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 Here sharp normestimates 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 θ ∪ {0} where θ ∈ (0, π) and θ := {λ ∈ C \ {0} : | arg(λ)| < θ}, and allowing a linear resolvent estimate (λ − A) −1 ≤ C/|λ| for λ ∈ C \ θ ∪ {0}. A homomorphism from the algebra A σ of holomorphic functions on σ , σ > θ, decaying sufficiently fast at zero and at infinity, A σ := f : σ → C holomorphic and | f (λ)| ≤ C |λ| ε 1 + |λ| 2ε (1.2) (where ε, C > 0 may depend 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 [27,44,58]. If −A generates a C 0 -semigroup (e −t A ) 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. [50,Chapter XV]. The calculus is given by a bounded homomorphism from the space of bounded Borel 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 [45] (see also [59]).
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 regularisation 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 regularisation 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 negative generator of a strongly continuous semigroup ( ∞ 0 e −s A 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 [69,Chapter 13], and also [8,38], for more on the Hirsch and Bochner-Phillips calculi, and [3,24] 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 normestimate 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 [27,45,58] 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. Examples of such a situation are generators of C 0 -contraction semigroups, where the uniform estimate for Lμ(A) arises from 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 [45,47,70,80,86].
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 E := g ∈ Hol(C + ) : g E 0 := sup α>0 α R |g (α + iβ)| dβ < ∞ . (1.4) Note that · E 0 is a seminorm on E vanishing on the constant functions. As observed in [80], 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 A : B → L(X , X * * ), 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 Sect. 2 for details of the spaces. To make our observations rigorous and to establish functional calculus properties of A , we prove several intermediate statements of independent interest leading to the following theorem which summarises the main results of this paper (see Sect. 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 norm-estimates are of interest even for semigroups which decay exponentially.
The use of Besov functions for functional calculus goes back to Peller's foundational paper [67] 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 [45,[77][78][79], 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 [67]. 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 [82] from 1989. He adapted a large part of Peller's estimates [67] 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 [67] in the semigroup setting, a calculus for the generators of bounded holomorphic semigroups on a Banach space X was constructed by Vitse in [80]. Vitse's results were put in a wider context of transference methods by Haase in [45]. In [45], 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 question of constructing a full Besov functional calculus for generators of bounded semigroups on Hilbert space and its compatibility to other calculi was raised explicitly in [45, p. 2992]. Other contributions were made in [45,47,70,82,86], 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 Sect. 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 [21,55,56], again by means of Fourier analysis.
Once the B-calculus has been established, it leads to a number of sharp normestimates. 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 [47, Theorem 1.1(c)] (see Corollary 5.6).
where h(s) := esssup |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 regularisations, 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 [45,47,70,80,84,86]. We refer to Sect. 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 terms Besov algebra for the Banach algebra B, and Besov functions for its elements, 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 (Sect. 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 the Littlewood-Paley dyadic decomposition. In [80], 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: if this limit exists in C.
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 + : S(R) denotes the Schwartz space on R, BUC(R) denotes the space of bounded, uniformly continuous, functions on R, with the sup-norm, Hol( ) denotes the space of holomorphic functions on an open subset of C + , M(R) denotes the Banach algebra of all bounded Borel measures on R under convolution, and M(R + ) denotes the corresponding algebra for R + . We shall identify L 1 (R + ) with a subalgebra of M(R + ) in the usual way. L denotes the Laplace transform applied to distributions, measures or functions on R + , F denotes the Fourier transform on R. For f ∈ L 1 (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.

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 , We will use the following standard facts.
There are absolute constants C n such that | f (n) (z)| ≤ C n f ∞ (Re z) n , n ∈ N. One may take C 1 = 1/2 and C 2 = 2/π . 4. Let a > 0, b > 0, and n > m ≥ 0. Then Proof (1) and (2) are standard facts; (3) is easily seen from Cauchy's integral formula. Then (4) can be deduced by applying (3) The proof of (3) via Cauchy's integral formula around a circular contour shows that C n can be taken to be n!.
exists, then f satisfies the Cauchy integral representation as a principal value integral. This may be seen by applying Cauchy's integral formula around large semi-circles. Similarly, or by differentiating the formula as an absolutely convergent integral, for any f ∈ H ∞ (C + ), and higher derivatives can be obtained by repeated differentiation of the formula. This shows that C n can be We shall use only the cases n = 1 and n = 2, giving C 1 = 1/2 and and its inverse is given in Lemma 2.1(1) [48, Section II. 1.5].
If f ∈ H ∞ (C + ) and f b ∈ L 1 (R), then f (∞) = 0 by the Poisson integral formula in Lemma 2.1(1). Then, for every z ∈ C + , the Cauchy representation becomes (see [48, p. 170]). Thus the Fourier-Laplace transform on the right hand side of (2.2) provides the analytic extension of f b to C + .

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 + , and so f b ∈ BUC(R).

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
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 Letting ε → 0+ gives The first two statements follow immediately. The remaining statements are straightforward.
We define a norm on B as follows: The proof of Proposition 2.2 shows that · B is equivalent to each norm of the form It is easy to see that B is an algebra and · B is an algebra norm. We shall use this norm on B unless stated otherwise. This choice is convenient for the theory developed in the rest of Sect. 2 and in Sect. 4, although the algebra norm is not optimal for the norm-estimates in Sect. 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.
We now show that the normed algebra (B, · B ) is complete, and establish two related properties. 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 . Clearly, f ∞ ≤ lim inf n→∞ f n ∞ . Moreover, for x + iy ∈ C + , From this and Fatou's Lemma, it follows that 2. This can be easily deduced from (1) above. Alternatively, let ( f n ) be a Cauchy sequence in B. Then ( f n ) converges pointwise to a function f , which is holomorphic on C + , by Vitali's theorem. Then as in (2.4) , we have and the claim follows.
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.

Proposition 2.4 If f is holomorphic on
Moreover, lim |z|→∞ f (z) exists in C.
. By a standard result [29, Theorem 11.10], g : Here the final inequality is an application of the half-plane analogue of Hardy's inequality [29, p. 198

Spectral decompositions
For any closed subset I of R + , we define the spectral subspace H ∞ (I ) of H ∞ (C + ) by 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. [48, p. 175, F]), one has where e σ (z) = e −σ z , reflecting the fact that shifts of the distributional Fourier transform correspond to multiplication by exponential functions. So Using the distributional Paley-Wiener-Schwartz theorem ([52, Section 7.4]; see also [18,Theorem 3.5]), a function f ∈ H ∞ (I ), where I ⊂ R + is compact, can be recovered from its boundary function f b by means of the distributional Fourier-Laplace transform: where e z (λ) = e −zλ and the right-hand side is well-defined since the distribution F −1 f b has compact support, 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 ( 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 [48,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 Applying this to f (z + x) gives We note also Bohr's inequality (see [61,Theorem 2]), which gives the following for f ∈ H ∞ [ε, ∞), ε > 0: We now show that H ∞ [ε, σ ] ⊂ B, with a continuous embedding, when 0 < ε < σ < ∞, directly from our definition of B in Sect. 2.2. Using an approach via dyadic decompositions (see the Appendix of this paper), a similar result was obtained in [82, Lemma 5.5.10] and related ideas appear in [45,80].
and then, by Bernstein's inequality (2.6), On the other hand, by Lemma 2.1(3), (2.10) Thus we obtain Then G is a subalgebra of B by Lemma 2.5, 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 2.10 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 2.10 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 2.10.
The following lemma is quite simple but it plays a crucial role here and in the development of the functional calculus in Sect. 4.

Lemma 2.6 Let
. 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.
3. This follows immediately from [4, Section 3.9] or from direct calculations. 4. It follows from (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. 5. This is simple to check.
We shall show in Proposition 4.6 that (T B (a)) a∈C + is a holomorphic C 0 -semigroup. 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 [23,Chapter 8] and [83]. To our knowledge this theory has not previously been applied to the study of holomorphic function spaces. Let For a closed subset I of R, the spectral subspace is defined to be There are numerous equivalent definitions; see [23,Section 8.2], [63, Section 2], [65, Section 8.1], for example. In particular, Let (S(t)) t∈R be the C 0 -group of shifts on BUC(R): (2.14) A simple calculation shows that Hence

Remark 2.7
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).
is the spectral subspace of L ∞ (R) corresponding to I = R + . We will not use that fact, but we will use Lemma 2.9 below which is related to it.
The following abstract result is a consequence of [65, Corollary 8.1.8] or [83, Corollary 3.5], 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 2.8
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 .
and set Moreover, using integration by parts we obtain Thus we infer that lim a→∞ x a = x. Now (2.16) 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 i A B respectively, as in Lemma 2.6.

Lemma 2.9
Let I be a compact subset of (0, ∞).
Proof Let (S(t)) t∈R be the C 0 -group of shifts on BUC(R), as in (2.14). Let K : B → BUC(R) be the injection given by and hence, for f ∈ B, On the other hand, if I is a compact subset of (0, ∞), then H ∞ (I ) ⊂ B by Lemma 2.5, and then (2.17) implies that H ∞ (I ) ⊂ B G (I ).

Proposition 2.10 The set
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 for every x > 0, and Then, by the dominated (or monotone) convergence theorem, This shows that the range of A B is dense in B 0 . It now suffices to apply Proposition 2.8 to (G(t)) t∈R on B 0 , and then apply Lemma 2.9.

Remark 2.11
We can now give a quantified version of Lemma 2.6(1), estimating 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 Applying this with h = g and with h = g , we obtain that Then So, we have

Laplace transforms of measures
Let LM be the Hille-Phillips algebra, which is the subalgebra of H ∞ (C + ) consisting of Laplace transforms Lμ of measures μ ∈ M(R + ). Let m = Lμ and m HP = μ . Then 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 2.12
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 tr a ∈ LM and e tr a B ≤ e 2|t|/x , t ∈ R. 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.
The following lemma is not new; for example, a similar argument is given in [80, p. 250]. It plays an important role in this subject, and we give a precise statement and proof here.
We will now consider some topological properties of B and LM. Lemma 2.14 Consider B with its norm-topology.
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 Let η be as in (2.19), and for δ > 0, let uniformly on compact sets, as δ → 0+. By Lemma 2.13, f δ ∈ LM. By Proposition 2.10, G is norm-dense in B 0 . Since 1 ∈ LM, the result follows.

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. Let g ∈ E, h ∈ L ∞ (C + ) and x > 0. The following simple estimate will be very useful: Functions in E have additional properties as follows.
Proof 1. By the definition of E, the function g a belongs to the Hardy space H 1 (C + ) and g a H 1 ≤ g E 0 /a. Then the first statement follows from Proposition 2.4. 2. This follows from the first statement and Proposition 2.2(1). 3. Replacing g(z) by g(z) − g(∞), we may assume that g(∞) = 0. Then, by [ We can now define a norm · E on E by Then E becomes a Banach space. To see this, let (g n ) be a Cauchy sequence in E. Then (g n (∞)) is convergent in C. Let a > 0, and f n (z) = g n (a + z) − g n (∞). By Proposition 2.15(1), (g n ) is Cauchy in B 0 , and hence in H ∞ (C + ). Thus, the limit f (z) := lim n→∞ f n (z) exists in C for every z ∈ C + , and f is holomorphic. As in the proof of Proposition 2.3 (2), one infers that f E ≤ lim inf n→∞ f n E , and Nevertheless 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 2.4).

Proposition 2.16
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 f E 0 ≤ f H 1 .
Moreover, the function a → r a is continuous from C + to E. Hence for any bounded Borel measure μ on C + , the function There is a (partial) duality between E and B given by It follows from the definition of B, and (2.21) for h = f , that the integral exists and 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 [80, p. 266] (where it is presented in slightly different form, but (2.23) can be converted into the formula in [80] by putting g(z) = G(z)). It was also noted in [80] that Green's formula on C + transforms (2.23) into for "good functions" (the argument in [80] requires a correction: In the notation of [80], one should set u 1 = x, u 2 = F · G). See [73, Lemma 17 and Remark, p. 456] 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 (2.24) only in cases where f and g have holomorphic extensions across iR and they and their derivatives decay at infinity sufficiently fast.
Then the following hold: 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 ±iy 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 (2.25). 3. Consider g = r 0 ∈ E, so g(z) = 1/z. If a > 0 and and hence 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, 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 (2.23) and the dominated convergence theorem, we obtain that

Representation of Besov functions
Here we show that any function in B can be represented in terms of the duality as in (2.23). When g = r z where r z (λ) = (z + λ) −1 for some z ∈ C + and (2.24) 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. Then where we have used that y → (x − iy + z) −2 is the inverse Fourier transform of t → 2π te −xt e −zt on R + (extended to R by 0).

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 (2.25).

Dual Banach spaces
Our partial duality induces a contractive map 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 Lemma 2.19 that the range of B is weak*-dense in B 0 * . However it is not norm-dense.

Proposition 2.22 The range of
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 The function is not measurable on R + . If it were measurable, the map a → χ(e b a ) 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 B , and also for the functional We now give an alternative proof of Proposition 2.22. Suppose that the range of 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 2.13 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 2.14(3). Thus we conclude that the range of B is not norm-dense in B 0 * . Let E 0 = {g ∈ E : g(∞) = 0}, with the norm · E which coincides with · E 0 on E 0 . Identify the dual E 0 * with the space of linear functionals in E * which annihilate the constant functions. Our duality then provides a contractive map E from B to E 0 * .

Proposition 2.23
The range of E is not norm-dense in E 0 * .

Proof
We argue by contradiction. Suppose that the range of E is norm-dense in E 0 * . Let (T B (t)) t≥0 and (T E (t)) t≥0 be the shift semigroups defined in Lemmas 2.6 and 2.18. Let g ∈ E. By (2.25) and Lemma 2.6, 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(3).

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 Sects. 4 and 5.

Holomorphic extensions to the left
In this subsection we consider some functions which have holomorphic extensions to a strip in the left half-plane. For fixed ω > 0, we let This implies that f ∈ B and and assume that Proof 1. Applying the estimate (3.1) for g we obtain that as required. 2. By (2.5), f = e τ g where e τ (z) = e −τ z and g ∈ H ∞ ω . Let ϕ(x) = e −τ x = |e τ (x + iy)|, x > 0, y ∈ R. Then e τ B = 2, and 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

Functions with decay on iR
In the following result we give an estimate for the B-norm of certain functions which decay at infinity in an appropriate sense.
Combining the two estimates above, we infer that and hence
As observed in Example 2.12(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 (see Remark 3.6). In Sect. 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).

Lemma 3.4 Letf
For t > 0, s > 1, let By a simple argument, Hence if t ∈ (0, 1], then and for t > 1 we obtain 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 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 (3.3) 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.
Then there is a constant C such that Note first that f t ∞ = 1 and Since 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 Let which is asymptotically equivalent to 2 log t for large t. Now the estimates (3.4) for small t, and (3.5) for large t, imply (3.3).

Remark 3.6
It is possible to deduce the estimate in Lemma 3.4 from the estimate (3.3), up to a multiplicative constant, sincẽ We have included the direct proof of Lemma 3.4 as it may be of interest to some readers. The estimate in Lemma 3.5 is sharp because for some c > 0.

Cayley transforms
As observed in Example 2.12(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 Sect. 5.5). The B-norms of these functions grow only logarithmically, as shown in the following lemma. Proof Note that f 1 B = 3, by Example 2.12 (3). For n ≥ 2, let a n = n − n 2 − 1, b n = a −1 n = n + n 2 − 1.

Remark 3.9
Note that 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 [35,72] that if A satisfies the assumptions above then −A is the generator of a bounded C 0 -semigroup (T (t)) t≥0 . The Hille-Yosida conditions can be obtained by applying Proposition 2.15(3) and using (4.2). Moreover, the following holds for all x ∈ X , x * ∈ X * and α > 0: This representation is not explicit in [35] or [72], 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). Let Putting α = 1/t in (4.5) and then using (4.4), we see that It follows that 1 ≤ K A ≤ eγ A /4. This implies that

7)
Examples 4.1 1. If −A generates a bounded C 0 -semigroup on a Hilbert space X , then (4.1) necessarily holds. This was observed in [35,72], but we give the argument here. Firstly, Plancherel's Theorem in the Hilbert space L 2 (R + , X ) gives, for any α > 0 and x, y ∈ X , Then, by Cauchy-Schwarz, 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 Sect. 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 Sect. 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, by (2.21) and (4.4), We will show that f (A) ∈ L(X ) and 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.

Lemma 4.2 Let f ∈ LM and f (A) be as defined as in (4.10). Then f (A) coincides with the operator x → R + T (t)x dμ(t) as defined in the Hille-Phillips functional calculus. In particular, f (A) ∈ L(X )
.
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 −t A in place of T (t).
Our next step is a version of Lemma 2.19 for operators. Let As in Example 2.12(4) and Lemma 2.6(5), η δ B = η B = 2. , and these functions form an approximate unit for L 1 (R + ) as δ → 0+. Since the Laplace transform is a bounded algebra homomorphism from L 1 (R + ) to B, it follows that Since f η δ ∈ LM and the HP-calculus is multiplicative on LM, it follows from Lemma 4.2 that Moreover, (4.11), and D(A) is dense in X , so it follows that lim δ→0+ ( f η δ )(A)x exists in X for all x ∈ X . By Lemma 2.19, for x ∈ X and x * ∈ X * , we now have Now we present the main result showing that the map A has the essential properties of a bounded functional calculus. We shall subsequently refer to A as the B-calculus for A.

Theorem 4.4 Under the assumptions (4.1), the map A : f → f (A) is a bounded algebra homomorphism from B into L(X ), which extends the Hille-Phillips calculus. Moreover,
then f η δ ∈ LM by Lemma 2.13, so Lemma 4.3 applies to all such f . In particular, To show that 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 4.3, we obtain Lemma 4.2 shows that this functional calculus agrees with the HP-calculus on LM. The final statement follows from (4.11).
The B-calculus defined by (4.10) and Theorem 4.4 applies to all operators satisfying (4.1), and to all functions in B. In particular it applies to all generators of bounded C 0 -semigroups on Hilbert space with estimates f (A) ≤ 2K 2 A f B , by (4.9). In that context White [ [45], and a smaller class in [82]). Moreover they used an equivalent norm arising from Littlewood-Paley decompositions (see the Appendix). We obtain the estimate for all functions in B.
Applications to bounded C 0 -semigroups on Hilbert spaces are given in Sect. 5.
We note two other simple properties of the B-calculus.
Proof The first statement follows from (4.10) and an application of (2.25) 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 Sects. 4.3 (compatibility with other calculi), 4.4 (convergence lemmas) and 4.5 (spectral inclusion).
So A ∈ Sect(θ ) ⊂ Sect(π/2−). Conversely, if A ∈ Sect(θ ) where θ ∈ [0, π/2), 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 2.6. A short proof of holomorphy appeals to abstract theory. 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 [4,Theorem A.7] that T B is holomorphic on C + . Another proof proceeds by applying Proposition 2.6(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 : Since the operators T B (is) are invertible isometries on B, we may assume that a > 0.
Let h = re iθ where 0 < r < a/2, and let Then by Lemma 2.1(3) By the same argument applied to g h , It follows that Hence Thus T B is a holomorphic function from C + to L(B).
We return to the general theory. Let A ∈ Sect(π/2−). Then (4.1) holds, because where M A is the sectoriality constant of A. This easily leads to (4.2) with γ A ≤ 2M 2 A . We will give a sharper estimate in Corollary 4.8 based on the following integral estimate.

Lemma 4.7 Let A ∈ Sect(π/2−). Then
We use the representation Using (4.14), we have We estimate these integrals separately.
First, note that and we obtain Observe that Thus, Taking into account that cos θ = 1/(2M A ) and M A ≥ 1, Thus, From this we deduce the following result which was first obtained by Schwenninger [70,Theorem 5.2] with an estimate of the same form as (4.16) (up to a multiplicative constant). Previously Vitse [80,Theorem 1.7] had obtained an estimate The arguments in [70,80] were based on the sectorial functional calculus and Littlewood-Paley decompositions (see the Appendix).

Corollary 4.8 Let A ∈ Sect(π/2−). Then
Hence, for all f ∈ B, Moreover, where the integral converges in the operator norm.
Some applications of the B-calculus for operators in Sect(π/2−) are discussed in Sect. 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 B-calculus 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 [8,Sections 4,5], [44,Proposition 3.3.2] and [58], 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 [44,58]) 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 [44,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 [44,Chapter 2,etc] and [10] respectively. We shall let f → A ( f ) stand for the (extended) sectorial holomorphic functional calculus, and f → * ( f ) stand for the (extended) half-plane calculus whenever they are defined as closed operators.
In considering compatibility, we have to take into account that functions in B are bounded functions on C + while the functions appearing in the sectorial calculus are defined on open sectors properly containing σ (A) \ {0}, and functions in the halfplane calculus are defined on open half-planes containing σ (A). Consequently for compatibility with each of the two calculi, we consider two situations; one where the spectrum of A is smaller than C + , and one where f ∈ B extends holomorpically to a larger domain. In the latter situation, the extension of f may not be bounded. The techniques of proof follow the pattern of an abstract result in [44, Proposition 1.2.7], but we present them here in more concrete forms. In particular, we do not present the details in the most general possible form, using arbitrary regularisers which depend on A and f , but we present the results for classes of functions f and specific regularisers which do not depend on A. The proofs can easily be extended to general regularisers.
Thus we give four compatibility results. The two results for the sectorial calculus are Proposition 4.9 where A is sectorial of angle less than π/2, and Proposition 4.10 for functions in B which extend holomorphically to a sector of angle greater than π/2. The two results for the half-plane calculus are both covered by Proposition 4.11, as it is easy to pass from one to the other, using shifts and replacing A by A + η and correspondingly for the functions.
For A ∈ Sect(π/2−), a complication is that a Besov function is not necessarily in the domain of A . If A is injective then B is contained in the domain of A , but this is not true if A is not injective (see [44,Lemma 2.3.8], [43,Example 5.2]). In that case the domain of the extended calculus is quite small, and there are technical problems when the sector is smaller than C + . Consequently our first result is confined to injective sectorial operators. It shows that the B-calculus and the extended sectorial calculus are compatible in that case. Variants of this result were shown by Vitse in [80], 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. Proposition 4.9 Let A ∈ Sect(π/2−) be injective.

It follows from this and the definition of the extended sectorial calculus that
In particular, Proposition 4.9 implies that the Composition Rule [44,Theorem 2.4.2] for the sectorial functional calculus can be used within the B-calculus for injective A ∈ Sect(π/2−).
Another compatibility result between the B-calculus and the sectorial calculus is the following. Here we consider functions defined on θ where θ ∈ (π/2, π). In this case the domain of the primary calculus is contained in B, which simplifies the arguments. We again denote the corresponding operators by A ( f ) and A ( f ), respectively. 1 Proposition 4.10 Assume that A satisfies (4.1). Let f ∈ B and assume that f extends to a function in Hol( θ ) for some θ ∈ (π/2, π) and A ( f ) is defined.

Now let f be in the domain of the extended sectorial calculus A . Then there is a regulariser h ∈ A θ ⊂ B such that A (h) is injective and h f ∈ A θ . Then
It follows from this that A ( f ) is defined on X and coincides with A ( f ).
When A is injective and f is bounded on θ one may take the regulariser h in the proof of Proposition 4.10 to be r 1 (1 − r 1 ). Using instead (r 1 (1 − r 1 )) k for k ≥ 1, one can apply the result if f is polynomially bounded on θ .
We now consider compatibility of the B-calculus A with the half-plane holomorphic calculus * A . Assume that σ (A) ⊂ C + , (4.1) holds, and f ∈ B. We may also assume either of the following additional conditions: . The conditions (C1) and (C2) are, loosely speaking, half-plane versions of the conditions in Propositions 4.9 and 4.10, respectively. However there are some differences from the sectorial situation. Firstly, B is contained in the domain of the extended halfplane calculus in both cases, irrespective of whether A is injective. Moreover it is easy to pass between the conditions (C1) and (C2) by shifts. Let η ∈ (0, ε). It follows from the definition of * A (g), Proposition 2.20 and Fubini's theorem that * From the primary functional calculus for half-plane operators, we obtain Re z=η For x ∈ X and x * ∈ X * , we have * Next we assume that (C2) holds for some ε > 0 (but (C1) does not hold), and initially we assume that f ∈ H ∞ (R −ε ). Let η ∈ (0, ε/2). The operator A + η satisfies (C1) with ε replaced by η. From the case above we obtain that By Lemmas 2.6 and 4.5, , as required. Next, we assume that (C2) holds and f is polynomially bounded in R −ε . Then we apply the case above with f replaced by r n 1 f for sufficiently large n, and the result follows. In the more general case, f would be replaced by h f for some regulariser h.

A convergence Lemma
The Convergence Lemma in the holomorphic functional calculus of sectorial operators [44,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 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 [44,58] 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 Sect. 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 4.11 hold uniformly for the approximating sequence. In particular all the functions must be holomorphic on the same open sector or half-plane.
Corollary 4.14 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. We shall deduce the Convergence Lemma from the following result about convergence in operator norm. 3 Proof By Proposition 2.3(1), f 0 ∈ B. Replacing f n by f n − f 0 , we may assume that f 0 = 0. By Proposition 2.4, g ∈ B 0 , and then g n ∈ B 0 , n ≥ 0. Let x ∈ X , x * ∈ X * with x = x * = 1, and n ≥ 1. Writing z = α + iβ and d S(z) for area measure on C + , we obtain from the definition (4.10) of g n (A) that Recall that if (4.1) holds, then −A generates a C 0 -semigroup (e −t A ) t≥0 in X such that K A = sup t≥0 e −t A < ∞. Using (4.1), (4.2), (4.4), and a Hille-Yosida estimate for −A, for fixed δ ∈ (0, 1), we have Since g ∈ B, the first two integrals in brackets above tend to 0 as n → ∞. Since f n (z) → 0 as n → ∞ for any z ∈ C + , and g ∈ H 1 (C + ), the third integral above tends also to 0 as n → ∞, by the dominated convergence theorem. So J n → 0 as n → ∞, uniformly for x and x * with x = x * = 1.
For r > 0 let D r = {z ∈ C + : |z| < r }. For I n , we consider separately the integrals over C + \ D r and over D r , where r is to be chosen. We consider first Using (4.2) again, where the right-hand side goes to zero as r → ∞ by the assumption that g(∞) = 0 and Proposition 2.4. Let ε > 0. We may take r > 0 so large that for all n ∈ N and all x ∈ X , x * ∈ X * with x = x * = 1. Now consider We have By the assumption (4.19), 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 (r , δ) be such that the second integral is less than ε for all n ≥ N . Then, summing up the estimates for K 1 (r , n) and K 2 (r , n), one has for all n ≥ N , and the estimate is uniform for x and x * with x = x * = 1. It follows that g n (A) → 0 as n → ∞, as required.
The dominated convergence theorem can be applied to this integral. 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 [50,Section 16.5] and [23,Section 2.2]. We may assume without loss of generality that First, assume that the resolvent of A is bounded on the left half-plane. By the resolvent identity, Then we may obtain the following formula in which the integral is absolutely convergent in the operator norm: For the general case, we proceed as follows. If ε > 0, λ + ε ∈ σ (A + ε). Applying the case above for A + ε (noting that A and γ do not change), A)). 4. Let A ∈ Sect(π/2−), let A be as above, let γ be any character of A, and let f ∈ B with f (∞) = 0. Applying γ to (4.17) gives If γ ((1 + A) −1 ) = 0, then γ ((z + A) −1 ) = 0 for all z ∈ C + , and then γ ( f (A)) = 0 = f (∞). Otherwise, there exists μ ∈ σ (A) such that γ ((z + A) −1 ) = (z +μ) −1 , and then we obtain

Remark 4.18
It may be possible to show that approximate λ-eigenvectors for A are approximate f (λ)-eigenvectors for f (A). This would give a more direct proof of Theorem 4.17 (3) and provide additional insight into the fine structure of σ ( f (A)). However this approach is not straightforward and we leave it as a topic for further research.

Applications of the B-calculus for operator norm-estimates
In this section we apply the B-norm estimates obtained in Sect. 3 to the B-calculus defined in Sect. 4, and we recover (and sometimes improve) various results in semigroup theory. Recall the definitions of γ A , K A and M A in (4.4), (4.6) and (4.13), and the inequalities in (4.9) and (4.15). We generally give the estimates in the form which arises from using the norm · B , but we note that in Corollaries 5.1-5.5, the relevant functions belong to B 0 , so the estimates could be slightly improved by using (4.11).

Functions in spectral subspaces
The following result was given in [80, Theorem 1.1] with a proof using the Littlewood-Paley decomposition and with a less precise norm-estimate. The proof is immediate from Theorem 4.4, Lemmas 2.5 and 4.7.

Corollary 5.4 Let −A be the generator of a bounded C 0 -semigroup on a Hilbert space.
Let ω > 0, α > 0, λ ∈ C + , and let g ∈ H ∞ ω . Then g(A)(λ + A) −α ∈ L(X ), and Proof For fixed α > 0 and λ ∈ C + , let We have So, 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 [44, 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.

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 0 -semigroup on a Hilbert space, so e −t A ≤ Me −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 [71] who considered the case when | f (is)| ≤ (log(|s| + e)) −α for some α > 1.

Corollary 5.6
Let −A be the generator of an exponentially stable C 0 -semigroup on a Hilbert space X , so that and assume that h satisfies the assumption (3.2) of Lemma 3.3. Then Proof Let g(z) = f (z + ω). By an elementary property of the half-plane calculus, f (A) = g(A − ω). By Lemma 3.3, g ∈ B. By compatibility of the two calculi (Proposition 4.11) and Lemma 3.3, we have Here we have used that e −t(A−ω) ≤ M and lim |s|→∞ f (is) = 0 , so f (∞) = 0. We then applied (4.11) with γ A−ω = 2M 2 .

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 given in [84] (and, implicitly, already in [54]). Moreover, essentially any growth of a semigroup (e −t A ) t≥0 rules out a positive answer, and the answer is also negative for bounded semigroups on L p -spaces when p = 2; see [32], or [40] 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 [84] 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 −t A −1 ) t≥0 is bounded. Thus long-time estimates of (e −t A −1 ) t≥0 when A −1 does generate a C 0 -semigroup are of value. The case when (e −t A ) t≥0 is exponentially stable is of particular interest, since then an integral representation for (e −t A −1 ) t≥0 in terms of (e −t A ) 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(−t A −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 Sect. 3.3, these functions are not uniformly continuous on C + , and hence they are not in B. We can derive [85,Theorem 2.2], the main result of [85], from the B-calculus, up to multiplication by an absolute constant.
and there exists A −1 ∈ L(X ). 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 only on A.
Proof. Let h(z) = z(1 + z) −1 and g t (z) = e −t/z , for t > 0. Then h 2 g t is the function f t as in Lemma 3.5, so f t ∈ B and f t B ≤ C(1 + log(1 + t)) for some constant C. For ε > 0, let f t,ε (z) = f t (z + ε) and g t,ε (z) = g t (z + ε). Then g t,ε ∈ B and g t,ε (A) = e −t(A+ε) −1 which is given by a power series as in the proof of Corollary 5.7. Using Lemma 4.5, we have Letting ε → 0+, using Lemma 2.6(1) and noting that A −1 is the limit in the operator norm of (A + ε) −1 , we obtain Hence (1 + t)).

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 [15,22,49,64]. 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 [49] 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 [22,64] (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 more demanding. While it is evident that V (A) is a contraction (so power bounded) if A is the generator of a contraction C 0 -semigroup 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 [5,36,42]). 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.

Bernstein functions
Sharp resolvent estimates for Bernstein functions are valuable in the Bochner-Phillips theory of subordination of C 0 -semigroups, 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 in the case of injective operators, as we show below.

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 Sects. 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 2.10 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 2.10 follows very easily from Proposition 6.2, as noted in [80] (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. Let with the topology induced from S(R), and let S ∞ (R) be the topological dual of S ∞ (R).
Consider a system of functions (ψ n ) n∈Z ⊂ S(R) such that • supp ψ n ⊂ t ∈ R : a2 n−1 ≤ |t| ≤ b2 n , n ∈ Z, for some a, b > 0, • n∈Z ψ n (t) = 1, t ∈ R \ {0}, and • sup t∈R sup n∈Z 2 kn |ψ  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, and, for every f ∈ B s p,q (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 [6,41,66,68] or [76] 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. For p, q ≥ 1, 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 B s p,q (R) . 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 S 0 (R) := ϕ ∈ S(R) : See [14,Proposition 8] for details of that. However, we prefer to avoid dealing with quotient classes.
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 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 Riesz projection, 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 [66]. 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: P(g)(t + is) := (P t * g)(s) = n∈Z P(t)(g * ϕ + n )(s), t ≥ 0, s ∈ R, (6.2) where P t and P(t), are the Poisson kernel and semigroup for the right half-plane 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 , (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 = P f b . Since G is dense in B 0 by Proposition 2.10, 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 [81, 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 2.5, 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 , semigroup (T E (a)) a≥0 is not strongly continuous on E (Lemma 2.18), so a result for E similar to Proposition 2.10 cannot be expected, and consequently it is not clear that the embedding is surjective. One may also consider the subspace E 0,dyad of E dyad defined by Then E 0,dyad is embedded continuously inẼ, as shown in statement (2) of [80, 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 [60,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 [80], 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.
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