat simpler counterexample.
The answer to the problem for bounded semigroups on Hilbert space remains unknown. The question arises in control theory (see [80] for a discussion), but it is also very natural from the viewpoint of general theory of functional calculus. It is known that if the answer is always positive, then the C 0 -semigroup (e −tA −1 ) t≥0 is bounded. Thus long-time estimates of (e −tA −1 ) t≥0 when A −1 does generate a C 0 -semigroup are of value. The case when (e −tA ) t≥0 is exponentially stable is of particular interest, since then an integral representation for (e −tA −1 ) t≥0 in terms of (e −tA ) t≥0 is available (see for example the recent survey article [37,Corollary 3.5]). The relevant estimates become simpler (and sometimes sharper), and one might hope for a positive solution at least in that restricted setting. A fuller and more detailed discussion of the problem can be found in [37].
The problem is essentially the same as the question whether the operators exp(−tA −1 ) (defined by half-plane functional calculus) are bounded. So we consider formally operators of the form f t (A) where f t (z) = exp(−t/z) for t > 0. As noted in Section 3.3, these functions are not uniformly continuous on C + , and hence they are not in B. We can derive [81, Theorem 2.2], the main result of [81], from the B-calculus, up to multiplication by an absolute constant.
Corollary 5.7. Let −A be the generator of an exponentially stable C 0semigroup on a Hilbert space X, so there exist M, ω > 0 such that Then Proof. We may assume that ω = 1 by replacing A by ω −1 A − I and t by ω −1 t. Then we consider the functionf t (z) = exp(−t/(z + 1)) as in Lemma 3.4. Thenf t = e −tr 1 ∈ B, so boundedness of the B-calculus gives The result follows from Lemma 3.4.
It is possible to improve Corollary 5.7 by replacing the assumption that the semigroup is exponentially stable by the weaker assumptions that the semigroup is bounded and A is invertible.
Corollary 5.8. Let −A be the generator of a bounded C 0 -semigroup on Hilbert space X, and assume that A has a bounded inverse. Then where C A is a constant depending on A.
Then g t,ε ∈ B and g t,ε (A) = e −t(A+ε) −1 which is given by a by power series as in the proof of Corollary 5.7. Using Lemma 4.5, we have Letting ε → 0+, using Lemma 2.5(1) and noting that A −1 is the limit in the operator norm of (A + ε) −1 , we obtain Hence We refer the reader to Remark 3.6 for discussion of the value of C and hence C A in Corollary 5.6 and its proof.

Cayley transforms.
If −A is the generator of a bounded C 0 -semigroup on a Banach space X we let V (A) be the Cayley transform (A − I)(A + I) −1 of A. Norm-estimates for powers of V (A) are of substantial value in numerical analysis, for example in stability analysis of the Crank-Nicolson scheme, see [47], [15], [22] and [61]. The first bound V (A) = O(n 3/2 ) was proved in [17], where applications to ergodic theory were treated. In the setting of generators of bounded semigroups on Banach spaces, the optimal estimate V (A) n = O(n 1/2 ) was obtained in [15] by means of the HP-functional calculus. This settled a conjecture by Hersh and Kato [47] who gave a slightly worse bound. Since V (A) = χ n (A), where χ(z) = (z − 1)(z + 1) −1 , the optimality of this bound implies that the HP-norm of χ n grows like n 1/2 (see [37,Section 9]).
Research on asymptotics of V (A) continued in a number of subsequent works, see the survey [37] and the references therein. By means of the holomorphic functional calculus, it was proved in [22] and [61] (see also [5]), that V (A) is power bounded if A generates a bounded holomorphic semigroup on X.
The corresponding question for bounded semigroups on Hilbert spaces has proved to be be more demanding. While it is evident that V (A) is a contraction (so power bounded) if A is the generator of a contraction C 0semigroup on a Hilbert space X, it is still unknown whether V (A) is power bounded whenever A generates a bounded C 0 -semigroup on X. The logarithmic estimate V (A) n = O(log n) proved in [36] remains the best so far in that case (see also [39]). The inverse generator problem and the problem of power boundedness of V (A) are strongly related (essentially equivalent) to each other. In particular, if A and A −1 are both generators of bounded C 0 -semigroups on a Hilbert space, then V (A) is power bounded (see [5], [42] and [36]). We refer the reader to [37] for more details.
The proof that V (A) n = O(log n) in [36] was based on intricate estimates of Laguerre polynomials. Here we obtain the logarithmic estimate as a direct and elementary application of the B-calculus and Lemma 3.7.
Corollary 5.9. Let −A be the generator of a bounded C 0 -semigroup on Hilbert space X. Then 5.6. Bernstein functions. Sharp resolvent estimates for Bernstein functions are valuable in the Bochner-Phillips theory of subordination of C 0semigroups, and in probability theory. In the 1980s, Kishimoto and Robinson asked whether subordination preserves holomorphy of semigroups, or in other words whether Bernstein functions transform Sect(θ) into itself, for θ ∈ [0, π/2). The positive answer in full generality was first obtained in [38], and then the result was reproved in [9] within a wider framework of functional calculus for Nevanlinna-Pick functions. The B-calculus offers a new streamlined and transparent proof of the result, as we show below.
If A ∈ Sect(π/2−) and f is a Bernstein function, then f (A) can be defined either by the sectorial calculus or by the Bochner-Phillips calculus, without ambiguity [38, Proposition 3.6].

Appendix: relation to Besov spaces
The main purpose of this section is to explain the connection between the spaces B (and E) (more precisely, B 0 and E 0 ) with (homogeneous, analytic) Besov spaces of functions on R. In Sections 2-5, we have given a largely self-contained account of the construction of the B-calculus and we have obtained all the relevant estimates, without any need for the theory of Besov spaces. We used Arveson's theory of spectral subspaces in the proof of Proposition 2.9 which played an important role in the construction, and we shall use the result again to show that B 0 can be identified with a Besov space by passing to the boundary functions (Proposition 6.2). Conversely, Proposition 2.9 follows very easily from Proposition 6.2, as noted in [76] (slightly incorrectly). More generally, the relations of the spaces in this paper to Besov spaces are instructive and potentially crucial for further work, so we set them out here.
Apparently (as remarked in [6, p.120]) there is no consensus surrounding the definition of homogeneous Besov spaces, let alone the definition of homogeneous analytic Besov spaces. The definitions given below reflect the aims of the paper, and they may differ slightly from other sources. Our aim is to study the holomorphic functions from B (and E) in terms of their boundary values, and to relate the boundary values to the Besov spaces defined on R. The theory of inhomogeneous Besov spaces is much better developed, but unfortunately it seems to be unsuitable for the study of spaces of holomorphic functions, since the spectrum of their elements does not split easily near zero.
It is crucial that the definition of B s p,q (R) does not depend of the choice of (ψ n ) n∈Z , up to equivalence of norms, and it is a Banach space for all 1 ≤ p, q ≤ ∞ and s ≥ 0. Moreover, S ∞ (R) ⊂ B s p,q (R) ⊂ S ′ ∞ (R) and, for every f ∈ B s p,q (R), f = n∈Z f n in S ′ ∞ (R). It is traditional to writeḂ s p,q (R) instead of B s p,q (R) to distinguish homogeneous and inhomogeneous Besov classes. Since we will not deal with inhomogeneous Besov spaces, we omit the dot. See [6], [41], [63], [65] or [72] for discussions of other properties of homogeneous Besov spaces.
Remark 6.1. One can show that S ′ ∞ (R) is topologically isomorphic to S ′ (R)/P, where P stands for the space of polynomials on R. The space B s p,q (R) can also be introduced by defining a norm on the equivalence class [f ] ∈ S ′ (R)/P as f + P := f dyad p,q . So long as we are interested only in the "smoothness index" s = 0 as in this paper, we could use an alternative isomorphic pair, S 0 (R)/C and S ′ 0 (R), where S ′ 0 (R) is the topological dual of See [14,Proposition 8] for details of that. However, we prefer to avoid dealing with quotient classes.
A particularly convenient way to choose the functions ψ n is as follows. Let ψ ∈ C ∞ (R) be such that Define ψ n (x) = ψ(2 −n |x|), ψ + n (x) = ψ(2 −n x), ϕ + n = Fψ + n . Then (ψ n ) n∈Z satisfy the required properties. From now onwards, we assume that ψ and (ψ n ) n∈Z have been chosen in this way.
Define the (homogeneous) analytic Besov space as B s+ p,q (R) = f ∈ B s p,q (R) : supp F −1 f ⊂ R + , and note that if f ∈ B s+ p,q (R) then f * ϕ n = f * ϕ + n , n ∈ Z. Since B s+ p,q (R) is a closed subspace of B s p,q (R), it is a Banach space. See also [2, Section 2] and [11, Section 3.2] for related constructions. Note that the Hilbert transform, given as a Fourier multiplier with the symbol 1 [0,∞) , is bounded on B s p,q (R) for all 1 ≤ p, q ≤ ∞ and s ≥ 0; see [41,Corollary 6.7.2] or [63]. Now we consider the analytic Besov space B dyad := B 0+ ∞,1 (R) equipped with the corresponding dyadic norm It is straightforward to show that B dyad ⊂ BUC(R), and for every f ∈ B dyad , f = n∈Z f * ϕ + n in BUC(R). We will show that B dyad can be identified with B 0 by showing that B dyad arises as the space of boundary values of functions in B 0 .
For g ∈ B dyad write its dyadic decomposition g = n∈Z g * ϕ + n , and define the (holomorphic) extension mapping P: where P t and P (t), are the Poisson kernel and semigroup for the right halfplane C + , as in Lemma 2.1. The mapping is well-defined, and P(g) ∈ H ∞ (C + ) since g ∈ H ∞ (R). Proof. Note that for g ∈ B dyad , and by [71, Corollary 3, p.285] (see also [72, Sections 5.2.3 and 2.12.2]) there exists C > 0 such that for every g ∈ B dyad , (6.5) Thus P yields an isomorphic embedding of B dyad into B 0 .
If f ∈ G, then f b * ϕ + n = 0 for all except finitely many n, so f b ∈ B dyad and f = Pf b . Since G is dense in B 0 by Proposition 2.9, we infer that P maps B dyad onto B 0 . The inverse map is given by f → f b , by well-known properties of the Poisson kernels (see Lemma 2.1).
The property (6.3) is standard, and it follows from the representation theory of holomorphic functions of slow growth as distributional Laplace transforms; see [77, Section 9 and Section 12.2, Corollary 4] or [12, Theorem 1 and Corollary].
Remark 6.3. The right-hand inequality in (6.5) may be deduced from Lemma 2.4, since P(g * ϕ + n ) ∈ H ∞ [2 n−1 , 2 n+1 ]. Using also the maximum principle, we obtain that, for every g ∈ B dyad , P(g * ϕ + n ) B 0 ≤ C g * ϕ + n ∞ , so This argument may also be used for other similar spaces.
Remark 6.4. The description of Besov spaces B s p,q (R) in terms of the boundary behaviour of the Poisson semigroup goes back to [70] (for s > 0); see also [72] and [63]. For the study of general Besov spaces in terms of spaces of holomorphic functions in C + , it seems more natural to use the half-plane extension of Besov spaces by the Fourier-Laplace transform, as in [11] (for s < 0). In this approach, one formally defines the holomorphic extension In particular, using the mapping I, certain holomorphic Besov spaces in a quite advanced setting were identified in [11] with Bergman spaces of holomorphic functions in a half-plane. We prefer to use Poisson semigroup since it fits better to this paper, and requires minimal developments of the theory of Besov spaces. An approach to Besov spaces on R as boundary values of spaces of harmonic functions on C + can be found in [16] (where the spaces are called Lipschitz spaces).
Remark 6.5. For constructions of functional calculi it is crucial to deal with Banach algebras of functions and to include the constant functions, so we have worked with the space B and the norm · B = · B 0 + · ∞ . (Recall from Lemma 2.2 that · B 0 ≥ · ∞ on B 0 .) The Banach algebra (B, · B ) can be considered as a modified Besov space, but it can hardly be identified with the conventional homogeneous Besov spaces. Thus the standard facts from the theory of Besov spaces (duality, isomorphisms, etc) cannot a priori be applied to (B, · B ). The algebra B has appeared, for example, in [76], and similar algebras were considered in [59]. Now we briefly discuss the homogeneous class E dyad := B 0+ 1,∞ (R) with the corresponding norm and its connections to the space E of holomorphic functions g on C + satisfying (2.19).
Recall from Section 2.5 that E is a Banach space with the norm in (2.20) and the subspace E 0 of functions g ∈ E with g(∞) = 0 is also a Banach space. The spaces E and B are related via the (partial) duality given in (2.21). Consequently, as discussed in Section 2.7, E and B are contractively embedded into B * 0 and E * 0 , respectively, but in each case the range is not norm-dense (Propositions 2.22 and 2.23). Nevertheless, it is natural to consider further the nature of E and any relations to B, in particular by looking at Besov classes that can be associated to E.
Parts of the proof of Proposition 6.2 can be applied in the context of E, taking account of the estimate in Proposition 2.15(3) and using the same results or arguments from [71], [77] and [12]. In this way one can see that E dyad is isomorphically embedded into E as a closed subspace, every f in the range of the embedding there exists a distributional boundary value f b , and f is the Fourier-Laplace transform of f b . However, the shift semigroup (T E (a)) a≥0 is not strongly continuous on E (Lemma 2.18), so a result for E similar to Proposition 2.9 cannot be expected, and consequently it is not clear that the embedding is surjective.
Then E 0,dyad is embedded isomorphically inẼ, as shown in statement (2) of [76, p.266]. It seems plausible thatẼ is isomorphic to E 0,dyad , and E 0 is isomorphic to E dyad in the sense of Proposition 6.2, and also that E * 0,dyad can be identified in a natural way with B dyad (see [56,Theorem 10.7], or [11, p.335-337] where several arguments are true for a larger class of Besov spaces than claimed). None of these identifications is needed for the results of this paper, and we have not worked out proofs (or corresponding counterexamples).
Remark 6.6. A different approach to Proposition 6.2 (and also to the very definition of holomorphic Besov spaces) was proposed in [76], but some of the arguments given in the Appendix of that paper (e.g., p.266, lines [16][17][18] appear to be incomplete. It is also stated there (p.265, last line) that E * 0,dyad coincides with B dyad , but no explanation is given.