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

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## Abstract

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.

## Mathematics Subject Classification

Primary 47A60 Secondary 30H25 46E15 47D03## 1 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*is to define a function algebra \({\mathcal {A}}\) on the spectrum \(\sigma (A)\) of

*A*and a homomorphism from \({\mathcal {A}}\) 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

*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.

*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 \(\Sigma _\theta \cup \{0\}\) where \(\theta \in (0,\pi )\) and \(\Sigma _\theta :=\{\lambda \in {\mathbb {C}}\setminus \{0\}: |\arg (\lambda )|<\theta \}\), and allowing a linear resolvent estimate \(\Vert (\lambda -A)^{-1}\Vert \le C/|\lambda |\) for \(\lambda \in {\mathbb {C}} \setminus \Sigma _{\theta }\cup \{0\}\). A homomorphism from the algebra \({\mathcal {A}}_\sigma \) of holomorphic functions on \(\Sigma _\sigma , \sigma >\theta \), decaying sufficiently fast at zero and at infinity,

*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].

*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 \(M({\mathbb {R}}_+)\) 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 \(z^\alpha \) 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 \(\mu \) in (1.3) is replaced by a convolution semigroup of measures \((\mu _t)_{t \ge 0}\), and a Bernstein function *f*(*A*) of *A* arises as the negative generator of a strongly continuous semigroup \((\int _{0}^{\infty } e^{-sA} \, d\mu _t(s))_{t \ge 0}\); 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 norm-estimate in terms of the representing measure \(\mu \) rather than \({\mathcal {L}} \mu \), 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 \(\Vert f(A)\Vert \le C_A \Vert f\Vert _\infty \) for all \(f \in {\mathcal {A}}_\sigma \), 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 \(H^\infty \)-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 \(H^\infty \)-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 \(\Vert {\mathcal {L}} \mu (A)\Vert \le C_A \Vert \mu \Vert \) 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 \({\mathcal {L}}\mu (A)\) 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 \(H^\infty \)-norm, but as close as possible to the \(H^\infty \)-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].

*X*or the generator of a bounded holomorphic semigroup on a Banach space

*X*. Then for all \(x \in X\) and \(x^* \in X^*\) its weak resolvent \(\langle (\cdot + A)^{-1}x, x^*\rangle \) belongs to the space \({\mathcal {E}}\) given by

*A*as above, we define a mapping \(\Phi _A : {\mathcal {B}} \rightarrow L(X,X^{**})\), \(\Phi _A(f)=f(A)\), as a \(w^*\)-integral:

*A*, since \({\mathcal {L}}(M({\mathbb {R}}_+))\) is not dense in \({\mathcal {B}}\). The duality \(\langle \cdot , \cdot \rangle _{{\mathcal {B}}}\) is only partial, since the spaces \({\mathcal {E}}\) and \({\mathcal {B}}\) have rather complicated structures; see Sect. 2 for details of the spaces. To make our observations rigorous and to establish functional calculus properties of \(\Phi _A\), 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

*X*or of a (sectorially) bounded holomorphic \(C_0\)-semigroup on a Banach space

*X*. Then the following hold.

- (a)The formula (1.8) defines a bounded algebra homomorphismand$$\begin{aligned} \Phi _A: {\mathcal {B}}\rightarrow L (X),\qquad \Phi _A (f):=f(A), \end{aligned}$$for a constant \(C_A\) depending only on$$\begin{aligned} \Vert f(A)\Vert \le C_A \Vert f\Vert _{{\mathcal {B}}} \end{aligned}$$(1.9)
*A*. - (b)
The \({\mathcal {B}}\)-calculus defined in a) (strictly) extends the HP-calculus, and it is compatible with the holomorphic functional calculi for sectorial and half-plane type operators.

- (c)
The spectral inclusion (spectral mapping, in the case of bounded holomorphic semigroups) theorem and a Convergence Lemma hold for \(\Phi _A\).

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 \(B^0_{\infty ,1}({\mathbb {D}})\) which is the analogue of \({\mathcal {B}}\) 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 \(B^0_{\infty ,1}({\mathbb {D}})\). 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 \(B^0_{\infty ,1}({\mathbb {D}})\), 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 \({\mathbb {C}}_+\). 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 \({\mathcal {L}} \mu \) for \(\mu \in M({\mathbb {R}}_+)\) 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 \(i{\mathbb {R}}\)) and for sectorial operators of zero angle were studied in [21, 55, 56], again by means of Fourier analysis.

Once the \({{\mathcal {B}}}\)-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 \({\mathcal {B}}\). As a sample result, we formulate the next statement which generalises results in [47, Theorem 1.1(c)] (see Corollary 5.6).

### Corollary 1.2

*f*. If \(-A\) is the generator of an exponentially stable \(C_0\)-semigroup on a Hilbert space

*X*, then \(f(A) \in L (X)\) and

*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 \((z-1)^n/(z+1)^n, n \in {\mathbb {N}}\). In the context of generators of bounded holomorphic semigroups, we give sharp constants for the norms of \({\mathcal {B}}\)-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 \({\mathcal {B}}\)-calculus. However, we emphasize that the most attractive feature of the \({\mathcal {B}}\)-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 \({\mathcal {B}}\), and *Besov functions* for its elements, we use only the definition of \({\mathcal {B}}\) 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 \({\mathcal {B}}\) (or more precisely, a subspace of codimension 1), is a realization of the analytic part of a conventional Besov space on \({\mathbb {R}}\) defined via the Littlewood–Paley dyadic decomposition. In [80], the algebra \({\mathcal {B}}\) arises as a half-plane realization of the analytic Besov algebra \(B^0_{\infty ,1}({\mathbb {C}}_+)\), and the space \({\mathcal {E}}\) is shown there to contain a continuously embedded analytic Besov space \(B^0_{1,\infty }({\mathbb {C}}_+)\).

**Notation**Throughout the paper, we shall use the following notation:

\({\mathbb {R}}_+ :=[0,\infty )\),

\({\mathbb {C}}_+ := \{z \in {\mathbb {C}}: {\text {Re}}z>0\}\), \({\overline{{\mathbb {C}}}}_+ = \{z \in {\mathbb {C}}: {\text {Re}}z\ge 0\}\),

\(\Sigma _\theta := \{z\in {\mathbb {C}}: z \ne 0, |\arg z|<\theta \}\) for \(\theta \in (0,\pi )\); \(\Sigma _0 := (0,\infty )\),

\(\mathrm {R}_a := \{z\in {\mathbb {C}}: {\text {Re}}z > a\}\).

\({\text {supp}}(f)\) denotes the support of a function or distribution

*f*on \({\mathbb {R}}\),- For \(f : {\mathbb {C}}_+ \rightarrow {\mathbb {C}}\), we writeif this limit exists in \({\mathbb {C}}\).$$\begin{aligned} f(\infty ) = \lim _{{\text {Re}}z \rightarrow \infty } f(z) \end{aligned}$$

\({\mathcal {S}}({\mathbb {R}})\) denotes the Schwartz space on \({\mathbb {R}}\),

\({\text {BUC}}({\mathbb {R}})\) denotes the space of bounded, uniformly continuous, functions on \({\mathbb {R}}\), with the sup-norm,

\({\text {Hol}}(\Omega )\) denotes the space of holomorphic functions on an open subset \(\Omega \) of \({\mathbb {C}}_+\),

\(M({\mathbb {R}})\) denotes the Banach algebra of all bounded Borel measures on \({\mathbb {R}}\) under convolution, and \(M({\mathbb {R}}_+)\) denotes the corresponding algebra for \({\mathbb {R}}_+\). We shall identify \(L^1({\mathbb {R}}_+)\) with a subalgebra of \(M({\mathbb {R}}_+)\) in the usual way.

\({\mathcal {L}}\) denotes the Laplace transform applied to distributions, measures or functions on \({\mathbb {R}}_+\),

- \({\mathcal {F}}\) denotes the Fourier transform on \({\mathbb {R}}\). For \(f \in L^1({\mathbb {R}})\),We shall also consider \({\mathcal {F}}\) applied to measures and distributions on \({\mathbb {R}}\), and \({\mathcal {F}}^{-1}\) will be the inverse Fourier transform on \({\mathbb {R}}\).$$\begin{aligned} ({\mathcal {F}}f)(s) := \int _{\mathbb {R}}f(t)e^{-ist} \,dt. \end{aligned}$$

*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*), \(\sigma (A)\) and \(\rho (A)\), respectively.

## 2 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 \(H^\infty ({\mathbb {C}}_+)\)

### Lemma 2.1

- 1.
\(\displaystyle f(t+is) = (P(t) f^b)(s) = \frac{1}{\pi } \int _{\mathbb {R}}\frac{t f^b(y)}{t^2 + (y-s)^2} \,dy\),

- 2.
\(\Vert f\Vert _\infty = \Vert f^b\Vert _{L^\infty ({\mathbb {R}})}\) (maximum principle);

- 3.
There are absolute constants \(C_n\) such that \(|f^{(n)}(z)| \le \dfrac{C_n \Vert f\Vert _\infty }{({\text {Re}}z)^n}\), \(n \in {\mathbb {N}}\). One may take \(C_1=1/2\) and \(C_2 = 2/\pi \).

- 4.Let \(a>0\), \(b>0\), and \(n> m \ge 0\). Then$$\begin{aligned} \sup _{{\text {Re}}z = a+b} |f^{(n)}(z)| \le \frac{C_{n-m}}{b^{n-m}} \sup _{{\text {Re}}z = a} |f^{(m)}(z)|. \end{aligned}$$

### Proof

(1) and (2) are standard facts; (3) is easily seen from Cauchy’s integral formula. Then (4) can be deduced by applying (3) to \(f^{(m)}(z+a)\). \(\square \)

*n*!. However if \(f \in H^\infty ({\mathbb {C}}_+)\) and \(f(\infty ):=\lim _{{\text {Re}}z \rightarrow \infty } f(z)\) exists, then

*f*satisfies the Cauchy integral representation

Let \(H^\infty ({\mathbb {R}})\) denote the space of \(g \in L^\infty ({\mathbb {R}})\) such that \({\text {supp}}({\mathcal {F}}^{-1}g) \subset {\mathbb {R}}_+\). Then the mapping \(f \mapsto f^b\) is an isometric isomorphism from \(H^\infty ({\mathbb {C}}_+)\) onto \(H^{\infty }({\mathbb {R}})\) and its inverse is given in Lemma 2.1(1) [48, Section II.1.5].

### 2.2 The Banach algebra \({\mathcal {B}}\)

*f*on \({\mathbb {C}}_+\) such that

### Proposition 2.2

- 1.
\(f(\infty ):= \lim _{{\text {Re}}z\rightarrow \infty } f(z)\) exists in \({\mathbb {C}}\).

- 2.
*f*is bounded, and \(\Vert f\Vert _\infty \le |f(\infty )| + \Vert f\Vert _{{\mathcal {B}}_0}\). - 3.
\(f(is) = \lim _{x\rightarrow 0+} f(x+is)\) uniformly for \(s \in {\mathbb {R}}\).

- 4.
The extended function

*f*is uniformly continuous on \({\overline{{\mathbb {C}}}}_+\), and so \(f^b \in {\text {BUC}}({\mathbb {R}})\). - 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 \(h \circ f \in {\mathcal {B}}\). - 6.
If

*f*is bounded away from 0, then \(1/f \in {\mathcal {B}}\). - 7.
Assume that the range of

*f*is contained in \(\Sigma _\pi \). If \(\beta >1\), then \(f^\beta (z) := f(z)^\beta \in {\mathcal {B}}\). If*f*is bounded away from 0, then \(f^\beta \in {\mathcal {B}}\) for all \(\beta \in {\mathbb {R}}\).

### Proof

*z*to \(x+\varepsilon ^{-1}y\), and then the horizontal half-line \([x+\varepsilon ^{-1}y,\infty )\) gives

The space \({\mathcal {B}}\) 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 \(({\mathcal {B}}, \Vert \cdot \Vert _{\mathcal {B}})\) is complete, and establish two related properties.

### Proposition 2.3

- 1.
The closed unit ball

*U*of \({\mathcal {B}}\) is compact in the topology of uniform convergence on compact subsets of \({\mathbb {C}}_+\). - 2.
\({\mathcal {B}}\) is a Banach algebra.

- 3.
For \(f \in {\mathcal {B}}\), the spectrum of

*f*in \({\mathcal {B}}\) is the closure of the range of*f*(considered as a function on \({\mathbb {C}}_+\)) and the spectral radius of*f*in \({\mathcal {B}}\) is \(\Vert f\Vert _\infty \).

### Proof

1. Since *U* is contained in the closed unit ball of \(H^\infty ({\mathbb {C}}_+)\), 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 \(f \in U\).

*f*, which is holomorphic on \({\mathbb {C}}_+\), by Vitali’s theorem. Then as in (2.4) , we have

3. This follows easily from Proposition 2.2(6). \(\square \)

*f*on \({\mathbb {C}}_+\) such that

### Proposition 2.4

*f*is holomorphic on \({\mathbb {C}}_+\) and \(f' \in H^1({\mathbb {C}}_+)\), then \(f \in {\mathcal {B}}\), \(f' \in L^1({\mathbb {R}}_+)\), and

### Proof

Now as \((f')^b \in L^1({\mathbb {R}})\) and *f* is uniformly continuous on \({\overline{{\mathbb {C}}}}_+\), we have \((f^b)'=i(f')^b\) a.e. by a simple limiting argument. Then the two limits \(\lim _{s \rightarrow \pm \infty }f^b(s)\) exist in \({\mathbb {C}}\), and they are equal, since \(g(0)=0\). By a consequence of the Phragmén–Lindelöf principle [74, Theorem 5.63], it follows that \(\lim _{|z| \rightarrow \infty }f(z)=\lim _{s \rightarrow \pm \infty } f^b(is)\). (The formulation of [74, Theorem 5.63] requires *f* to be holomorphic on \({\overline{{\mathbb {C}}}}_+\), but it clearly extends to the case when *f* is continuous on \({\overline{{\mathbb {C}}}}_+\) and holomorphic on \({\mathbb {C}}_+\).) \(\square \)

In Sects. 2.4 and 3 we shall consider some other classes of functions which are in \({\mathcal {B}}\).

### 2.3 Spectral decompositions

*I*of \({\mathbb {R}}_+\), we define the spectral subspace \(H^{\infty }(I)\) of \(H^\infty ({\mathbb {C}}_+)\) by

*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

*f*(

*z*) by \(f(z+is)\). It follows, from the Poisson formula (Lemma 2.1) for example, that if \(f \in H^\infty (I)\), then \(y \mapsto f(x+iy)\) and \(y \mapsto f'(x+iy)\) (for fixed \(x>0\)) also have distributional inverse Fourier transforms with support in

*I*. For similar statements for more general distributions see [18, Proposition 2.10].

*g*extends to an entire function of exponential type not exceeding \(\sigma \) [48, Chapter II.5.7]. This can be easily transformed into a characterization of \(H^\infty (I)\) 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 \(H^\infty (I)\) has

*g*extends to \({\mathbb {C}}_+\) with the representation

### Lemma 2.5

### Proof

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 \({\mathcal {B}}\). 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

- 1.For each \(f \in {\mathcal {B}}\),$$\begin{aligned} \Vert T_{\mathcal {B}}(a)f\Vert _{\mathcal {B}}\le \Vert f\Vert _{\mathcal {B}}, \quad \lim _{a\in {{\overline{{\mathbb {C}}}}}_+, a\rightarrow 0}\,\Vert T_{\mathcal {B}}(a)f-f\Vert _{\mathcal {B}}=0. \end{aligned}$$
- 2.Let \(-A_{\mathcal {B}}\) be the generator of the \(C_0\)-semigroup \((T_{\mathcal {B}}(t))_{t\ge 0}\) on \({\mathcal {B}}\). Then$$\begin{aligned} D(A_{\mathcal {B}}) = \{ f \in {\mathcal {B}}: f' \in {\mathcal {B}}\}, \quad A_{\mathcal {B}}f = -f'. \end{aligned}$$
- 3.
The generator of the \(C_0\)-group \((T_{\mathcal {B}}(-is))_{s\in {\mathbb {R}}}\) is \(iA_{\mathcal {B}}\).

- 4.
\(\sigma (A_{\mathcal {B}}) = {\mathbb {R}}_+\).

- 5.
Let \(f\in {\mathcal {B}}\) and \((S_{\mathcal {B}}(b)f)(z) = f(bz)\), \(b>0\). Then \(S_{\mathcal {B}}(b)f \in {\mathcal {B}}\) and \(\Vert S_{\mathcal {B}}(b)f\Vert _{\mathcal {B}}= \Vert f\Vert _{\mathcal {B}}\).

### Proof

1. It is clear from the definition of \(\Vert \cdot \Vert _{\mathcal {B}}\) that \(\Vert T_{\mathcal {B}}(t)f\Vert _{\mathcal {B}}\le \Vert f\Vert _{\mathcal {B}}\).

*f*is uniformly continuous on \({{\overline{{\mathbb {C}}}}}_+\), \(\Vert T_{\mathcal {B}}(a)f - f\Vert _\infty \rightarrow 0\) as \(a\rightarrow 0\). For \(x>0\) and \(y \in {\mathbb {R}}\), integrating \(f''\) along a line-segment and applying Lemma 2.1(3) gives

3. This follows immediately from [4, Section 3.9] or from direct calculations.

4. It follows from (2) and (3) that \(\sigma (A_{\mathcal {B}}) \subset {\mathbb {R}}_+\). On the other hand, for any \(a\in {\mathbb {R}}_+\), \(e_a\) is an eigenvector of \(A_{\mathcal {B}}\) with eigenvalue *a*.

5. This is simple to check. \(\square \)

We shall show in Proposition 4.6 that \((T_{\mathcal {B}}(a))_{a\in {\mathbb {C}}_+}\) 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.

*X*, with generator

*A*, so the spectrum \(\sigma (A) \subset i{\mathbb {R}}\). For \(x \in X\) define the spectrum \({\text {sp}}_T(x)\) of \(x \in X\) as

*I*of \({\mathbb {R}}\), the spectral subspace is defined to be

*T*-invariant subspace

*Y*of

*X*such that \(\sigma (A_Y) \subset \{is: s\in I\}\), where \(A_Y\) is the generator of

*T*restricted to

*Y*. Moreover \(\{s \in {\mathbb {R}}: is \in \sigma (A)\}\) is the smallest closed set \(J \subset {\mathbb {R}}\) such that \(X_T(J) = X\) (see [4, Remark 4.6.2, Lemma 4.6.8], [23, Lemma 8.17]).

### Remark 2.7

The notion of spectral subspaces also applies to the duals of \(C_0\)-groups, and in particular to the shifts on \(L^\infty ({\mathbb {R}})\) regarded as the dual of a \(C_0\)-group on \(L^1({\mathbb {R}})\). Since (2.15) is also valid for \(f \in L^\infty ({\mathbb {R}})\), the space \(H^\infty ({\mathbb {R}})\) is the spectral subspace of \(L^\infty ({\mathbb {R}})\) corresponding to \(I = {\mathbb {R}}_+\). 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

*X*, with generator

*A*, and assume that the range of

*A*is dense in

*X*.

- 1.
The set \(\bigcup \{X_T(I) : I \; \mathrm{compact}, 0\notin I \}\) is dense in

*X*. - 2.
If, in addition, \(\sigma (A)\subset i [0,\infty )\), then \(\bigcup \{X_T(I) : I \; \mathrm{compact}, I \subset (0,\infty ) \}\) is dense in

*X*.

### Proof

*A*imply both claims. \(\square \)

### Lemma 2.9

Let *I* be a compact subset of \((0,\infty )\). Then \({\mathcal {B}}_G(I) = H^\infty (I)\).

### Proof

*I*of \({\mathbb {R}}_+\).

On the other hand, if *I* is a compact subset of \((0,\infty )\), then \(H^\infty (I) \subset {\mathcal {B}}\) by Lemma 2.5, and then (2.17) implies that \(H^\infty (I) \subset {\mathcal {B}}_G(I)\). \(\square \)

### Proposition 2.10

The set \({\mathcal {G}}:=\bigcup _{0<\varepsilon <\delta } H^{\infty }[\varepsilon ,\delta ]\) is dense in \({\mathcal {B}}_0\).

### Proof

We consider the restrictions of *G*(*t*) to \({\mathcal {B}}_0\) denoted by the same symbol, and note that this does not change the spectral subspaces \({\mathcal {B}}_G[\varepsilon ,\delta ]\).

### Remark 2.11

We can now give a quantified version of Lemma 2.6(1), estimating \(\Vert T_{\mathcal {B}}(t)f-f\Vert _{\mathcal {B}}\) 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).

### 2.4 Laplace transforms of measures

Since the Laplace transform \(\mu \mapsto {\mathcal {L}}\mu \) is a bounded algebra homomorphism from \(M({\mathbb {R}}_+)\) to \({\mathcal {B}}\), \({{\mathcal {L}}}{{\mathcal {M}}}\) is a subalgebra of \({\mathcal {B}}\). The convolution-exponential \(\exp _*(\mu )\) of \(\mu \) in \(M({\mathbb {R}}_+)\) may be calculated in \({{\mathcal {L}}}{{\mathcal {M}}}\), and its Laplace transform is the function \(z \mapsto e^{m(z)}\).

### Examples 2.12

- 1.
For \(a \in {\mathbb {R}}_+\), let \(e_a(z) = e^{-az}\). Then \(e_a\) is the Laplace transform of the Dirac delta measure \(\delta _a\), \(\Vert e_a\Vert _{\mathcal {B}}= 2\) if \(a>0\), and \(\Vert e_a- e_b\Vert _{\mathcal {B}}= 4\) whenever

*a*,*b*are distinct and non-zero. - 2.
For \(a = x+iy \in {\mathbb {C}}_+\), let \(r_a(z) = (z+a)^{-1}\). Then \(r_a\) is the Laplace transform of the function \(e_a \in L^1({\mathbb {R}}_+)\) and \(\Vert r_a\Vert _{\mathcal {B}}= 2/x\). Hence \(e^{ t r_a} \in {{\mathcal {L}}}{{\mathcal {M}}}\) and \(\Vert e^{t r_a}\Vert _{\mathcal {B}}\le e^{2|t|/x}, \, t \in {\mathbb {R}}\). We shall give a sharper estimate in Lemma 3.4.

- 3.Let \(\chi = 1 - 2r_1\), soThen \(\chi \) is the Laplace transform of the measure \(\delta _0 - 2e^{-t}\), and \(\Vert \chi \Vert _{\mathcal {B}}= 3\).$$\begin{aligned} \chi (z) = \frac{z-1}{z+1}. \end{aligned}$$
- 4.LetThen \(\eta \) is the Laplace transform of Lebesgue measure on [0, 1], and \(\Vert \eta \Vert _{\mathcal {B}}= 2\).$$\begin{aligned} \eta (z):=\frac{1-e^{-z}}{z}. \end{aligned}$$(2.19)

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.

### Lemma 2.13

Let \(f \in H^\infty [0,\sigma ]\), \(g \in {{\mathcal {L}}}{{\mathcal {M}}}\) with \({\mathcal {F}}^{-1}g^b \in L^2[0,\delta ]\), where \(\sigma , \delta >0\). Then \(f g \in {{\mathcal {L}}}{{\mathcal {M}}}\).

### Proof

Since \(f \in H^\infty [0,\sigma ]\), \({\mathcal {F}}^{-1}f^b\) is a distribution on \({\mathbb {R}}\) with support in \([0,\sigma ]\). Then \(\psi := {\mathcal {F}}^{-1}f^b * {\mathcal {F}}^{-1}g^b\) is a distribution on \({\mathbb {R}}\) with support in \([0,\sigma +\delta ]\), and \({\mathcal {F}}\psi = f^b g^b\). Since \(f^b\) is bounded and \(g^b \in L^2({\mathbb {R}})\) by Plancherel, \(f^b g^b \in L^2({\mathbb {R}})\). Then by Plancherel, \(\psi \in L^2({\mathbb {R}})\) and \(fg = {\mathcal {L}}\psi \). Since \(\psi \) has support in \([0,\sigma +\delta ]\), \(\psi \in L^1 ({\mathbb {R}}_+)\), so \(fg \in {{\mathcal {L}}}{{\mathcal {M}}}\). \(\square \)

We will now consider some topological properties of \({\mathcal {B}}\) and \({{\mathcal {L}}}{{\mathcal {M}}}\).

### Lemma 2.14

- 1.
The Banach space \({\mathcal {B}}\) is not separable.

- 2.
The subspace \({{\mathcal {L}}}{{\mathcal {M}}}\) is not closed in \({\mathcal {B}}\).

- 3.
\({{\mathcal {L}}}{{\mathcal {M}}}\) is not dense in \({\mathcal {B}}\) in the norm-topology.

- 4.
\({{\mathcal {L}}}{{\mathcal {M}}}\) is dense in \({\mathcal {B}}\) in the topology of uniform convergence on compact subsets of \({\mathbb {C}}_+\).

### Proof

- 1.
The first statement is an immediate consequence of Example 2.12(1).

- 2.
If \({{\mathcal {L}}}{{\mathcal {M}}}\) were closed in \({\mathcal {B}}\), then \(\Vert \cdot \Vert _{\text {HP}}\) and \(\Vert \cdot \Vert _{\mathcal {B}}\) would be equivalent norms on \({{\mathcal {L}}}{{\mathcal {M}}}\). We shall show in Sects. 3.4 and 5.5 that this is not so.

- 3.For this we recall some facts about spaces of functions on \({\mathbb {R}}\) and apply them to boundary functions. For each measure \(\mu \in M({\mathbb {R}})\), \({\mathcal {F}}^{-1}\mu \) is weakly almost periodic [30, Theorem 11.2], [28, Corollary 4.2.4]. The weakly almost periodic functions form a proper closed subspace of \({\text {BUC}}({\mathbb {R}})\) [30, Theorems 5.3, 12.1]. Moreover, the space of functions in \({\text {BUC}}({\mathbb {R}})\) which are restrictions to \({\mathbb {R}}\) of entire functions of exponential type is dense in \({\text {BUC}}({\mathbb {R}})\), by the Bernstein–Kober theorem (see [13, Theorem 12.11.1]). Hence there exist an entire function
*G*of exponential type and \(\delta >0\) such that, for all \(\mu \in M({\mathbb {R}})\),where \(G_{\mathbb {R}}\) is the restriction of$$\begin{aligned} \Vert G_{\mathbb {R}}- {\mathcal {F}}^{-1}\mu \Vert _{L^\infty ({\mathbb {R}})} \ge \delta , \end{aligned}$$*G*to \({\mathbb {R}}\). Let \(\sigma \) be greater than the exponential type of*G*, so that the support of \({\mathcal {F}}^{-1}G_{\mathbb {R}}\) is contained in \([-\sigma ,\sigma ]\). Let \(g(z)=e^{-2i\sigma z} G(z)\) and note that \({\text {supp}} ({\mathcal {F}}^{-1}g_{\mathbb {R}})\subset [\sigma , 3\sigma ]\). For \(\mu \in M({\mathbb {R}})\), define \(\mu _\sigma \) by \(\mu _\sigma (A)=\mu (A + 2\sigma )\) for each Borel subset*A*of \({\mathbb {R}}\). ThenNow let \(f(z) = g(-iz)\) for \(z \in {\mathbb {C}}_+\). Then \(f \in H^\infty [\sigma , 3\sigma ] \subset {\mathcal {B}}\), and \(\Vert f- m\Vert _{\mathcal {B}}\ge \Vert f- m\Vert _\infty \ge \delta \) for all \(m \in {{\mathcal {L}}}{{\mathcal {M}}}\).$$\begin{aligned} \Vert g_{\mathbb {R}}- {\mathcal {F}}^{-1}\mu \Vert _{L^\infty ({\mathbb {R}})} = \Vert G_{\mathbb {R}}-{\mathcal {F}}^{-1}\mu _\sigma \Vert _{L^\infty ({\mathbb {R}})} \ge \delta . \end{aligned}$$ - 4 .
Let \(\eta \) be as in (2.19), and for \(\delta >0\), let \(\eta _\delta (z) = \eta (\delta z)\). For \(f \in {\mathcal {G}}\), let \(f_\delta (z) = f(z) \eta _\delta (z)\). Then \(\eta _\delta (z) \rightarrow 1\) and \(f_\delta (z) \rightarrow f(z)\) uniformly on compact sets, as \(\delta \rightarrow 0+\). By Lemma 2.13, \(f_\delta \in {{\mathcal {L}}}{{\mathcal {M}}}\). By Proposition 2.10, \({\mathcal {G}}\) is norm-dense in \({{\mathcal {B}}_0}\). Since \(1 \in {{\mathcal {L}}}{{\mathcal {M}}}\), the result follows. \(\square \)

### 2.5 Duality and weak approximation

*g*on \({\mathbb {C}}_+\) such that

### Proposition 2.15

- 1.
For each \(a>0\), \(g_a \in {\mathcal {B}}\) and \(\Vert g_a\Vert _{{\mathcal {B}}_0} \le \Vert g\Vert _{{\mathcal {E}}_0}/(2a)\).

- 2.
\(g(\infty ) := \lim _{{\text {Re}}z\rightarrow \infty } g(z)\) exists in \({\mathbb {C}}\).

- 3.
There exists \(h \in L^\infty ({\mathbb {R}}_+)\) such that \(g(z) = g(\infty ) + {\mathcal {L}}{h}(z)\).

- 4.
There exists \(C_g>0\) such that \(|g(z) - g(\infty )| \le C_g({\text {Re}}z)^{-1}\) and \(|g^{(n)}(z)| \le n!C_g ({\text {Re}}z)^{-(n+1)}, \, z \in {\mathbb {C}}_+,\, n\ge 1\) .

### Proof

- 1.
By the definition of \({\mathcal {E}}\), the function \(g_a'\) belongs to the Hardy space \(H^1({\mathbb {C}}_+)\) and \(\Vert g_a'\Vert _{H^1} \le \Vert g\Vert _{{\mathcal {E}}_0}/a\). 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 \(g(z)-g(\infty )\), we may assume that \(g(\infty )=0\). Then, by [57, Theorem 2.2],*g*satisfies Widder’s growth conditionBy Widder’s theorem [4, Theorem 2.2.1], \(g = {\mathcal {L}}h\) for some \(h \in L^\infty ({\mathbb {R}}_+)\).$$\begin{aligned} \sup _{x>0,k\ge 0} \frac{x^{k+1}}{k!} \big |g^{(k)}(x) \big | < \infty . \end{aligned}$$ - 4.
This is immediate from (2), with \(C_g = \Vert h\Vert _\infty \).

*f*is holomorphic. As in the proof of Proposition 2.3(2), one infers that \(\Vert f\Vert _{{\mathcal {E}}} \le \liminf _{n \rightarrow \infty }\Vert f_n\Vert _{{\mathcal {E}}}\), and

The proof of Proposition 2.15 has shown a relation between \({\mathcal {E}}\) and \(H^1({\mathbb {C}}_+)\), and the following proposition gives another relation between them (cf. Proposition 2.4).

### Proposition 2.16

If \(f \in H^1({\mathbb {C}}_+)\), then \(f \in {\mathcal {E}}\).

### Proof

### Examples 2.17

- 1.The functions \(r_a(z) := (z+a)^{-1} \quad (a \in {{\overline{{\mathbb {C}}}}}_+)\) are in \({\mathcal {E}}\) and \(\Vert r_a\Vert _{{\mathcal {E}}_0} = \pi \). Moreover, the function \(a \mapsto r_a\) is continuous from \({\mathbb {C}}_+\) to \({\mathcal {E}}\). Hence for any bounded Borel measure \(\mu \) on \({\mathbb {C}}_+\), the functionis in \({\mathcal {E}}\).$$\begin{aligned} f(z) := \int _{{\mathbb {C}}_+} \frac{d\mu (a)}{z+a} \end{aligned}$$
- 2.
The functions \(e_a(z) := e^{-az} \quad (a \in {{\overline{{\mathbb {C}}}}}_+)\) are not in \({\mathcal {E}}\).

- 3.
The function \((z+1)^{-2}\log z\) is in \(H^1({\mathbb {C}}_+) \subset {\mathcal {E}}\), but it is unbounded near 0.

- 4.
The function \(e^{-1/z}\) is in \({\mathcal {E}}\) and is bounded on \({\mathbb {C}}_+\). It is not in \({\mathcal {B}}\) as it is not uniformly continuous near 0.

*f*and

*g*have holomorphic extensions across \(i{\mathbb {R}}\) and they and their derivatives decay at infinity sufficiently fast.

### Lemma 2.18

- 1.
\(T_{\mathcal {E}}(a) g \in {\mathcal {E}}\) and \(\Vert T_{\mathcal {E}}(a) g\Vert _{\mathcal {E}}\le \Vert g\Vert _{\mathcal {E}}\).

- 2.Let \(f \in {\mathcal {B}}\) and \(T_{\mathcal {B}}(a)f\) be as in Lemma 2.6. Then$$\begin{aligned} \langle T_{\mathcal {E}}(a)g,f \rangle _{\mathcal {B}}= \langle g, T_{\mathcal {B}}(a)f \rangle _{\mathcal {B}}. \end{aligned}$$(2.25)
- 3.
The semigroup \((T_{\mathcal {E}}(t))_{t\ge 0}\) is not strongly continuous on \(({\mathcal {E}}, \Vert \cdot \Vert _{\mathcal {E}})\).

- 4.
\(S_{\mathcal {E}}(b)g \in {\mathcal {E}}\) and \(\Vert S_{\mathcal {E}}(b)g\Vert _{{\mathcal {E}}} = b^{-1}\Vert g\Vert _{\mathcal {E}}\).

### Proof

- 1.
This is very simple.

- 2.
For \(a = is \in i{\mathbb {R}}\), the statement follows from a simple change of variable. So we may assume that \(a>0\).

Let \(f \in {\mathcal {B}}\), and \(x>0\). Let \(h(z) := g'(a+x-z) f'(x+z)\) for \(-x< {\text {Re}}z < a+x\). Since \(f \in {\mathcal {B}}\) and \(g \in {\mathcal {E}}\), the integrals of*h*along \(i{\mathbb {R}}\) and \(a+i{\mathbb {R}}\) are absolutely convergent, andBy applying Cauchy’s theorem to the integral of$$\begin{aligned} \int _{\mathbb {R}}\left| \int _0^a h(t+iy) \, dt\right| \,dy&\le \int _0^a \frac{\Vert g\Vert _{{\mathcal {E}}}\Vert f\Vert _\infty }{(a+x-t)(x+t)} \,dt < \infty . \end{aligned}$$*h*around the rectangles with vertices at \(\pm iy_n\) and \(a \pm iy_n\), for suitable \(y_n\rightarrow \infty \), we conclude that the integrals of*h*along \(i{\mathbb {R}}\) and \(a+i{\mathbb {R}}\) coincide, so thatMultiplying by$$\begin{aligned} \int _{\mathbb {R}}g'(a+x-iy) f'(x+iy) \, dy = \int _{\mathbb {R}}g'(x-iy) f'(a+x+iy) \, dy. \end{aligned}$$*x*and integrating with respect to*x*over \({\mathbb {R}}_+\) gives (2.25). - 3.Consider \(g = r_0 \in {\mathcal {E}}\), so \(g(z)= 1/z\). If \(a>0\) andthen \(xJ(x;a)=sJ(s;1),\) where \(s=x/a\). So$$\begin{aligned} J(x;a):=\int _{{\mathbb {R}}}\left| \frac{1}{(x+iy)^2}-\frac{1}{(x+a+iy)^2}\right| \,dy, \end{aligned}$$and hence \({\mathbb {R}}_+ \ni a \mapsto T_{\mathcal {E}}(a) r_0\) is not continuous at 0.$$\begin{aligned} \Vert T_{\mathcal {E}}(a)r_0 - r_0\Vert _{\mathcal {E}}= \sup _{x>0}\,xJ(x;a)=\sup _{s>0} sJ(s,1)>0, \end{aligned}$$
- 4.
This is very simple.\(\square \)

Next we show that we can approximate functions in \({\mathcal {B}}\) 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.

### Lemma 2.19

### Proof

*z*and \(\delta \), and

### 2.6 Representation of Besov functions

### Proposition 2.20

### Proof

### Remark 2.21

### 2.7 Dual Banach spaces

Our partial duality induces a contractive map \(\Psi _{\mathcal {B}}: {\mathcal {E}}\rightarrow {{\mathcal {B}}_0}^*\), where \({{\mathcal {B}}_0}^*\) can be identified in the natural way with the space of functionals in \({\mathcal {B}}^*\) which annihilate the constant functions. It follows from Lemma 2.19 that the range of \(\Psi _{\mathcal {B}}\) is weak*-dense in \({{\mathcal {B}}_0}^*\). However it is not norm-dense.

### Proposition 2.22

The range of \(\Psi _{\mathcal {B}}\) is not norm-dense in \({{\mathcal {B}}_0}^*\).

### Proof

For all \(f \in {\mathcal {B}}\), \(f^b \in {\text {BUC}}({\mathbb {R}})\) and \(\Vert f^b\Vert _\infty \le \Vert f\Vert _{\mathcal {B}}\). For \(a \in {\mathbb {R}}_+\), the function \(e_a\) of Example 1.8(1) belongs to \({\mathcal {B}}\), and its boundary function is \(e_a^b(y) = e^{-iay}\) which belongs to the Banach algebra \({\text {AP}}({\mathbb {R}})\) of almost periodic functions on \({\mathbb {R}}\).

For any \(g \in {\mathcal {E}}\), \(a \mapsto \langle g, e_a \rangle _{\mathcal {B}}\) is measurable, and so \(a \mapsto \varphi (e_a)\) is measurable for any \(\varphi \) which is in the norm-closure of the range of \(\Psi _{\mathcal {B}}\), and also for the functional \(\xi (f) = f(\infty )\) on \({\mathcal {B}}\). Since \({\mathcal {B}}^*\) is spanned by \({{\mathcal {B}}_0}^* \cup \{\xi \}\), the norm-closure of the range of \(\Psi _{\mathcal {B}}\) cannot be \({{\mathcal {B}}_0}^*\). \(\square \)

We now give an alternative proof of Proposition 2.22. Suppose that the range of \(\Psi \) is norm-dense in \({{\mathcal {B}}_0}^*\) (for a contradiction). Let \(f \in {\mathcal {G}}\). Then Lemma 2.19 would imply that \(f_\delta \) tends to *f* weakly in \({{\mathcal {B}}_0}\), and Lemma 2.13 shows that \(f_\delta \in {{\mathcal {L}}}{{\mathcal {M}}}\). Hence \({{\mathcal {L}}}{{\mathcal {M}}}\) would be weakly dense in \({\mathcal {G}}\) and then in \({{\mathcal {B}}_0}\). By Mazur’s Theorem, \({{\mathcal {L}}}{{\mathcal {M}}}\) would be norm-dense in \({\mathcal {B}}\). This would contradict Lemma 2.14(3). Thus we conclude that the range of \(\Psi _{\mathcal {B}}\) is not norm-dense in \({{\mathcal {B}}_0}^*\).

Let \({\mathcal {E}}_0 = \{ g \in {\mathcal {E}}: g(\infty ) = 0\}\), with the norm \(\Vert \cdot \Vert _{\mathcal {E}}\) which coincides with \(\Vert \cdot \Vert _{{\mathcal {E}}_0}\) on \({\mathcal {E}}_0\). Identify the dual \({{\mathcal {E}}_0}^*\) with the space of linear functionals in \({\mathcal {E}}^*\) which annihilate the constant functions. Our duality then provides a contractive map \(\Psi _{\mathcal {E}}\) from \({\mathcal {B}}\) to \({{\mathcal {E}}_0}^*\).

### Proposition 2.23

The range of \(\Psi _{\mathcal {E}}\) is not norm-dense in \({{\mathcal {E}}_0}^*\).

### Proof

## 3 Norm-estimates for some subclasses of \({\mathcal {B}}\)

In this section we obtain estimates of the \({\mathcal {B}}\)-norms of some more specific classes of functions in \({\mathcal {B}}\). 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.

### 3.1 Holomorphic extensions to the left

### Lemma 3.1

### Proof

### Lemma 3.2

- 1.Let \(f \in {\mathcal {B}}\), letand assume that$$\begin{aligned} \varphi (x) = \sup _{y\in {\mathbb {R}}} |f(x+iy)| , \qquad x>0, \end{aligned}$$Let \(g \in H^\infty _\omega \) for some \(\omega >0\). Then \(fg \in {\mathcal {B}}\) and$$\begin{aligned} \int _0^\infty \frac{\varphi (x)}{1+x}\,dx < \infty . \end{aligned}$$$$\begin{aligned} \Vert fg\Vert _{\mathcal {B}}\le \Vert f\Vert _{\mathcal {B}}\Vert g\Vert _\infty + \frac{\Vert g\Vert _{H^{\infty }_\omega }}{2} \int _0^\infty \frac{\varphi (x)}{\omega +x} \,dx. \end{aligned}$$
- 2.Let \(f \in H^\infty [\tau ,\infty ) \cap H_\omega ^\infty \), for some \(\tau ,\omega >0\). Then \(f \in {\mathcal {B}}\) and$$\begin{aligned} \Vert f\Vert _{\mathcal {B}}\le e^{-\omega \tau }\left( 2 + \frac{1}{2} \log \left( 1 + \frac{1}{\tau \omega } \right) \right) \Vert f\Vert _{H_\omega ^\infty }. \end{aligned}$$

### Proof

*g*we obtain that

### 3.2 Functions with decay on \(i{\mathbb {R}}\)

In the following result we give an estimate for the \({\mathcal {B}}\)-norm of certain functions which decay at infinity in an appropriate sense.

### Lemma 3.3

### Proof

The assumption (3.2) and monotonicity of *h* imply that \(f(is) \rightarrow 0\) as \(|s|\rightarrow \infty \). Then by the Poisson integral formula, \(f(\infty ) =0\) and therefore \(g(\infty )=0\).

### 3.3 Exponentials of inverses

Let \(f(z) = \exp (-1/z)\). Then \(f \in H^\infty ({\mathbb {C}}_+)\), but it does not have a continuous extension to \({\overline{{\mathbb {C}}}}_+\), and hence \(f \notin {\mathcal {B}}\). This can also be seen by observing that \(|f'(x+iy)| = (ex)^{-1}\) when \(x \in (0,1)\) and \(y^2 = x - x^2\).

As observed in Example 2.12(2), the function \(e^{- t r_1}(z) = \exp (-t/(z+1)) \in {{\mathcal {L}}}{{\mathcal {M}}}\) for \(t\in {\mathbb {R}}\). In Lemma 3.4 we estimate the \({\mathcal {B}}\)-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

### Proof

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 \({{\mathcal {L}}}{{\mathcal {M}}}\). In the next lemma, we consider the function \(f_t := g_{t/2}^2 \in {{\mathcal {L}}}{{\mathcal {M}}}\). We estimate \(\Vert f_t\Vert _{\mathcal {B}}\) 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.

### Lemma 3.5

*C*such that

### Proof

*t*. Now the estimates (3.4) for small

*t*, and (3.5) for large

*t*, imply (3.3). \(\square \)

### Remark 3.6

### 3.4 Cayley transforms

As observed in Example 2.12(3), the function \(\chi = 1 - 2r_1 : z \mapsto (z-1)/(z+1)\) belongs to \({{\mathcal {L}}}{{\mathcal {M}}}\) and therefore so do its powers. Estimating \(\Vert \chi ^n\Vert _{\text {HP}}\) 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 \({\mathcal {B}}\)-norms of these functions grow only logarithmically, as shown in the following lemma.

### Lemma 3.7

### Proof

### 3.5 Bernstein to Besov

*Bernstein function*is a holomorphic function \(f :{\mathbb {C}}_+ \rightarrow {\mathbb {C}}_+\) of the form

*f*:

- (B1)
*f*maps \(\Sigma _\theta \) to \(\Sigma _\theta \) for each \(\theta \in [0,\pi /2]\) [69, Proposition 3.6]. - (B2)
On \((0,\infty )\),

*f*is increasing and \(f'\) is decreasing. - (B3)On \({\mathbb {C}}_+\),$$\begin{aligned} {\text {Re}}f(z)&= a + b {\text {Re}}z + \int _{(0,\infty )} \left( 1 -{\text {Re}}e^{-zs} \right) \,d\mu (s) \\&\ge a + b {\text {Re}}z + \int _{(0,\infty )} \left( 1 - e^{-{\text {Re}}zs} \right) \,d\mu (s) = f({\text {Re}}z), \\ \left| f'(z)\right|&= \Big |b + \int _{(0,\infty )} s e^{-s z} \,d\mu (s) \Big | \\&\le b + \int _{(0,\infty )} s e^{- s {\text {Re}}z} \,d\mu (s) = f'({\text {Re}}z). \end{aligned}$$

### Proposition 3.8

*f*be a Bernstein function, \(\alpha \in (0,1)\), \(\beta \in (1,1/\alpha ]\), \(\theta \in (0,\pi /2)\), \(\lambda \in \Sigma _\theta \). Define

*g*and

*h*from \({\mathbb {C}}_+\) to \({\mathbb {C}}_+\) by

### Proof

*g*maps \({\mathbb {C}}_+\) into \(\Sigma _{\alpha \beta \pi /2} \subset {\mathbb {C}}_+\). Hence

### 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].

## 4 Functional calculus for \({\mathcal {B}}\)

In this section *X* is a complex Banach space, \(X^*\) is its Banach dual, and the duality between *X* and \(X^*\) is written as \(\langle x,x^* \rangle \) for \(x \in X,\, x^* \in X^*\). We shall write \(z \in {\mathbb {C}}_+\) as \(z = \alpha + i\beta \) in this section, as we shall be using *x* to denote vectors in *X*.

### 4.1 Definition

*A*be a closed operator on a Banach space

*X*, with dense domain

*D*(

*A*). We assume that the spectrum \(\sigma (A)\) is contained in \({{\overline{{\mathbb {C}}}}}_+\) and

*c*such that

*c*such that (4.2) holds, so

*A*satisfies the assumptions above then \(-A\) is the generator of a bounded \(C_0\)-semigroup \((T(t))_{t \ge 0}\). The Hille–Yosida conditions can be obtained by applying Proposition 2.15(3) and using (4.2). Moreover, the following holds for all \(x \in X\), \(x^* \in X^*\) and \(\alpha >0\):

*t*, and then take Laplace transforms of each side with respect to

*t*. The resulting functions of

*z*are both \(\langle (z+A)^{-2}x, x^* \rangle \), so uniqueness of Laplace transforms implies (4.5) (see [19, p. 505] for the full argument).

### Examples 4.1

*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 \(L^2({\mathbb {R}}_+,X)\) gives, for any \(\alpha >0\) and \(x, y \in X\),

Assume that (4.1) holds, so that the functions \(g_{x,x^*}\) in (4.3) are in \({\mathcal {E}}\), and let \(f \in {\mathcal {B}}\). 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 \({\mathcal {B}}\) and \({\mathcal {E}}\) considered in Sect. 2.5.

*A*be as above. Define

### Lemma 4.2

Let \(f \in {{\mathcal {L}}}{{\mathcal {M}}}\) and *f*(*A*) be as defined as in (4.10). Then *f*(*A*) coincides with the operator \(x \mapsto \int _{{\mathbb {R}}_+} T(t)x \,d\mu (t)\) as defined in the Hille–Phillips functional calculus. In particular, \(f(A) \in L(X)\).

### Proof

In particular, Lemma 4.2 implies that \((\lambda +A)^{-1} = r_\lambda (A)\), where \(\lambda \in {\mathbb {C}}_+\), 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*).

### Lemma 4.3

Let \(f \in {{\mathcal {B}}_0}\), and assume that \( f \eta _\delta \in {{\mathcal {L}}}{{\mathcal {M}}}\) for each \(\delta >0\). Then \(\lim _{\delta \rightarrow 0+} (f \eta _\delta )(A)x\) exists in *X* for every \(x \in X\). Moreover \(f(A)x=\lim _{\delta \rightarrow 0+} (f \eta _\delta )(A)x\) for all \( x \in X,\) and thus \(f(A)\in L(X)\).

### Proof

*D*(

*A*) is dense in

*X*, so it follows that \(\lim _{\delta \rightarrow 0+} (f \eta _\delta )(A)x\) exists in

*X*for all \(x \in X\).

Now we present the main result showing that the map \(\Phi _A\) has the essential properties of a bounded functional calculus. We shall subsequently refer to \(\Phi _A\) as the \({\mathcal {B}}\)-*calculus* for *A*.

### Theorem 4.4

Under the assumptions (4.1), the map \(\Phi _A : f \mapsto f(A)\) is a bounded algebra homomorphism from \({\mathcal {B}}\) into \(L(X)\), which extends the Hille–Phillips calculus. Moreover, \(\Vert \Phi _A\Vert \le \gamma _A\).

### Proof

The mapping \(\Phi _A: f \mapsto f(A)\) is bounded from \({\mathcal {B}}\) into \(L(X,X^{**})\). If \(f \in {\mathcal {G}}\), then \(f\eta _\delta \in {{\mathcal {L}}}{{\mathcal {M}}}\) by Lemma 2.13, so Lemma 4.3 applies to all such *f*. In particular, \(\Phi _A\) maps \({\mathcal {G}}\) into \(L(X)\). Since \({\mathcal {G}}\) is norm-dense in \({{\mathcal {B}}_0}\), it follows that \(\Phi _A(f) \in L(X)\) whenever \(f(\infty )=0\). Since \(\Phi _A(1) = I\), it follows that \(\Phi _A\) maps \({\mathcal {B}}\) into \(L(X)\).

Lemma 4.2 shows that this functional calculus agrees with the HP-calculus on \({{\mathcal {L}}}{{\mathcal {M}}}\). The final statement follows from (4.11). \(\square \)

The \({\mathcal {B}}\)-calculus defined by (4.10) and Theorem 4.4 applies to all operators satisfying (4.1), and to all functions in \({\mathcal {B}}\). In particular it applies to all generators of bounded \(C_0\)-semigroups on Hilbert space with estimates \(\Vert f(A)\Vert \le 2K_A^2 \Vert f\Vert _{\mathcal {B}}\), by (4.9). In that context White [82, Section 5.5] and Haase [45, Theorem 5.3, Corollary 5.5] independently had obtained estimates of the form \(\Vert f(A)\Vert \le CK_A^2 \Vert f\Vert _{\mathcal {B}}\) but only for subclasses of \({\mathcal {B}}\) which are not norm-dense (for \({{\mathcal {L}}}{{\mathcal {M}}}\) in [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 \({\mathcal {B}}\). Applications to bounded \(C_0\)-semigroups on Hilbert spaces are given in Sect. 5.

We note two other simple properties of the \({\mathcal {B}}\)-calculus.

### Lemma 4.5

- 1.
\(f(A+a) = f_a(A)\).

- 2.
\(f(aA) = g_a(A)\).

### 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): \(t = a\alpha , \, s=a\beta \). \(\square \)

We shall consider some more general properties of the \({\mathcal {B}}\)-calculus in Sects. 4.3 (compatibility with other calculi), 4.4 (convergence lemmas) and 4.5 (spectral inclusion).

### 4.2 Bounded holomorphic semigroups

*A*on a Banach space

*X*is

*sectorial of angle*\(\theta \in [0,\pi /2)\) if \(\sigma (A) \subset {{\overline{\Sigma }}}_\theta \) and, for each \(\theta ' \in (\theta ,\pi ]\), there exists \(C_{\theta '}\) such that

*X*. Some authors call these semigroups

*sectorially*bounded holomorphic semigroups, but we shall adopt the common convention that bounded holomorphic semigroups are bounded on sectors. We refer to [44] for the general theory of sectorial operators, and to [4, Section 3.7] for the theory of holomorphic semigroups.

*A*be an operator and assume that \(\sigma (A) \subset \overline{{\mathbb {C}}}_+\) and

*A*, which we call the

*sectoriality constant*of

*A*.

Before discussing bounded holomorphic semigroups in general, we show that the semigroup \((T_{\mathcal {B}}(a))_{a\in {\mathbb {C}}_+}\) of shifts is a bounded holomorphic semigroup on \({\mathcal {B}}\), as mentioned after Lemma 2.6. A short proof of holomorphy appeals to abstract theory. Since the functionals \(f \mapsto f(z)\) for \(z \in {\mathbb {C}}_+\) form a separating subspace of \({\mathcal {B}}^*\) and since the functions \(a \mapsto (T_{\mathcal {B}}(a)f)(z) = f(z+a)\) are holomorphic on \({\mathbb {C}}_+\), and \(T_{\mathcal {B}}(a)\) is a contraction, it follows from [4, Theorem A.7] that \(T_{\mathcal {B}}\) is holomorphic on \({\mathbb {C}}_+\). 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.

### Proposition 4.6

The family \((T_{\mathcal {B}}(a))_{a\in {\mathbb {C}}_+}\) of shifts is a holomorphic \(C_0\)-semigroup of contractions on \({\mathcal {B}}\).

### Proof

We return to the general theory. Let \(A \in {\text {Sect}}(\pi /2-)\). Then (4.1) holds, because \(\Vert (\alpha +i\beta +A)^{-1}\Vert \le M_A|\alpha +i\beta |^{-1}\), where \(M_A\) is the sectoriality constant of *A*. This easily leads to (4.2) with \(\gamma _A \le 2M_A^2\). We will give a sharper estimate in Corollary 4.8 based on the following integral estimate.

### Lemma 4.7

### Proof

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 \(\Vert f(A)\Vert \le 31M_A^3\Vert f\Vert _{\mathcal {B}}\). The arguments in [70, 80] were based on the sectorial functional calculus and Littlewood–Paley decompositions (see the Appendix).

### Corollary 4.8

### Proof

Lemma 4.7 shows that (4.15) holds, and the estimate (4.16) follows immediately. The convergence in operator-norm of the integral in (4.17) follows from either the estimate \(\Vert (\alpha -i\beta +A)^{-2}\Vert \le M_A^2 (\alpha ^2+\beta ^2)^{-1}\) or from Lemma 4.7, and then (4.17) follows from (4.10). \(\square \)

Some applications of the \({\mathcal {B}}\)-calculus for operators in \({\text {Sect}}(\pi /2-)\) are discussed in Sect. 5.

### 4.3 Compatibility with other calculi

It is important to compare the \({\mathcal {B}}\)-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 \({\mathcal {B}}\)-calculus does not deviate from that general principle, as we will see below.

We first remark that according to Theorem 4.4 the \({\mathcal {B}}\)-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 \({\mathcal {B}}\)-calculus restricted to the subalgebra \({{\mathcal {L}}}{{\mathcal {M}}}\) of \({\mathcal {B}}\). 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 \({\mathcal {B}}\)-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 \(f \mapsto \Psi _A(f)\) stand for the (extended) sectorial holomorphic functional calculus, and \(f \mapsto \Psi ^*(f)\) 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 \({\mathcal {B}}\) are bounded functions on \({\overline{{\mathbb {C}}}}_+\) while the functions appearing in the sectorial calculus are defined on open sectors properly containing \(\sigma (A) \setminus \{0\}\), and functions in the half-plane calculus are defined on open half-planes containing \(\sigma (A)\). Consequently for compatibility with each of the two calculi, we consider two situations; one where the spectrum of *A* is smaller than \({\overline{{\mathbb {C}}}}_+\), and one where \(f \in {\mathcal {B}}\) 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 \(\pi /2\), and Proposition 4.10 for functions in \({\mathcal {B}}\) which extend holomorphically to a sector of angle greater than \(\pi /2\). 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 \(A+\eta \) and correspondingly for the functions.

For \(A \in {\text {Sect}}(\pi /2-)\), a complication is that a Besov function is not necessarily in the domain of \(\Psi _A\). If *A* is injective then \({\mathcal {B}}\) is contained in the domain of \(\Psi _A\), 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 \({\mathbb {C}}_+\). Consequently our first result is confined to injective sectorial operators. It shows that the \({\mathcal {B}}\)-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 \({\mathcal {B}}\)-calculus and Vitse’s calculus, and hence with sectorial calculus, initially on \({\mathcal {G}}\) and then by approximation arguments on \({\mathcal {B}}\). Here we give a more direct argument.

### Proposition 4.9

Let \(A \in {\text {Sect}}(\pi /2-)\) be injective. Then \(\Phi _A(f) = \Psi _A(f)\) for all \(f \in {\mathcal {B}}\).

### Proof

*A*be an injective operator in \({\text {Sect}}(\theta )\), where \(\theta \in [0,\pi /2)\). Then, for any \(\theta '\in (\theta ,\pi /2)\),

*X*and \(\Psi _A(f) = \Phi _A(f)\). \(\square \)

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 \({\mathcal {B}}\)-calculus for injective \(A \in {\text {Sect}}(\pi /2-)\).

Another compatibility result between the \({\mathcal {B}}\)-calculus and the sectorial calculus is the following. Here we consider functions defined on \(\Sigma _\theta \) where \(\theta \in (\pi /2,\pi )\). In this case the domain of the primary calculus is contained in \({\mathcal {B}}\), which simplifies the arguments. We again denote the corresponding operators by \(\Phi _A(f)\) and \(\Psi _A(f)\), respectively.^{1}

### Proposition 4.10

Assume that *A* satisfies (4.1). Let \(f \in {\mathcal {B}}\) and assume that *f* extends to a function in \( {\text {Hol}}(\Sigma _\theta )\) for some \(\theta \in (\pi /2,\pi )\) and \(\Psi _A(f)\) is defined. Then \(\Phi _A(f) = \Psi _A(f)\).

### Proof

For \(\theta \in (\pi /2,\pi ]\), \({\mathcal {A}}_\theta \subset {{\mathcal {L}}}{{\mathcal {M}}}\), by [44, Lemma 3.3.1] or [4, Theorem 2.6.1]. If \(f \in {\mathcal {A}}_\theta \), then \(\Phi _A(f)\) and \(\Psi _A(f)\) both agree with the Hille–Phillips calculus applied to *f*, by Lemma 4.2 and [44, Proposition 3.3.2], and hence \(\Psi _A(f) = \Phi _A(f)\).

*f*be in the domain of the extended sectorial calculus \(\Psi _A\). Then there is a regulariser \(h\in {\mathcal {A}}_\theta \subset {\mathcal {B}}\) such that \(\Psi _A(h)\) is injective and \(h f \in {\mathcal {A}}_\theta \). Then

*X*and coincides with \(\Phi _A(f)\). \(\square \)

When *A* is injective and *f* is bounded on \(\Sigma _\theta \) one may take the regulariser *h* in the proof of Proposition 4.10 to be \(r_1(1-r_1)\). Using instead \(\left( r_1(1-r_1)\right) ^k\) for \(k \ge 1\), one can apply the result if *f* is polynomially bounded on \(\Sigma _\theta \).

- (C1)
Assume that for some \(\varepsilon >0\), \(\sigma (A) \subset \mathrm {R}_\varepsilon := \{z\in {\mathbb {C}}: {\text {Re}}z > \varepsilon \}\) and \(\Vert (z+A)^{-1}\Vert \) is bounded for \(z \in \mathrm {R}_{-\varepsilon }\); or

- (C2)
Assume that \(f \in {\text {Hol}}(\mathrm {R}_{-\varepsilon })\) for some \(\varepsilon >0\), there exist \(C,\alpha \ge 0\) such that \(|f(z)| \le C(1+|z|)^\alpha \) for \(z \in \mathrm {R}_{-\varepsilon }\) (or more generally, that \(\Psi ^*_A(f)\) is defined).

*A*is injective. Moreover it is easy to pass between the conditions (C1) and (C2) by shifts.

### Proposition 4.11

Assume that \(\sigma (A) \subset {\overline{{\mathbb {C}}}}_+\), (4.1) holds, \(f \in {\mathcal {B}}\) and either (C1) or (C2) above holds. Then \(\Psi _A^*(f) = \Phi _A(f)\).

### Proof

First, we assume that (C1) holds. Let \(f \in {\mathcal {B}}\) and \(g = r_1^2 f\). Then \(g \in {\mathcal {B}}\), \(|g(z)| \le C(1+|z|^2)^{-1}\) and \(|g'(z)| \le C ({\text {Re}}z)^{-1} (1+|z|^2)^{-1}\) for some *C*.

*X*and \(\Psi _A^*(f) = \Phi _A(f)\).

*f*on \(\mathrm {R}_{-\varepsilon }\), and the family \(\{\Psi _A^*(f_\eta ) : \eta \in (0,\varepsilon /2)\}\) is uniformly bounded in \(L(X)\). By the Convergence Lemma for the half-plane calculus [10, Theorem 3.1], \(\Psi _A^*(f_\eta ) \rightarrow \Psi _A^*(f)\) in the strong operator topology. Hence \(\Psi _A^*(f) = \Phi _A(f)\), as required.

Next, we assume that (C2) holds and *f* is polynomially bounded in \(\mathrm {R}_{-\varepsilon }\). Then we apply the case above with *f* replaced by \(r_1^n f\) for sufficiently large *n*, and the result follows. In the more general case, *f* would be replaced by *hf* for some regulariser *h*. \(\square \)

### Remarks 4.12

- 1.
An alternative proof

^{2}of case (C2) proceeds as follows. Let*f*satisfy (C2), and let \(g = r_1^2 f\). Then \(|g(z)| \le C(1+|z|^2)^{-1}\) and \(|g''(z)| \le C(1+|z|^2)^{-1}\) for \(z \in {\overline{{\mathbb {C}}}}_+\). Then \({\mathcal {F}}^{-1}g^b \in L^1({\mathbb {R}})\) and has support in \({\mathbb {R}}_+\), so \(g \in {{\mathcal {L}}}{{\mathcal {M}}}\). By [46, Theorem 8.20], \(\Psi ^*_A(g)\) coincides with the Hille–Phillips calculus of*g*and hence with \(\Phi _A(g)\). Then \(\Psi _A^*(f) = \Phi _A(f)\). - 2.Proposition 4.11 suggests an alternative way to define the \({\mathcal {B}}\)-calculus. For \(f \in {\mathcal {B}}\), one could define \(f(A+\varepsilon )\) by the half-plane functional calculus, and then show that \(f(A+\varepsilon )\) satisfies (4.10). This implies that \(f(A+\varepsilon ) \in L(X)\) and \(\Vert f(A+\varepsilon )\Vert \le C\Vert f\Vert _{\mathcal {B}}\) for \(f \in {\mathcal {B}}\). Then one can conclude from Lemma 2.6 thatexists in the operator norm, and this could become the definition of$$\begin{aligned} \lim _{\varepsilon \rightarrow 0+} f(A+\varepsilon ) \end{aligned}$$
*f*(*A*). This process reverses some of the steps that we have taken.

### 4.4 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 \(A \in {\text {Sect}}(\pi /2-)\) has dense range, and *A* admits bounded \(H^\infty ({\mathbb {C}}_+)\)-calculus (so \(f(A) \in L(X)\) for all \(f \in H^\infty ({\mathbb {C}}_+)\)). Then pointwise convergence of \((f_n)_{n \ge 1}\subset H^{\infty }({\mathbb {C}}_+)\) to *f* on \({\mathbb {C}}_+\), together with \(\sup _{n \ge 1}\Vert f_n\Vert _{\infty }<\infty \), implies strong convergence of \(f_n(A)\) to *f*(*A*). Moreover, \(f_n(A) \rightarrow f(A)\) in the operator norm if \(f_n \rightarrow f\) in \(H^\infty ({\mathbb {C}}_+)\). The Convergence Lemma allows one to replace convergence of \((f_n)_{n \ge 1}\) in \(H^\infty ({\mathbb {C}}_+)\) by convergence of \((f_n)_{n \ge 1}\) in a weaker topology at the price of getting only strong convergence of \((f_n(A))_{n \ge 1}\). 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 \(H^\infty ({\mathbb {C}}_+)\)-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 \({\mathcal {B}}\)-calculus is compatible with the sectorial and half-plane calculi the Convergence Lemmas for those calculi can be applied to the \({\mathcal {B}}\)-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 \({\mathcal {B}}\)-calculus. It applies to operators whose spectrum may be as large as \({{\overline{{\mathbb {C}}}}}_+\) and functions which may not extend beyond \({{\overline{{\mathbb {C}}}}}_+\), and pointwise convergence of \((f_n)_{n\ge 1}\) is assumed only on \({{\overline{{\mathbb {C}}}}}_+\). This is compensated by adding an assumption on the behaviour of \((f_n')_{n \ge 1}\) near the imaginary axis. We shall deduce the Convergence Lemma from the following result about convergence in operator norm.^{3}

### Theorem 4.13

*A*be a densely defined operator on a Banach space

*X*such that \(\sigma (A)\subset {\overline{{\mathbb {C}}}}_{+}\) and (4.1) holds. Let \((f_n)_{n\ge 1}\subset {\mathcal {B}}\) be such that \( \sup _{n\ge 1}\,\Vert f_n\Vert _{{\mathcal {B}}}<\infty . \) Assume that for every \(z\in {\mathbb {C}}_{+}\) there exists

*n*. Let \(g \in \mathrm{Hol}\,({\mathbb {C}}_+)\) be such that \(g'\in H^1({\mathbb {C}}_+)\) and \(g(\infty )=0\), and let \( g_n(z):=f_n(z)g(z), n\ge 0. \) Then \(f_0\in {\mathcal {B}}\), \(g_n\in {\mathcal {B}}_0, n \ge 0,\) and

### Proof

By Proposition 2.3(1), \(f_0\in {\mathcal {B}}\). Replacing \(f_n\) by \(f_n-f_0,\) we may assume that \(f_0=0\). By Proposition 2.4, \(g \in {\mathcal {B}}_0\), and then \(g_n\in {\mathcal {B}}_0, n \ge 0\).

*dS*(

*z*) for area measure on \({\mathbb {C}}_{+}\), we obtain from the definition (4.10) of \(g_n(A)\) that

*X*such that \(K_A=\sup _{t \ge 0}\Vert e^{-tA}\Vert <\infty \). Using (4.1), (4.2), (4.4), and a Hille–Yosida estimate for \(-A\), for fixed \(\delta \in (0,1)\), we have

*x*and \(x^*\) with \(\Vert x\Vert =\Vert x^{*}\Vert =1\).

*r*is to be chosen. We consider first

*r*and \(\delta \), the second integral converges to zero as \(n\rightarrow \infty \). Let \(N=N(r,\delta )\) be such that the second integral is less than \(\varepsilon \) for all \(n\ge N\). Then, summing up the estimates for \(K_1(r,n)\) and \(K_2(r,n),\) one has

*x*and \(x^*\) with \(\Vert x\Vert =\Vert x^{*}\Vert =1\). It follows that \(\Vert g_n(A)\Vert \rightarrow 0\) as \(n\rightarrow \infty \), as required. \(\square \)

### Corollary 4.14

*A*, \(f_n\) and \(f_0\) be as in Theorem 4.13. Then, for every \(x\in X\),

### Proof

We assume that \(f_0=0\). Let \(g(z)=(1+z)^{-1}\). By Theorem 4.13, \(\lim _{n\rightarrow \infty } \Vert f_n(A)(1+A)^{-1}\Vert = 0\). It follows that \(\lim _{n \rightarrow \infty } f_n(A)x = 0\), for each \(x \in D(A)\). By the boundedness of the \({\mathcal {B}}\)-calculus we have \(\sup _{n\ge 1} \Vert f_n(A)\Vert <\infty \). Since *D*(*A*) is dense in *X*, we have \(\lim _{n\rightarrow \infty } f_n(A) = 0\) strongly. \(\square \)

### Remarks 4.15

- 1.
The assumption (4.19) in Theorem 4.13 implies that \(f_n(i\beta )=\lim _{\alpha \rightarrow 0+}\,f_n(\alpha +i\beta )\) uniformly in

*n*, and hence (4.18) holds also for \(z\in i{\mathbb {R}}\). On the other hand, the assumptions do not imply that \(f_0(\infty ) = \lim _{n\rightarrow \infty } f_n(\infty )\), as Example 4.16a) below shows. - 2.When \(g(z) = (1+z)^{-1}\), the estimation of \(J_n\) in the proof of Theorem 4.13 can be simplified. Applying (2.21) with \(g = g_{x,x^*}\) and \(h(z) = f_n(z)(1+z)^{-2}\), and using \(|1+z| \ge 1+\alpha \) and (4.4), we haveThe dominated convergence theorem can be applied to this integral.$$\begin{aligned} |J_n|&\le \frac{2}{\pi }\int _{{\mathbb {C}}_{+}} \frac{\alpha }{(1+\alpha )^{3/2}} \left| \langle ({\overline{z}}+A)^{-2}x, x^*\rangle \right| \left| \frac{f_n(z)}{(1+z)^{1/2}}\right| \,dS(z) \\&\le 2 \gamma _A \int _0^\infty (1+\alpha )^{{-3/2}}\sup _{{\text {Re}}w = \alpha } \left| \frac{f_n(w)}{(1+w)^{1/2}}\right| \,d\alpha . \end{aligned}$$

Now we illustrate Corollary 4.14 for two natural choices of \((f_n)_{n \ge 1}\).

### Examples 4.16

Let \(f \in {\mathcal {B}}\) and let *A* be densely defined and satisfy (4.1).

*A*is bounded if \(e^{-A/n}\rightarrow I\) in

*L*(

*X*).

### 4.5 Spectral inclusion and mapping

With a few exceptions, given a semigroup generator \(-A\), the spectral “mapping” theorem for a functional calculus \(\Upsilon _A\) usually takes the form of the spectral inclusion \(f(\sigma (A))\subset \sigma (\Upsilon _A(f))\). In general, equality fails dramatically, even for the very natural functions \(e^{-tz}\) and for rather simple operators *A*; see [31, Section IV.3], for example. While one may expect only the spectral inclusion as above, the equality \(f(\sigma (A))\cup \{f(\infty )\} = \sigma (\Upsilon _A(f))\cup \{f(\infty )\}\) sometimes holds if *A* inherits some properties of bounded operators such as sharp resolvent estimates. The statement below shows that the \({\mathcal {B}}\)-calculus possesses the standard spectral features.

### Theorem 4.17

*A*be a densely defined operator on a Banach space

*X*such that \(\sigma (A) \subset {\overline{{\mathbb {C}}}}_+\) and (4.1) holds. Let \(f \in {\mathcal {B}}\) and \(\lambda \in {\mathbb {C}}\).

- 1.
If \(x \in D(A)\) and \(Ax = \lambda x\), then \(f(A)x = f(\lambda )x\).

- 2.
If \(x^* \in D(A^*)\) and \(A^*x^* = \lambda x^*\), then \(f(A)^*x^* = f(\lambda )x^*\).

- 3.
If \(\lambda \in \sigma (A)\) then \(f(\lambda ) \in \sigma (f(A))\).

- 4.
If \(A \in {\text {Sect}}(\pi /2-)\), then \(\sigma (f(A)) \cup \{f(\infty )\} = f(\sigma (A)) \cup \{f(\infty )\}\).

### Proof

- 1.From (4.10) and Proposition 2.20, for all \(x^* \in X^*\), we have$$\begin{aligned} \langle f(A)x,x^* \rangle&= f(\infty ) \langle x,x^* \rangle - \frac{2}{\pi } \int _0^\infty \int _{\mathbb {R}}\frac{\alpha \langle x,x^* \rangle }{(\alpha -i\beta +\lambda )^2}f'(\alpha +i\beta ) \,d\beta \, d\alpha \\&= f(\lambda ) \langle x,x^* \rangle . \end{aligned}$$
- 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 \(f(\infty )=0\). Let \({\mathcal {A}}\) be the bicommutant of \(\{(\lambda -A)^{-1} : \lambda \in \rho (A)\}\) in \(L(X)\), so the spectrum of

*f*(*A*) in \({\mathcal {A}}\) 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, \(\Vert (z+A)^{-2}(1+A)^{-2}\Vert \le C(1+|z-1|)^{-2}\) for \(z \in {\mathbb {C}}_+\). Then we may obtain the following formula in which the integral is absolutely convergent in the operator norm:Applying (4.10) with$$\begin{aligned} f(A)(1+A)^{-2} = - \frac{2}{\pi } \int _0^\infty \int _{\mathbb {R}}\alpha (\alpha -i\beta +A)^{-2} (1+A)^{-2} f'(\alpha +i\beta ) \,d\beta \,d\alpha .\nonumber \\ \end{aligned}$$(4.20)*x*replaced by \((1+A)^{-2}x\) gives a weak form of (4.20), and then this strong form follows.Let \(\lambda \in \sigma (A)\). Then \((1+\lambda )^{-1} \in \sigma ((1+A)^{-1})\), so there exists a character \(\gamma \) of \({\mathcal {A}}\) such that \(\gamma ((1+A)^{-1})= (1+\lambda )^{-1}\). It follows from the resolvent identity that \(\gamma ((z+A)^{-1}) = (z+\lambda )^{-1}\) whenever \(-z \in \rho (A)\). Applying \(\gamma \) to (4.20) and using Proposition 2.20, we obtainThus \(f(\lambda ) = \gamma (f(A)) \in \sigma (f(A))\).$$\begin{aligned}&\gamma (f(A)) (1+\lambda )^{-2} \\&\quad = - \frac{2}{\pi } \int _0^\infty \int _{\mathbb {R}}\alpha (\alpha -i\beta +\lambda )^{-2} (1+\lambda )^{-2} f'(\alpha +i\beta ) \,d\beta d\alpha = f(\lambda ) (1+\lambda )^{-2}. \end{aligned}$$For the general case, we proceed as follows. If \(\varepsilon >0\), \(\lambda +\varepsilon \in \sigma (A+\varepsilon )\). Applying the case above for \(A+\varepsilon \) (noting that \({\mathcal {A}}\) and \(\gamma \) do not change), \(f(\lambda +\varepsilon ) = \gamma (f(A+\varepsilon ))\). Now \(\Vert f(A+\varepsilon ) - f(A)\Vert \rightarrow 0\) as \(\varepsilon \rightarrow 0+\), by Lemmas 2.6 and 4.5. So \(f(\lambda ) = \gamma (f(A)) \in \sigma (f(A))\).

- 4.Let \(A \in {\text {Sect}}(\pi /2-)\), let \({\mathcal {A}}\) be as above, let \(\gamma \) be any character of \({\mathcal {A}}\), and let \(f \in {\mathcal {B}}\) with \(f(\infty )=0\). Applying \(\gamma \) to (4.17) givesIf \(\gamma ((1+A)^{-1})=0\), then \(\gamma ((z+A)^{-1}) = 0\) for all \(z \in {\mathbb {C}}_+\), and then \(\gamma (f(A)) = 0 = f(\infty )\). Otherwise, there exists \(\mu \in \sigma (A)\) such that \(\gamma ((z+A)^{-1}) = (z+\mu )^{-1}\), and then we obtain$$\begin{aligned} \gamma (f(A)) = - \frac{2}{\pi } \int _0^\infty \int _{\mathbb {R}}\alpha \gamma \left( (\alpha -i\beta +A)^{-1}\right) ^2 f'(\alpha +i\beta ) \,d\beta \, d\alpha . \end{aligned}$$\(\square \)$$\begin{aligned} \gamma (f(A)) = f(\mu ) \in f(\sigma (A)). \end{aligned}$$

### Remark 4.18

It may be possible to show that approximate \(\lambda \)-eigenvectors for *A* are approximate \(f(\lambda )\)-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 \(\sigma (f(A))\). However this approach is not straightforward and we leave it as a topic for further research.

## 5 Applications of the \({\mathcal {B}}\)-calculus for operator norm-estimates

In this section we apply the \({\mathcal {B}}\)-norm estimates obtained in Sect. 3 to the \({\mathcal {B}}\)-calculus defined in Sect. 4, and we recover (and sometimes improve) various results in semigroup theory. Recall the definitions of \(\gamma _A\), \(K_A\) 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 \(\Vert \cdot \Vert _{\mathcal {B}}\), but we note that in Corollaries 5.1–5.5, the relevant functions belong to \({\mathcal {B}}_0\), so the estimates could be slightly improved by using (4.11).

### 5.1 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 5.1

The following corollary covers some ideas which are included in [82, Theorem 5.5.12, etc] using Peller’s method, and in [45] using transference methods. The proof is immediate from Theorem 4.4, Lemma 2.5 and (4.9).

### Corollary 5.2

### 5.2 Holomorphic extensions to the left

Here we show how results of Haase and Rozendaal [47, Theorem 1.1] (and a preliminary result of Zwart [86]) for bounded semigroups on Hilbert space are corollaries of results in Sect. 3.

Corollary 5.3 coincides with [47, Theorem 1.1(a)], and a similar result for bounded holomorphic semigroups on Banach spaces was given by Schwenninger [70]. The result is an immediate consequence of Lemma 3.2(2), because \(H^\infty [\tau ,\infty ) \cap H^\infty _\omega = e_{\tau } H^\infty _\omega \), by (2.5).

### Corollary 5.3

Corollary 5.4 below coincides with [47, Theorem 1.1(b)]. For \(g \in H^\infty _\omega \), *g*(*A*) is a closed operator defined by the half-plane calculus.

### Corollary 5.4

### Proof

Corollary 5.5 was originally obtained in [10, Theorem 7.1] and then reproved in [47, Theorem 1.1(c), Corollary 4.4]. It follows directly from Lemma 3.1.

### Corollary 5.5

Similar results to Corollaries 5.3, 5.4 and 5.5 could be deduced in the same way for \(A \in {\text {Sect}}(\pi /2-)\). However those results can be proved directly by defining *f*(*A*), \(f(A)(\lambda +A)^{-\alpha }\) 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*), \(f(z)(\lambda +z)^{-\alpha }\) and \(f'(z)\) respectively all decay at least at a polynomial rate along rays with arguments in \((-\pi /2,\pi /2)\) 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 \(C_0\)-semigroup on a Hilbert space, so \(\Vert e^{-tA}\Vert \le Me^{-\omega t}, \, t\ge 0\). In this situation, one may apply the corollaries above with *A* replaced by \(A - \omega \) and \(f \in H^\infty ({\mathbb {C}}_+)\) replaced by \(f(\cdot +\omega ) \in H^\infty _\omega \). For example, the conclusion of this version of Corollary 5.4 becomes that \(\Vert f(A)(\lambda +A)^{-\alpha }\Vert \le CM^2\Vert f\Vert _\infty \) for all \(f \in H^\infty ({\mathbb {C}}_+)\), 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 \(|f(is)| \le (\log (|s|+e))^{-\alpha }\) for some \(\alpha >1\).

### Corollary 5.6

*X*, so that

*h*satisfies the assumption (3.2) of Lemma 3.3. Then

### Proof

### 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 \(C_0\)-semigroup on a Banach space. Then *A* is sectorial of angle \(\theta \in [0,\pi /2)\) 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 \((e^{-tA})_{t \ge 0}\) rules out a positive answer, and the answer is also negative for bounded semigroups on \(L^p\)-spaces when \(p \ne 2\); 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 \((e^{-tA^{-1}})_{t \ge 0}\) is bounded. Thus long-time estimates of \((e^{-tA^{-1}})_{t \ge 0}\) when \(A^{-1}\) does generate a \(C_0\)-semigroup are of value. The case when \((e^{-tA})_{t \ge 0}\) is exponentially stable is of particular interest, since then an integral representation for \((e^{-tA^{-1}})_{t \ge 0}\) in terms of \((e^{-tA})_{t \ge 0}\) 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 Sect. 3.3, these functions are not uniformly continuous on \({\mathbb {C}}_+\), and hence they are not in \({\mathcal {B}}\). We can derive [85, Theorem 2.2], the main result of [85], from the \({\mathcal {B}}\)-calculus, up to multiplication by an absolute constant.

### Corollary 5.7

*X*, so there are \(M, \omega >0\) such that

### Proof

*A*by \(\omega ^{-1}A - I\) and

*t*by \(\omega ^{-1}t\). Then we consider the function \({\tilde{f}}_t(z) = \exp (-t/(z+1))\) as in Lemma 3.4. Then \({{\tilde{f}}}_t = e^{-tr_1} \in {\mathcal {B}}\), so boundedness of the \({\mathcal {B}}\)-calculus gives

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

*X*, and assume that

*A*has a bounded inverse. Then

*A*.

### Proof

*C*. For \(\varepsilon >0\), let \(f_{t,\varepsilon }(z) = f_t(z+\varepsilon )\) and \(g_{t,\varepsilon }(z) = g_t(z+\varepsilon )\). Then \(g_{t,\varepsilon } \in {\mathcal {B}}\) and \(g_{t,\varepsilon }(A) = e^{-t(A+\varepsilon )^{-1}}\) which is given by a power series as in the proof of Corollary 5.7. Using Lemma 4.5, we have

### 5.5 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 \(\Vert V(A)\Vert =O(n^{3/2})\) 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 \(\Vert V(A)^n\Vert =O(n^{1/2})\) 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 \(V(A) = \chi ^n(A)\), where \(\chi (z) = (z-1)(z+1)^{-1}\), the optimality of this bound implies that the HP-norm of \(\chi ^n\) 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 \(\Vert V(A)^n\Vert =O(\log n)\) 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 \(\Vert V(A)^n\Vert =O(\log n)\) in [36] was based on intricate estimates of Laguerre polynomials. Here we obtain the logarithmic estimate as a direct and elementary application of the \({\mathcal {B}}\)-calculus and Lemma 3.7.

### Corollary 5.9

*X*. Then

### 5.6 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 \({\text {Sect}}(\theta )\) into itself, for \(\theta \in [0,\pi /2)\). 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 \({\mathcal {B}}\)-calculus offers a new streamlined and transparent proof of the result in the case of injective operators, as we show below.

If \(A \in {\text {Sect}}(\pi /2-)\) 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].

### Corollary 5.10

Let *X* be a Banach space, \(A \in {\text {Sect}}(\omega )\) for some \(\omega \in [0,\pi /2)\), and *A* be injective. Let *f* be a Bernstein function. Then \(f(A) \in {\text {Sect}}(\omega )\).

### Proof

It is known that \(-f(A)\) generates a bounded \(C_0\)-semigroup [69, Theorem 13.6], so \(f(A) \in {\text {Sect}}(\pi /2)\). We need to improve the angle.

*A*, but not on

*f*, and not on \(\lambda \) subject to \(\lambda \in \Sigma _\theta \). By compatibility of the functional calculi (Proposition 4.9), \(h(A^{1/\alpha })\) in the \({\mathcal {B}}\)-calculus is the same operator as in the sectorial calculus.

## 6 Appendix: Relation to Besov spaces

The main purpose of this section is to explain the connection between the spaces \({\mathcal {B}}\) (and \({\mathcal {E}}\)) (more precisely, \({{\mathcal {B}}_0}\) and \({\mathcal {E}}_0\)) with (homogeneous, analytic) Besov spaces of functions on \({\mathbb {R}}\). In Sects. 2–5, we have given a largely self-contained account of the construction of the \({\mathcal {B}}\)-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 \({{\mathcal {B}}_0}\) 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 \({\mathcal {B}}\) (and \({\mathcal {E}}\)) in terms of their boundary values, and to relate the boundary values to the Besov spaces defined on \({\mathbb {R}}\). 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.

\({\text {supp}}\psi _n \subset \left\{ t\in {\mathbb {R}}: a2^{n-1}\le |t|\le b 2^{n}\right\} , \quad n \in {\mathbb {Z}}\), for some \(a, b >0\),

\(\sum _{n\in {\mathbb {Z}}} \psi _n(t)=1, \quad t \in {\mathbb {R}} \setminus \{0\}\), and

\(\sup _{t \in {\mathbb {R}}} \sup _{n \in {\mathbb {Z}}} 2^{kn} |\psi ^{(k)}_n(t)|<\infty , \quad k\ge 0\).

*homogeneous*Besov space \(B^s_{p,q}({\mathbb {R}})\) as the set of \(f \in S'_{\infty }({\mathbb {R}})\) such that

It is traditional to write \({\dot{B}}^s_{p,q}({\mathbb {R}})\) instead of \(B^s_{p,q}({\mathbb {R}})\) 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

*analytic Besov space*as

*P*(

*t*), are the Poisson kernel and semigroup for the right half-plane \({\mathbb {C}}_+\), as in Lemma 2.1. The mapping is well-defined, and \({\mathcal {P}} (g) \in H^\infty ({\mathbb {C}}_+)\) since \(g \in H^\infty ({\mathbb {R}})\).

### Proposition 6.2

### Proof

If \(f \in {\mathcal {G}}\), then \(f^b * \varphi _n^+ = 0\) for all except finitely many *n*, so \(f_b \in {\mathcal {B}}_{\mathrm{dyad}}\) and \(f = {\mathcal {P}}f^b\). Since \({\mathcal {G}}\) is dense in \({\mathcal {B}}_0\) by Proposition 2.10, we infer that \({\mathcal {P}}\) maps \({\mathcal {B}}_{\mathrm{dyad}}\) onto \({{\mathcal {B}}_0}\). The inverse map is given by \(f \mapsto f^b\), 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]. \(\square \)

### Remark 6.3

### Remark 6.4

### 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 \({\mathcal {B}}\) and the norm \(\Vert \cdot \Vert _{{\mathcal {B}}}=\Vert \cdot \Vert _{{\mathcal {B}}_0}+\Vert \cdot \Vert _{\infty }\). (Recall from Proposition 2.2 that \(\Vert \cdot \Vert _{{\mathcal {B}}_0}\ge \Vert \cdot \Vert _{\infty }\) on \({\mathcal {B}}_0\).) The Banach algebra \(({\mathcal {B}}, \Vert \cdot \Vert _{{\mathcal {B}}})\) 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 \(({\mathcal {B}}, \Vert \cdot \Vert _{{\mathcal {B}}})\). The algebra \({\mathcal {B}}\) has appeared, for example, in [80], and similar algebras were considered in [62].

*g*on \({\mathbb {C}}_+\) satisfying (2.20).

Recall from Sect. 2.5 that \({\mathcal {E}}\) is a Banach space with the norm in (2.22) and the subspace \({\mathcal {E}}_0\) of functions \(g \in {\mathcal {E}}\) with \(g(\infty )=0\) is also a Banach space. The spaces \({\mathcal {E}}\) and \({\mathcal {B}}\) are related via the (partial) duality given in (2.23). Consequently, as discussed in Sect. 2.7, \({\mathcal {E}}\) and \({\mathcal {B}}\) are contractively embedded into \({\mathcal {B}}_0^*\) and \({\mathcal {E}}_0^*\), 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 \({\mathcal {E}}\) and any relations to \({\mathcal {B}}\), in particular by looking at Besov classes that can be associated to \({\mathcal {E}}\).

Parts of the proof of Proposition 6.2 can be applied in the context of \({\mathcal {E}}\), taking account of the estimate in Proposition 2.15(3) and using the same results or arguments from [12, 75, 81]. In this way one can see that \({\mathcal {E}}_{\mathrm{dyad}}\) is isomorphically embedded into \({\mathcal {E}}\) 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 \((T_{{\mathcal {E}}}(a))_{a \ge 0}\) is not strongly continuous on \({\mathcal {E}}\) (Lemma 2.18), so a result for \({\mathcal {E}}\) similar to Proposition 2.10 cannot be expected, and consequently it is not clear that the embedding is surjective.

### 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 16–18) appear to be incomplete. It is also stated there (p. 265, last line) that \({\mathcal {E}}_{0, \mathrm{dyad}}^*\) coincides with \({\mathcal {B}}_{\mathrm{dyad}}\), but no explanation is given.

## Footnotes

## Notes

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