# Introducing and solving generalized Black-Scholes PDEs through the use of
functional calculus

Jesús Oliva-Maza  J. Oliba-Maza: Departamento de Matematicas, Instituto
Universitario de Matematicas y Aplicaciones, Universidad de Zaragoza, Zaragoza
50009, Spain<EMAIL_ADDRESS>and  Mahamadi Warma  M. Warma: Department of
Mathematical Sciences and the Center for Mathematics and Artificial
Intelligence (CMAI), George Mason University, Fairfax, VA 22030, USA
<EMAIL_ADDRESS>
###### Abstract.

We introduce some families of generalized Black–Scholes equations which
involve the Riemann-Liouville and Weyl space-fractional derivatives. We prove
that these generalized Black–Scholes equations are well-posed in
$(L^{1}-L^{\infty})$-interpolation spaces. More precisely, we show that the
elliptic type operators involved in these equations generate holomorphic
semigroups. Then, we give explicit integral expressions for the associated
solutions. In the way to obtain well-posedness, we prove a new connection
between bisectorial operators and sectorial operators in an abstract setting.
Such a connection extends some known results in the topic to a wider family of
both operators and the functions involved.

###### Key words and phrases:

Bisectorial operators, sectorial operators, functional calculus, holomorphic
semigroups, Riemann-Liouville space-fractional derivative, Weyl space-
fractional derivative, generalized Black-Scholes equations

###### 2010 Mathematics Subject Classification:

26A33, 47A60, 47B12, 47D06

This work has been done when the first author was visiting the Department of
Mathematical Sciences at George Mason University in Fairfax, Virginia. He is
grateful for the financial aid of Warma’s AFOSR research grant. The work of J.
Oliva-Maza is partially supported by Project PID2019-105979GB-I00 of the
MICINN, and by BES-2017-081552, MINECO, Spain and FSE. The work of M. Warma is
partially supported by AFOSR under Award NO: FA9550-18-1-0242 and US Army
Research Office (ARO) under Award NO: W911NF-20-1-0115.

##  1\. Introduction

Bisectorial operators play a central role in the theory of abstract
inhomogeneous first order differential equations on the whole real line, like

$u^{\prime}(t)=Au(t)+f(t),\qquad t\in\mathbb{{R}},$ (1.1)

where $A$ is a bisectorial operator on a Banach space $X$. The theory has been
an active topic of research during the past years. We refer to [4, 24] and
their references for a complete overview, the importance, and applications of
such operators.

It is well-known that sectorial operators are related to the theory of
abstract first order homogeneous differential equations on the positive real
axis with initial conditions. Namely, equations of the form

$v^{\prime}(t)=Av(t),\quad t>0,\qquad v(0)=g.$ (1.2)

We refer to the monographs [1, 12] and their references for a precise
description of this fact.

Both families of operators are closely related. Indeed, an operator $A$ is
bisectorial if and only if both $aI+A$ and $aI-A$ are sectorial for some real
number $a\geq 0$, where $I$ denotes the identity operator. As one may expect,
both families share multiple properties and, in particular, one can define
similar functional calculus for the two families of operators. Indeed, in this
work we make use of the meromorphic functional calculus from sectorial
operators completely developed in the excellent monograph [18] and we adapt it
carefully to the theory of bisectorial operators, which extends the functional
calculi considered in [22] and the references therein.

Another connection between these two families lies in the fact that $A^{2}$ is
sectorial whenever $A$ is a bisectorial operator, see e.g. [4, Proposition
5.1]. In the particular case that $A$ generates an exponentially bounded
group, then $A^{2}$ generates a holomorphic semigroup (see for instance [3,
Theorem 1.15] or [12, Corollary 4.9]). An application of this result is to
study differential equations on the positive real line like Equation (1.2) in
terms of a possibly simpler equation on the real line Equation (1.1). One may
find a concrete example of this fact in [2], where the authors obtained
properties of the classical Black–Scholes equation

$u_{t}=x^{2}u_{xx}+xu_{x},\qquad t,x>0,$ (BS)

through the simpler and elegant partial differential equation

$u_{t}=-xu_{x},\qquad t,x>0.$

Since the seminal work [9], the Black–Scholes equation (BS) has been an active
topic of research in mathematical finance due to its importance in the
modeling of pricing options contracts, see for instance [14] and the
references therein. One of our purposes in the present paper is to study
fractional differential equations which extend the classical equation (BS),
using the theory of bisectorial operators established here. At this point, we
wish to observe that part of our contribution is to show how the relations
between bisectorial operators and sectorial operators that we develop here can
be used successfully to solve several of those equations extending (BS). We
are not dealing with mathematical finance in this paper, but on the other hand
we are confident that our new generalized Black–Scholes partial differential
equations could be used to understand the disturbing and anomalous behavior of
the financial market.

Let us explain the method we follow to extend (BS) to a fractional
differential equation. As stated before, the classical Black–Scholes equation
is studied by means of the following degenerate differential operator:

$(Jf)(x)\mathrel{\mathop{\ordinarycolon}}=-xf^{\prime}(x),\qquad x>0,$ (1.3)

on $(L^{1}-L^{\infty})$ interpolation spaces that we shall explain in Section
5. In [2], the authors used the connection between the operator $J$ and the
classical Cesàro operator $\mathcal{C}$ given by

$\displaystyle(\mathcal{C}f)(x)$
$\displaystyle=\frac{1}{x}\int_{0}^{x}f(y)\,dy,\quad x>0$

and its adjoint Cesàro operator $\mathcal{C}^{\ast}$. This connection had been
first pointed out in [11] to study the Cesàro operator $\mathcal{C}$ on the
half-plane. In addition, the differential operator $J$ was also related in
[21] to the generalized fractional version of the Cesàro operator
$\mathcal{C}_{\alpha}$ on $L^{p}$-spaces, for real numbers $\alpha>0$, given
by

$(\mathcal{C}_{\alpha}f)(x)=\frac{\alpha}{x^{\alpha}}\int_{0}^{x}(x-y)^{\alpha-1}f(y)\,dy=\frac{\Gamma(\alpha+1)}{x^{\alpha}}(D^{-\alpha}f)(x),\quad
x>0$ (1.4)

and the associated adjoint Cesàro operator $\mathcal{C}_{\alpha}^{\ast}$ given
by

$(\mathcal{C}_{\alpha}^{\ast}f)(x)=\alpha\int_{x}^{\infty}\frac{(y-x)^{\alpha-1}}{y^{\alpha}}f(y)\,dy=\Gamma(\alpha+1)(W^{-\alpha}(y^{-\alpha}f))(x),\quad
x>0,$ (1.5)

where $D^{-\alpha}$ and $W^{-\alpha}$ denote the Riemann-Liouville and Weyl
fractional integrals of order $\alpha$, respectively, see Section 2 for more
details. We notice that $D^{-\alpha}$ is usually denoted by $I^{-\alpha}$ in
most of the literature on fractional calculus. The connection between $J$ and
$\mathcal{C}_{\alpha}$, $\mathcal{C}_{\alpha}^{\ast}$ is given in terms of the
exponentially bounded group $(G(s))_{s\in\mathbb{{R}}}$ generated by $J$, with
expression $(G(s)f)(x)=f(e^{-s}x)$. More precisely, we have that

$\displaystyle(\mathcal{C}_{\alpha}f)(x)$
$\displaystyle=\alpha\int_{0}^{\infty}e^{-s}(1-e^{-s})^{\alpha-1}(G(s)f)(x)\,ds,\quad
x>0,$ (1.6) $\displaystyle(\mathcal{C}_{\alpha}^{\ast}f)(x)$
$\displaystyle=\alpha\int_{-\infty}^{0}(1-e^{s})^{\alpha-1}(G(s)f)(x)\,ds,\quad
x>0.$

Furthermore, (1.6) yields a representation in functional calculus of the type
$\mathcal{C}_{\alpha}=f_{\alpha}(J)$,
$\mathcal{C}_{\alpha}^{\ast}=f_{\alpha}^{\ast}(J)$ for some suitable
holomorphic functions $f_{\alpha},\,f_{\alpha}^{\ast}$ which have some
fractional-powers like behavior that we shall specify later in the paper.
Therefore, as in the same way one can write (BS) in terms of the operators
$\mathcal{C},\,\mathcal{C}^{\ast}$, it seems natural to construct some
families of generalized Black–Scholes equations which can be written in terms
of $\mathcal{C}_{\alpha},\,\mathcal{C}_{\alpha}^{\ast}$, operators which,
respectively involve fractional Riemann-Liouville derivatives of order
$\alpha$, $D^{\alpha}$, and fractional Weyl derivatives of order $\alpha$,
$W^{\alpha}$. This method will give rise to generalized fractional
Black–Scholes equations of the following three forms for $x$, $t>0$:

$\displaystyle u_{t}$
$\displaystyle=\frac{1}{\Gamma(\alpha+1)^{2}}D^{\alpha}(x^{\alpha}D^{\alpha}(x^{\alpha}u))-\frac{2}{\Gamma(\alpha+1)}D^{\alpha}(x^{\alpha}u)+u,$
(1.7) $\displaystyle u_{t}$
$\displaystyle=\frac{1}{\Gamma(\alpha+1)^{2}}x^{\alpha}W^{\alpha}(x^{\alpha}W^{\alpha}u),$
$\displaystyle u_{t}$
$\displaystyle=-\frac{1}{\Gamma(\alpha+1)^{2}}D^{\alpha}(x^{2\alpha}W^{\alpha}u)+\frac{1}{\Gamma(\alpha+1)}x^{\alpha}W^{\alpha}u,$

for suitable values of $\alpha$, see Section 5 for more details. The theory of
bisectorial operators developed here will allow us to obtain the well-
posedness and an explicit integral expressions of solutions of these
fractional versions of the Black–Scholes equation, of course by incorporating
initial and boundary conditions. Also, at the limiting case $\alpha=1$, we
recover all the classical known results.

In order to get these results, we first give a significant extension of the
above mentioned results regarding a bisectorial operator $A$ and its square
$A^{2}$. Indeed, we are able to prove that if $A$ is a bisectorial operator
and $g$ is a suitable function whose range is contained in a sector, and which
has a fractional-power like behavior at the singularity points of the spectra
of $A$ and $g(A)$, then $g(A)$ is a sectorial operator (see Theorem 3.10). As
one may expect, the setting for sectorial operators serves again as an
inspiration for this result, more precisely the scaling property, i.e., for a
sectorial operator $A$, $A^{\alpha}$ is sectorial for small enough $\alpha>0$,
see [19, Theorem 2]. We are also able to give supplementary results of special
importance when $g(A)$ generates a semigroup, such as the characterization of
the closure of its domain $\overline{\mathcal{D}(g(A))}$ or an integral
expression for the semigroup it generates in terms of the functional calculus
of $A$.

Interestingly, as a consequence of our results one obtains that, if $A$ is
bisectorial of angle $\frac{\pi}{2}$ and half-width $a\geq 0$ (in particular,
if it generates an exponentially bounded group), then either $(A+a)^{\alpha}$
or $-(A+a)^{\alpha}$ generate a holomorphic semigroup for all $\alpha>0$ with
$\alpha\neq 1,3,5,\ldots$ This is a remarkable discovery that generalizes the
already known results in this directions for the case $\alpha=2$, see for
example [3, Theorem 1.15] and [4, Proposition 5.1], or in the case where $A$
generates a bounded group, see [7, Theorem 4.6].

As a final remark, we mention that fractional versions of the Black–Scholes
equation have been proposed and analyzed in a number of papers, see for
instance [13, 20, 25, 27]. In most of these references, authors are only able
to deal with time-fractional derivatives, that is, replacing the time
derivative $u_{t}$ in (BS) with a Riemann-Liouville or a Caputo type time
fractional derivative. Of course in this setting the authors cannot get some
generation of semigroups results due to the limitation of time-fractional
derivatives. Spatial-fractional derivatives are indeed more difficult to deal
with, since spatial terms are more complex than the time ones in the equation
(BS). A spatial fractional Black–Scholes model can be found in [10] without
given further details. The fractional Black–Scholes equation we propose here
also contains fractional powers acting as multiplication, yielding to
equations which are definitely difficult to solve by more classical methods
such as the Laplace transform or the Fourier transform. Therefore, we observe
that the fractional versions of (BS) proposed in the present paper, which are
solved through the theory we developed here, seem to be notably difficult to
be solved with the classical methods.

In short, the contributions of the present paper can be regarded as centered
around two facts:

  1. (1)

The introduction of new (generalized) fractional Black–Scholes equations
arising from fractional Cesàro operators in a natural manner. As we have
already observed, such equations are difficult -maybe not possible- to solve
by classical methods.

  2. (2)

In order to overcome the quoted failure of usual methods, we establish a new
connection, in an abstract setting, between bisectorial operators and
sectorial operators. Such a connection extends notably previous results in the
field. Actually our approach is based on the proof of the scaling property
given in [5, Proposition 5.2], but it requires quite more general functions to
operate in functional calculi defined on the basis of more intricate
integration paths, as well as nontrivial, more involved, approximation tools.
This requires more work to be proven than the latter one. This is because we
consider more general functions that require working with approximation tools,
and moreover our family of functional calculus presents more intricate
integration paths.

The rest of the paper is organized as follows. In Section 2.1 we fix some
notations and introduce some definitions. In Section 2.2 we develop the
functional calculus of bisectorial operators and its extensions. Section 3 is
devoted to the passage from bisectorial to sectorial operators, where we also
give a note regarding holomorphic functional calculus for sectorial operators.
In Section 4 we give some generation of bounded holomorphic semigroups results
by the general sectional operators we have constructed in Section 3. The
complete theory and applications of the generalized Black–Scholes equation are
contained in Section 5. We conclude the paper with A where some useful results
have been proven.

##  2\. Notations and preliminary results

In this section, we fix some notations, give some definitions and introduce
some well-known results as they are used throughout the remainder of the
paper. We start with some notations and definitions.

###  2.1. Notations and definitions

Given any $\varphi\in(0,\pi)$, we denote the sector

$\displaystyle
S_{\varphi}\mathrel{\mathop{\ordinarycolon}}=\Big{\\{}z\in\mathbb{{C}}\mathrel{\mathop{\ordinarycolon}}\left|\mbox{arg}(z)\right|<\varphi\Big{\\}},$
(2.1)

and set $S_{0}\mathrel{\mathop{\ordinarycolon}}=(0,\infty)$. If
$\lambda\in\mathbb{{C}}$, we shall let $\Re\lambda$ denote the real part of
$\lambda$.

Let $A$ with domain $\mathcal{D}(A)$ be a closed linear operator in a Banach
space $X$. We shall denote by $\sigma(A)$ the spectrum of $A$, $\sigma_{p}(A)$
the point spectrum of $A$, and by
$\rho(A)\mathrel{\mathop{\ordinarycolon}}=\mathbb{{C}}\setminus\sigma(A)$, the
resolvent set of $A$. We also let
$\widetilde{\sigma}(A)\mathrel{\mathop{\ordinarycolon}}=\sigma(A)\cup\\{\infty\\}$.
For $\lambda\in\rho(A)$ we shall let
$R(\lambda,A)\mathrel{\mathop{\ordinarycolon}}=(\lambda I-A)^{-1}$ be the
resolvent operator of $A$. Then, $R(\lambda,A)$ is a bounded linear operator
from $X$ into $X$. By $\mathcal{R}(A)$, we shall mean the range space of $A$
and $\mathcal{N}(A)$ shall denote the null set (or the kernel) of the operator
$A$.

We shall denote by ${\mathcal{L}(X)}$ the space of all bounded linear
operators from a Banach space $X$ into $X$. If $B\in\mathcal{L}(X)$, we shall
let $\|B\|_{\mathcal{L}(X)}$ be the operator norm of $B$.

Finally, we use throughout the notation $h\lesssim g$ to denote $h\leq Mg$,
for some constant $M$, when the dependence of the constant $M$ on some
physical parameters is not relevant, and so it is suppressed.

######  Definition 2.1.

A closed linear operator $A$ with domain $\mathcal{D}(A)$ in a Banach space
$X$ is said to be a sectorial operator if there exists $\varphi\in[0,\pi)$
such that $\sigma(A)\subset\overline{S_{\varphi}}$ and, for every
$\varphi^{\prime}\in(\varphi,\pi)$, there exists a constant
$M_{\varphi^{\prime}}>0$ such that

$\displaystyle\|\lambda R(\lambda,A)\|_{\mathcal{L}(X)}\leq
M_{\varphi^{\prime}}\;\mbox{ for all }\lambda\in\mathbb{{C}}\setminus
S_{\varphi^{\prime}}.$

Next, for any $\omega\in(0,\pi/2]$ and $a\geq 0$ a real number, we set

$BS_{\omega,a}\mathrel{\mathop{\ordinarycolon}}=\begin{cases}(-a+S_{\pi-\omega})\cap(a-S_{\pi-\omega})\;\;&\mbox{
if }\omega\in(0,\pi/2)\mbox{ and }a\geq 0,\\\ \\\ i\mathbb{{R}}&\mbox{ if
}\omega=\pi/2\mbox{ and }a=0.\end{cases}$

Figure 1. Illustration of the spectrum of a sectorial operator (left) and
bisectorial operator (right).

######  Definition 2.2.

Let $(\omega,a)\in(0,\pi/2]\times[0,\infty)$ and let $A$ be a closed linear
operator in a Banach space $X$. We will say that $A$ is a bisectorial operator
of angle $\omega$ and half-width $a$ if the following conditions hold:

  * •

$\sigma(A)\subset\overline{BS_{\omega,a}}$.

  * •

For all $\omega^{\prime}\in(0,\omega)$, $A$ satisfies the resolvent bound

$\displaystyle\sup\Big{\\{}\min\\{|\lambda-a|,|\lambda+a|\\}\|R(\lambda,A)\|_{\mathcal{L}(X)}\,\mathrel{\mathop{\ordinarycolon}}\,\lambda\notin\overline{BS_{\omega^{\prime},a}}\Big{\\}}<\infty.$

Given a Banach space $X$, we will denote the set of all bisectorial operators
of angle $\omega$ and half-width $a$ in $X$ by $\text{{BSect}}(\omega,a)$. We
omit an explicit mention to $X$ for the sake of simplicity in our notations.
Notice that a closed operator $A\in\text{{BSect}}(\omega,a)$ if and only if
both $aI+A$ and $aI-A$ are sectorial of angle $\pi-\omega$.

Next, we introduce the notion of bounded holomorphic semigroups in the sense
of [1, Definition 3.7.3].

######  Definition 2.3.

Let $\delta\in(0,\pi/2]$. A mapping
$T\mathrel{\mathop{\ordinarycolon}}S_{\delta}\to\mathcal{L}(X)$ is called a
bounded holomorphic semigroup (of angle $\delta$) if it has the following
properties:

  1. (a)

The semigroup property: $T(z)T(z^{\prime})=T(z+z^{\prime})$ for all
$z,z^{\prime}\in S_{\delta}$.

  2. (b)

The mapping $T\mathrel{\mathop{\ordinarycolon}}S_{\delta}\to\mathcal{L}(X)$ is
holomorphic.

  3. (c)

The mapping $T$ satisfies: $\sup_{w\in
S_{\delta^{\prime}}}\|T(w)\|_{\mathcal{L}(X)}<\infty$ for every
$\delta^{\prime}\in(0,\delta)$.

It is known that every bounded holomorphic semigroup $T$ on a Banach space $X$
is determined by its generator $A$, which is a closed linear operator given by

$\begin{cases}\displaystyle\mathcal{D}(A)=\left\\{x\in\
X\mathrel{\mathop{\ordinarycolon}}\lim_{t\downarrow
0}\frac{T(t)x-x}{t}\;\mbox{exists}\right\\}\\\ \displaystyle
Ax=\lim_{t\downarrow 0}\frac{T(t)x-x}{t},\quad x\in\mathcal{D}(A).\end{cases}$
(2.2)

Another way to define the infinitesimal generator is through its resolvent
$R(\lambda,A)$ by using the Laplace transform. Namely,

$\displaystyle R(\lambda,A)=\int_{0}^{\infty}e^{-\lambda
t}T(t)\,dt,\quad\Re\lambda>0.$ (2.3)

For any $\delta\in(0,\pi/2]$, it is well known that a closed linear operator
$A$ is the generator of a bounded holomorphic semigroup of angle $\delta$ if
and only if $-A$ is a sectorial operator of angle $\pi/2-\delta$. Even more,
one has that

$\displaystyle T(w)=\exp_{-w}(A),\quad w\in S_{\delta},$

where we make use of the primary functional calculus of sectorial operators
and where $\exp_{-w}(z)\mathrel{\mathop{\ordinarycolon}}=\exp(-wz)$.

######  Definition 2.4.

For $\delta\in(0,\pi/2]$, we say that a mapping
$T\mathrel{\mathop{\ordinarycolon}}S_{\delta}\to\mathcal{L}(X)$ is an
exponentially bounded holomorphic semigroup if it satisfies the bounded
semigroup property, is holomorphic (namely (a) and (b) in Definition 2.3 are
satisfied) and it holds that the set
$\\{\|T(w)\|_{\mathcal{L}(X)}\mathrel{\mathop{\ordinarycolon}}w\in
S_{\delta^{\prime}},\,|w|\leq 1\\}$ is bounded for every
$\delta^{\prime}\in(0,\delta)$.

In particular, for each $\delta^{\prime}\in(0,\delta)$ one can find
$M_{\delta^{\prime}},\rho_{\delta^{\prime}}\geq 0$ such that
$\|T(w)\|_{\mathcal{L}(X)}\leq M_{\delta^{\prime}}e^{\rho_{\delta^{\prime}}\Re
w}$ for all $w\in S_{\delta^{\prime}}$. As a consequence, one can define the
generator $A$ of an exponentially bounded holomorphic semigroup $T$ in the
same way as in (2.2) or (2.3) for all $\lambda$ with $\Re\lambda$ large
enough.

It is readily seen that $A$ generates an exponentially bounded holomorphic
semigroup if and only if there is some $\rho\in\mathbb{{R}}$ such that
$A-\rho$ generates a bounded holomorphic semigroup (of possibly strictly
smaller angle).

######  Definition 2.5.

The space $\mathbb{D}_{T}$ of strong continuity of a (holomorphic) semigroup
$T$ of angle $\delta$ is defined by

$\displaystyle\mathbb{D}_{T}\mathrel{\mathop{\ordinarycolon}}=\left\\{x\in
X\mathrel{\mathop{\ordinarycolon}}\;\lim_{S_{\delta^{\prime}}\ni w\to
0}T(w)x=x\quad\text{for all }\delta^{\prime}\in(0,\delta)\right\\}.$

It is well known that the space $\mathbb{D}_{T}$ is precisely the closure of
the domain of its generator, that is,
$\mathbb{D}_{T}=\overline{\mathcal{D}(A)}$. We refer the reader to [18,
Section 3.4] and [12, Section II.4.a] for more details about holomorphic
semigroups.

Next, we introduce the notion of $(L^{1}-L^{\infty})$-interpolation spaces.
Let $E$ be a Banach space consisting of functions with domain $(0,\infty)$,
for which the inclusions $(L^{1}(0,\infty)\cap L^{\infty}(0,\infty))\subset
E\subset(L^{1}(0,\infty)+L^{\infty}(0,\infty))$ hold and are continuous. We
will say that $E$ is a $(L^{1}-L^{\infty})$-interpolation space if, for every
linear operator
$S\mathrel{\mathop{\ordinarycolon}}(L^{1}(0,\infty)+L^{\infty}(0,\infty))\to(L^{1}(0,\infty)+L^{\infty}(0,\infty))$
that restricts to bounded operators
$S|_{L^{1}(0,\infty)}\mathrel{\mathop{\ordinarycolon}}L^{1}(0,\infty)\to
L^{1}(0,\infty),\,S|_{L^{\infty}(0,\infty)}\mathrel{\mathop{\ordinarycolon}}L^{\infty}(0,\infty)\to
L^{\infty}(0,\infty)$, then it holds that the restriction to $E$,
$S|_{E}\mathrel{\mathop{\ordinarycolon}}E\to E$, is well defined and bounded.
This class includes many of the classical function spaces (e.g.
$L^{p}$-spaces, Orlicz spaces, Lorenz spaces, Marcinkiewiecz spaces). Also,
$E$ is said to have an order continuous norm if $\|f_{n}\|_{E}\to 0$ for every
sequence of functions $f_{n}\in E$ converging to $0$ almost everywhere and for
which $|f_{n}|$ is non-increasing. For more details about
$(L^{1}-L^{\infty})$-interpolation spaces, we refer to the monograph [8].

We conclude this section by recalling the definition of the Riemann-Liouville
and Weyl fractional integrals and derivatives. Let $\alpha>0$ be a real number
and $f$ a suitable function defined on $(0,\infty)$. The Riemann-Liouville
fractional integral of order $\alpha$ of $f$, denoted $D^{-\alpha}f$, and the
Weyl fractional integral of order $\alpha$ of $f$, denoted $W^{-\alpha}f$, are
respectively given by

$\displaystyle
D^{-\alpha}f(x)\mathrel{\mathop{\ordinarycolon}}=\frac{1}{\Gamma(\alpha)}\int_{0}^{x}(x-y)^{\alpha-1}f(y)\,dy,\quad
x>0,$ (2.4)

and

$\displaystyle
W^{-\alpha}f(x)\mathrel{\mathop{\ordinarycolon}}=\frac{1}{\Gamma(\alpha)}\int_{x}^{\infty}(y-x)^{\alpha-1}f(y)\,dy,\quad
x>0,$ (2.5)

where $\Gamma$ denotes the usual Euler-Gamma function.

The Riemann-Liouville fractional derivative of order $\alpha$ of $f$, denoted
$D^{\alpha}f$, and the Weyl fractional derivative of order $\alpha$ of $f$,
denoted $W^{\alpha}f$, are respectively given by

$\displaystyle
D^{\alpha}f(x)\mathrel{\mathop{\ordinarycolon}}=\frac{d^{n}}{dx^{n}}D^{-(n-\alpha)}f(x),\quad
x>0,$ (2.6)

and

$\displaystyle
W^{\alpha}f(x)\mathrel{\mathop{\ordinarycolon}}=(-1)^{n}\frac{d^{n}}{dx^{n}}W^{-(n-\alpha)}f(x),\quad
x>0,$ (2.7)

where $n$ is the smallest integer greater than or equal to $\alpha$. We refer
to the monograph [23] and the references therein for the class of functions
for which the above expressions (2.4)-(2.7) exist.

###  2.2. The natural functional calculus (NFC) for bisectorial operators

Next, we proceed to develop a functional calculus for bisectorial operators
which is analogous to the functional calculus for sectorial operators
presented in [18]. This functional calculus for bisectorial operators will
extend the ones considered in [22, 24] and the references therein.

For any $(\varphi,a)\in(0,\pi/2]\times[0,\infty)$, we have the algebra of
holomorphic functions

$\displaystyle\mathcal{E}_{0}(BS_{\varphi,a})\mathrel{\mathop{\ordinarycolon}}=$
$\displaystyle\Bigg{\\{}f\in
H^{\infty}(BS_{\varphi,a})\,\mathrel{\mathop{\ordinarycolon}}\,\int_{\partial(BS_{\omega^{\prime},a})}\frac{|f(z)|}{\min\\{|z-a|,|z+a|\\}}|dz|<\infty,$
$\displaystyle\hfill\qquad\qquad\qquad\qquad\qquad\text{for all
}\varphi<\omega^{\prime}\leq\pi/2\,\text{ and
}\lim_{z\to-a,a,\infty}f(z)=0\Bigg{\\}},$

where $H^{\infty}(\Omega)$ denotes the algebra of bounded holomorphic
functions in an open subset $\Omega\subset\mathbb{{C}}$. It is readily seen
that $\mathcal{E}_{0}(BS_{\varphi,a})$ is an ideal of
$H^{\infty}(BS_{\varphi,a})$.

Given a bisectorial operator $A\in\text{{BSect}}(\omega,a)$ for some
$\omega>\varphi$, we define the algebraic homomorphism
$\Phi\mathrel{\mathop{\ordinarycolon}}\mathcal{E}_{0}(BS_{\varphi,a})\to\mathcal{L}(X)$
given by

$\displaystyle\Phi(f)\mathrel{\mathop{\ordinarycolon}}=f(A)\mathrel{\mathop{\ordinarycolon}}=\frac{1}{2\pi
i}\int_{\Gamma}f(z)R(z,A)\,dz,$

where $\Gamma$ is the positively oriented boundary of the bisector
$BS_{\omega^{\prime},a}$ for any $\omega^{\prime}\in(\varphi,\omega)$ (see
Figure 1). It is readily seen that $f(A)$ is well defined and does not depend
on the choice of $\omega^{\prime}$. Moreover, one obtains the following
result.

######  Lemma 2.6.

The mapping
$\Phi\mathrel{\mathop{\ordinarycolon}}\mathcal{E}_{0}(BS_{\varphi,a})\to\mathcal{L}(X)$
satisfies the following properties:

  1. (a)

$\Phi$ is a homomorphism of algebras.

  2. (b)

If $T\in\mathcal{L}(X)$ commutes with the resolvent operator $R(\lambda,A)$ of
$A$, then it also commutes with each operator $\Phi(f)$, where
$f\in\mathcal{E}_{0}(BS_{\varphi,a})$.

  3. (c)

For $\lambda\notin\overline{BS_{\varphi,a}}$ and
$f\in\mathcal{E}_{0}(BS_{\varphi,a})$, one has

$\displaystyle
R(\lambda,A)\Phi(f)=\Phi(f)R(\lambda,A)=\left(\frac{f(z)}{\lambda-z}\right)(A).$

  4. (d)

If

$f(z)=\frac{z}{(\lambda-z)(\mu-z)}\;\mbox{ for
}\lambda,\mu\notin\overline{BS_{\varphi,a}},$

then $f\in\mathcal{E}_{0}(BS_{\varphi,a})$ and

$\displaystyle f(A)=AR(\lambda,A)R(\mu,A)\quad\text{for
}\lambda,\mu\notin\overline{BS_{\varphi,a}}.$

###### Proof.

The lemma is an immediate consequence of straightforward applications of
Cauchy’s theorem and the resolvent identity. See [16, Proposition 2.2] for an
analogous result about sectorial operators. ∎

Next, we add some functions in order to make our functional calculus to
contain certain class of resolvents. Set

$\displaystyle\mathcal{E}(BS_{\varphi,a})\mathrel{\mathop{\ordinarycolon}}=\mathcal{E}_{0}(BS_{\varphi,a})\oplus\mathbb{{C}}\frac{1}{b+z}\oplus\mathbb{{C}}\frac{1}{b-z}\;\mbox{
for any }b\in\mathbb{{C}}\backslash\overline{BS_{\varphi,a}}.$

It is easy to see that the definition of $\mathcal{E}(BS_{\varphi,a})$ is
independent on the particular choice of $b$.

Then, one can extend $\Phi$ from $\mathcal{E}_{0}(BS_{\varphi,a})$ to
$\mathcal{E}(BS_{\varphi,a})$ by defining

$\Phi\left(\frac{1}{b+z}\right)=-R(-b,A)\mbox{ and
}\Phi\left(\frac{1}{b-z}\right)=R(b,A),$

so that
$\Phi\mathrel{\mathop{\ordinarycolon}}\mathcal{E}(BS_{\varphi,a})\to\mathcal{L}(X)$
is an algebraic homomorphism. Moreover,
$(\mathcal{E}(BS_{\varphi,a}),\mathcal{M}(BS_{\varphi,a}),\Phi)$ is an
abstract functional calculus (see e.g. [16]), where $\mathcal{M}(\Omega)$
denoted the algebra of meromorphic functions with domain an open set
$\Omega\subset\mathbb{{C}}$.

######  Definition 2.7.

Let $(\varphi,a)\in(0,\pi/2]\times[0,\infty)$. We introduce the following
notions.

  1. (a)

A function $f\in\mathcal{M}(BS_{\varphi,a})$ is said to be regularizable by
$\mathcal{E}(BS_{\varphi,a})$ if there exists
$e\in\mathcal{E}(BS_{\varphi,a})$ such that $e(A)$ is injective and
$ef\in\mathcal{E}(BS_{\varphi,a})$.

  2. (b)

For any regularizable $f\in\mathcal{M}(BS_{\varphi,a})$ we set

$\displaystyle f(A)\mathrel{\mathop{\ordinarycolon}}=e(A)^{-1}(ef)(A).$

By [16, Lemma 3.2], one has that this definition is independent of the
regularizer $e$, and that $f(A)$ is a well defined closed operator.

Finally, we set

$\displaystyle\mathcal{M}(BS_{\varphi,a})_{A}$
$\displaystyle\mathrel{\mathop{\ordinarycolon}}=\\{f\in\mathcal{M}(BS_{\varphi,a})\,\mathrel{\mathop{\ordinarycolon}}\,f\text{
is regularizable by }\mathcal{E}(BS_{\varphi,a})\\},$

and

$\displaystyle H(A)$
$\displaystyle\mathrel{\mathop{\ordinarycolon}}=\\{f\in\mathcal{M}(BS_{\varphi,a})_{A}\,\mathrel{\mathop{\ordinarycolon}}\,f(A)\in\mathcal{L}(X)\\}.$

This meromorphic functional calculus satisfies the following properties.

######  Lemma 2.8.

Let $(\varphi,a)\in(0,\pi/2]\times[0,\infty)$ and let also
$f\in\mathcal{M}(BS_{\varphi,a})_{A}$ with $A\in\text{{BSect}}(\omega,a)$.
Then the following assertions hold.

  1. (a)

If $S\in\mathcal{L}(X)$ commutes with $A$, that is, $SA\subset AS$, then $S$
also commutes with $f(A)$, i.e. $Sf(A)\subset f(A)S$.

  2. (b)

$\mathbf{1}(A)=I$ and $z(A)=A$.

  3. (c)

Let $g\in\mathcal{M}(BS_{\varphi,a})_{A}$. Then

$\displaystyle f(A)+g(A)\subset(f+g)(A),\;\mbox{ and
}\;f(A)g(A)\subset(fg)(A).$ (2.8)

Furthermore,
$\mathcal{D}(f(A)g(A))=\mathcal{D}((fg)(A))\cap\mathcal{D}(g(A))$, and one has
equality in (2.8) if $g(A)\in\mathcal{L}(X)$.

  4. (d)

$\Phi\mathrel{\mathop{\ordinarycolon}}H(A)\to\mathcal{L}(X)$ is a homomorphism
of unital algebras.

  5. (e)

Let $g\in H(A)$ such that $g(A)$ is bounded and injective. Then
$f(A)=g(A)^{-1}f(A)g(A)$.

  6. (f)

Let $\lambda\in\mathbb{{C}}$. Then

$\displaystyle\frac{1}{\lambda-f(z)}\in\mathcal{M}(BS_{\varphi,a})_{A}\iff\lambda-f(A)\text{
is injective}.$

If this is the case, $(\lambda-f(z))^{-1}(A)=(\lambda-f(A))^{-1}$. In
particular, $\lambda\in\rho(A)$ if and only if $(\lambda-f(z))^{-1}\in H(A)$.

###### Proof.

The lemma is a direct consequence of the results contained in [16, Section 3
]. ∎

Since the inclusions
$\mathcal{E}(BS_{\varphi,a})\subset\mathcal{E}(BS_{\varphi^{\prime},a}),\,\mathcal{M}(BS_{\varphi,a})\subset\mathcal{M}(BS_{\varphi^{\prime},a})$
hold for any $\varphi<\varphi^{\prime}<\omega$, we can form the inductive
limits

$\displaystyle\mathcal{E}[BS_{\omega,a}]\mathrel{\mathop{\ordinarycolon}}=\bigcup_{0<\varphi<\omega}\mathcal{E}(BS_{\varphi,a}),\quad\mathcal{M}[BS_{\omega,a}]\mathrel{\mathop{\ordinarycolon}}=\bigcup_{0<\varphi<\omega}\mathcal{M}(BS_{\varphi,a}),\quad\mathcal{M}[BS_{\omega,a}]_{A}\mathrel{\mathop{\ordinarycolon}}=\bigcup_{0<\varphi<\omega}\mathcal{M}(BS_{\varphi,a})_{A}.$

If $f\in\mathcal{M}[BS_{\omega,a}]_{A}$, that is, if $f$ is regularizable by
$\mathcal{E}[BS_{\omega,a}]$, then we say that “$f(A)$ is defined by the
natural functional calculus (NFC) for bisectorial operators”.

###  2.3. Extensions of the natural functional calculus (NFC)

Let $d\in\\{-a,a\\}$ with $a>0$ and consider an operator
$A\in\text{{BSect}}(\omega,a)$ for which $dI-A$ is invertible, that is,
$d\notin\sigma(A)$. Fix $0<\varphi<\omega$. Then, there is an
$\varepsilon\in(0,a)$ such that the ball
$\overline{B_{\varepsilon}(d)}\subset\rho(A)$, where
$B_{\varepsilon}(d)\mathrel{\mathop{\ordinarycolon}}=\\{z\in\mathbb{{C}}\mathrel{\mathop{\ordinarycolon}}\,|z-d|<\varepsilon\\}$.
Consider the algebra

$\displaystyle\mathcal{E}_{0}(BS_{\varphi,a,d,\varepsilon})\mathrel{\mathop{\ordinarycolon}}=\Bigg{\\{}f\in
H^{\infty}(BS_{\varphi,a}\backslash\overline{B_{\varepsilon}(d)})\mathrel{\mathop{\ordinarycolon}}\,\lim_{z\to-d,\infty}f(z)=0\text{
and}$
$\displaystyle\qquad\qquad\qquad\qquad\int_{\partial(BS_{\omega^{\prime},a}\backslash\overline{B_{\varepsilon}(d)})}\frac{|f(z)|}{|d-z|}\,|dz|<\infty\quad\text{
for all }\varphi<\omega^{\prime}\leq\frac{\pi}{2}\Bigg{\\}},$

and set
$\mathcal{E}(BS_{\varphi,a,d,\varepsilon})\mathrel{\mathop{\ordinarycolon}}=\mathcal{E}_{0}(BS_{\varphi,a,d,\varepsilon})\oplus\mathbb{{C}}(1/(b-z))$
for any $b>a$. One can extend our elementary functional calculus $\Phi$ from
$\mathcal{E}(BS_{\varphi,a})$ to the algebra
$\mathcal{E}(BS_{\varphi,a,d,\varepsilon})$ by integrating on a positively
oriented parametrization of the boundary
$\partial(BS_{\omega^{\prime},a}\backslash\overline{B_{\varepsilon}(d)})$ for
any $\varphi<\omega^{\prime}\leq\frac{\pi}{2}$, where we avoid integration
near the point $d$ and therefore the functions in this new algebra do not need
to satisfy any regularity conditions near $d$. This yields a new abstract
functional calculus
$(\mathcal{E}(BS_{\varphi,a,d,\varepsilon}),\mathcal{M}(BS_{\varphi,a}),\Phi)$
which is an extension of the former.

Since the new algebra $\mathcal{E}(BS_{\varphi,a,d,\varepsilon})$ is larger
than $\mathcal{E}(BS_{\varphi,a})$, more functions
$f\in\mathcal{M}(BS_{\varphi,a})$ become regularizable. If $f$ is
regularizable by $\mathcal{E}(BS_{\varphi,a,d,\varepsilon})$, we say that
$f(A)$ is defined by the NFC for $d-$invertible operators.

A similar extension is possible when $A\in\text{{BSect}}(\omega,a)$ is
bounded. Again, let $0<\varphi<\omega$ and let $R>\max\\{a,r(A)\\}$, where
$r(A)$ denotes the spectral radius of $A$. Then one considers the algebra

$\displaystyle\Bigg{\\{}f\in H^{\infty}(BS_{\varphi,a}\cap
B_{R}(0))\mathrel{\mathop{\ordinarycolon}}\,\lim_{z\to-a}f(z)=\lim_{z\to
a}f(z)=0\mbox{ and}$ $\displaystyle\qquad\int_{\partial
BS_{\omega^{\prime},a},|z|<R/2}\frac{|f(z)|}{\min\\{|z-a|,|z+a|\\}}|dz|<\infty\text{
for all }\varphi<\omega^{\prime}\leq\frac{\pi}{2}\Bigg{\\}},$

and adds the spaces $\mathbb{{C}}(1/(b-z))$ and $\mathbb{{C}}(1/(b+z))$ for
any $b>a$. The elementary calculus of this algebra is constructed along the
lines of the former elementary calculus. If a function $f$ is regularizable by
this algebra, we say that $f(A)$ is defined by the NFC for bounded bisectorial
operators.

Similar remarks apply when $A\in\text{{BSect}}(\omega,a)$ satisfies any
combination of the above properties, avoiding integration near the appropriate
selection of the points in $\\{-a,a,\infty\\}$. A minor difference appears if
$a=0$ and $A$ is invertible: in that case the two top and bottom branches
merge without further consequences. Clearly, Lemma 2.8 remains valid when it
is adapted appropriately to any of this NFC.

######  Definition 2.9.

Let $A\in\text{{BSect}}(\omega,a)$ and set
$M_{A}\mathrel{\mathop{\ordinarycolon}}=\\{-a,a,\infty\\}\cap\widetilde{\sigma}(A)$.
From now on, we will denote by $\mathcal{M}[\Omega_{A}]$ the class of
meromorphic functions defined on an open set related to the larger primary
functional calculus of $A$. Moreover, $\mathcal{M}_{A}$ will denote the class
of regularizable functions $\mathcal{E}_{A}$ of the larger natural functional
calculus of $A$. For instance, if $M_{A}=\\{-a,\infty\\}$, then
$\mathcal{M}[\Omega_{A}]$ and $\mathcal{M}_{A}$ will refer to the calculus of
$a-$invertible bisectorial operators; if $M_{A}=\\{-a\\}$, then they will
refer to the calculus of $a-$invertible and bounded bisectorial operators, and
so on.

Finally, if $f\in\mathcal{M}_{A}$, that is, if $f$ is regularizable by
$\mathcal{E}_{A}$ for the appropriate primary functional calculus of $A$, then
we say that $f(A)$ is defined by the natural functional calculus (NFC) of $A$.

Next, our goal is to give a sufficient condition for a meromorphic function
$f\in\mathcal{M}(BS_{\varphi,a})$ to be defined by the NFC of $A$. We will ask
$f$ to satisfy a regularity property near the singular points
$M_{A}\mathrel{\mathop{\ordinarycolon}}=\\{-a,a,\infty\\}\cap\widetilde{\sigma}(A)$.

######  Definition 2.10.

Let $A\in\text{{BSect}}(\omega,a)$, $f\in\mathcal{M}[\Omega_{A}]$ and $d\in
M_{A}\cap\\{-a,a\\}$.

  1. (a)

We say that $f$ is regular at $d$ if $\lim_{z\to
d}f(z)=\mathrel{\mathop{\ordinarycolon}}c_{d}\in\mathbb{{C}}$ exists and, for
small enough $\varepsilon>0$ and some $\varphi<\omega$,

$\displaystyle\int_{\partial(BS_{\omega^{\prime},a}\cap
B_{\varepsilon}(d))}\left|\frac{f(z)-c_{d}}{z-d}\right||dz|<\infty,\quad\text{for
all }\omega^{\prime}\in\left(\varphi,\frac{\pi}{2}\right).$

  2. (b)

Similarly, if $\infty\in M_{A}$, we say that $f$ is regular at $\infty$ if
$\lim_{z\to\infty}f(z)=\mathrel{\mathop{\ordinarycolon}}c_{\infty}\in\mathbb{{C}}$
exists and, for large enough $R>0$, and some $\varphi<\omega$,

$\displaystyle\int_{\partial
BS_{\omega^{\prime},a},|z|>R}\left|\frac{f(z)-c}{z}\right||dz|<\infty,\quad\text{for
all }\omega^{\prime}\in\left(\varphi,\frac{\pi}{2}\right).$

  3. (c)

We say that $f$ is quasi-regular at $d\in M_{A}$ if $f$ or $1/f$ is regular at
$d$.

  4. (d)

Finally, we say that $f$ is (quasi)-regular at $M_{A}$ if $f$ is
(quasi)-regular at each point of $M_{A}$.

######  Remark 2.11.

Note that if $f$ is regular at $M_{A}$ with every limit being not equal to
$0$, then $1/f$ is also regular at $M_{A}$. If $f$ is quasi-regular at
$M_{A}$, then $\mu-f$ and $1/f$ are also quasi-regular at $M_{A}$ for each
$\mu\in\mathbb{{C}}$. A function $f$ which is quasi-regular at $M_{A}$ has
well defined limits in $\mathbb{{C}}_{\infty}$ as $z$ tends to each point of
$M_{A}$.

######  Lemma 2.12.

Let $A\in\text{{BSect}}(\omega,a)$ and $f\in\mathcal{M}[\Omega_{A}]$. Assume
that $f$ is regular at $M_{A}$ and that all the poles of $f$ are contained in
$\mathbb{{C}}\backslash\sigma_{p}(A)$. Then, $f(A)$ is defined by the NFC of
$A$, that is, $f\in\mathcal{M}_{A}$. More precisely, the following assertions
hold true.

  1. (a)

If $M_{A}=\\{-a,a,\infty\\}$, then $f(A)$ is defined by the NFC for
bisectorial operators.

  2. (b)

If $M_{A}=\\{-a,\infty\\}$, then $f(A)$ is defined by the NFC for
$a$-invertible bisectorial operators. Analogous statement is true if
$M_{A}=\\{a,\infty\\}$.

  3. (c)

If $M_{A}=\\{-a,a\\}$, then $f(A)$ is defined by the NFC for bounded
bisectorial operators.

  4. (d)

If $M_{A}=\\{\infty\\}$, then $f(A)$ is defined by the NFC for $a$-invertible
and $-a$-invertible bisectorial operators.

  5. (e)

If $M_{A}=\\{-a\\}$, then $f(A)$ is defined by the NFC for $a$-invertible and
bounded bisectorial operators. Analogous statement is true if $M_{A}=\\{a\\}$.

  6. (f)

If $M_{A}=\emptyset$, then $f(A)$ is defined by the NFC for bounded,
$a$-invertible and $-a$-invertible bisectorial operators.

Moreover, if the poles of $f$ are contained in $\rho(A)$, then
$f(A)\in\mathcal{L}(X)$.

###### Proof.

The proof is analogous to the corresponding result for sectorial operators,
see [17, Lemma 6.2]. We omit the details for the sake of brevity. ∎

We conclude this section by giving a spectral inclusion result for bisectorial
operators.

######  Proposition 2.13.

Let $A\in\text{{BSect}}(\omega,a)$, and take $f\in\mathcal{M}_{A}$ to be
quasi-regular at $M_{A}$. Then

$\displaystyle\widetilde{\sigma}(f(A))\subset f(\widetilde{\sigma}(A)).$

###### Proof.

The proof follows as the case of sectorial operators, see [17, Proposition
6.3]. We omit the details for the sake of brevity. ∎

##  3\. From bisectorial to sectorial operators and the NFC for sectional
operators

We start with relationship between bisectorial and sectorial operators.

###  3.1. From bisectorial to sectorial operators

In this section, we present a connection between bisectorial operators and
sectorial operators, namely Theorem 3.10. The method to prove this connection
is based on the proof of the scaling property for sectorial operators given in
[5, Proposition 5.2], but its proof requires longer and more sophisticated
techniques because of the more intricate setting.

We start with a new definition referring to the limit behavior of a function
at the singular points $\\{-a,a,\infty\\}$. By $|f(z)|\sim|g(z)|$ as $z\to
d\in\mathbb{{C}}_{\infty}$, we mean that there is some neighborhood $\Omega$
of $d$ such that $|g(z)|\lesssim|f(z)|\lesssim|g(z)|$ for all $z\in\Omega$.

######  Definition 3.1.

Let $A\in\text{{BSect}}(\omega,a)$, $f\in\mathcal{M}[\Omega_{A}]$,
$d\in\overline{\mathcal{D}(f)}$, and $c\in\mathbb{{C}}$.

  1. (a)

We say that $f(z)\to c$ exactly polynomially as $z\to d$ if there is
$\alpha>0$ such that $|f(z)-c|\sim|z-d|^{\alpha}$ as $z\to d$ if
$d\in\mathbb{{C}}$, or such that $|f(z)-c|\sim|z|^{-\alpha}$ as $z\to\infty$
if $d=\infty$.

  2. (b)

We say that $f(z)\to\infty$ exactly polynomially as $z\to d$ if $(1/f)(z)\to
0$ exactly polynomially as $z\to d$.

Now, we fix some notations for the rest of this section. From now on, $A$ will
be a bisectorial operator on a Banach space $X$ of angle $\omega\in(0,\pi/2]$
and half-width $a\geq 0$, i.e. $A\in\text{{BSect}}(\omega,a)$, and recall that
$M_{A}=\\{a,-a,\infty\\}\cap\widetilde{\sigma}(A)$. For any
$\lambda\in\mathbb{{C}},\,f\in\mathcal{M}[\Omega_{A}]$, we let
$R_{f}^{\lambda}\in\mathcal{M}[\Omega_{A}]$ be given by

$\displaystyle
R_{f}^{\lambda}(z)\mathrel{\mathop{\ordinarycolon}}=\frac{\lambda}{\lambda-f(z)},\qquad
z\in\mathcal{D}(f).$ (3.1)

Moreover, we will also consider $\gamma\in[0,\pi)$ and a function
${\widetilde{g}}\in\mathcal{M}_{A}$ satisfying the following conditions:

  1. (a)

$\mathcal{R}({\widetilde{g}})\subset\overline{S_{\gamma}}\cup\\{\infty\\}$.

  2. (b)

${\widetilde{g}}$ is quasi-regular at $M_{A}$. In particular, it has limits in
$M_{A}$, which we denote by $c_{d}\in\mathbb{{C}}_{\infty}$ for $d\in M_{A}$.

  3. (c)

${\widetilde{g}}$ has exactly polynomial limits at
$M_{A}\cap{\widetilde{g}}^{-1}(\\{0,\infty\\})$.

######  Remark 3.2.

By the open mapping theorem, Property (a) implies that ${\widetilde{g}}$ does
not have any zeros (unless ${\widetilde{g}}=0$) or poles in
$\mathcal{D}({\widetilde{g}})$. In particular, if ${\widetilde{g}}\neq 0$,
then both ${\widetilde{g}}$ and ${\widetilde{g}}^{-1}$ are holomorphic.

Next, we present a family of functions which will be crucial to prove our main
result of this section. Recall that a meromorphic function $f$ belongs to
$H(A)\subset\mathcal{M}_{A}$ if and only if $f(A)$ is a bounded operator.

######  Definition 3.3.

Assume that we have a family of functions
$(f^{\lambda})_{\lambda\notin\overline{S_{\gamma}}}\subset H(A)$. We say that
$(f^{\lambda})_{\lambda\notin\overline{S_{\gamma}}}$ makes
$(R_{\widetilde{g}}^{\lambda})_{\lambda\notin\overline{S_{\gamma}}}$
$\varepsilon-$uniformly bounded at $d\in M_{A}$ with respect to the NFC of $A$
if, for any $\varepsilon\in(0,\pi-\gamma)$, it satisfies the following
properties:

  1. (a)

$|f^{\lambda}(z)|$ is uniformly bounded for all $z\in\mathcal{D}(f^{\lambda})$
and $\lambda\notin\overline{S_{\gamma+\varepsilon}}$.

  2. (b)

$\|f^{\lambda}(A)\|_{\mathcal{L}(X)}$ is uniformly bounded for all
$\lambda\notin\overline{S_{\gamma+\varepsilon}}$.

  3. (c)

Let $\Gamma$ be an integration path for the NFC of $A$ (see Section 2.3).
Then, there exists a neighborhood $\Omega_{d^{\prime}}$ containing each
$d^{\prime}\in M_{A}\backslash\\{d\\}$ for which

$\displaystyle\sup_{\lambda\notin\overline{S_{\gamma+\varepsilon}}}\int_{\Gamma\cap\Omega_{d^{\prime}}}|f^{\lambda}(z)|\|R(z,A)\|_{\mathcal{L}(X)}\,|dz|<\infty\quad\text{for
each }d^{\prime}\in M\backslash\\{d\\}.$

  4. (d)

Let $\Gamma$ be as above. Then, there exists a neighborhood $\Omega_{d}$
containing $d$ for which

$\displaystyle\sup_{\lambda\notin\overline{S_{\gamma+\varepsilon}}}\int_{\Gamma\cap\Omega_{d}}|R_{\widetilde{g}}^{\lambda}(z)-f^{\lambda}(z)|\|R(z,A)\|_{\mathcal{L}(X)}\,|dz|<\infty.$

The following lemmas will be useful in finding the family of functions that
makes $(R_{\widetilde{g}}^{\lambda})_{\lambda\notin\overline{S_{\gamma}}}$
$\varepsilon-$uniformly bounded. Let $d(z,\Omega)$ denote the distance between
a point $z\in\mathbb{{C}}$ and a set $\Omega\subset\mathbb{{C}}$.

######  Lemma 3.4.

Let $\varepsilon>0$. We have that
$d(z,\mathbb{{C}}\backslash\overline{S_{\gamma+\varepsilon}})\gtrsim|z|$ for
all $z\in\overline{S_{\gamma}}$ and that
$d(w,\overline{S_{\gamma}})\gtrsim|w|$ for all
$w\notin\overline{S_{\gamma+\varepsilon}}$. As a consequence, $|w/(w-z)|$ and
$|z/(w-z)|$ are uniformly bounded for all $z\in\overline{S_{\gamma}}$ and
$w\notin\overline{S_{\gamma+\varepsilon}}$.

###### Proof.

The first two inequalities follow from the fact that
$d(z,\mathbb{{C}}\backslash\overline{S_{\gamma+\varepsilon}})=|z|\sin(\gamma+\varepsilon-|\arg(z)|)\geq|z|\sin\varepsilon$
and $d(w,\overline{S_{\gamma}})=|w|\sin(|\arg
w|-\gamma)\geq|w|\sin\varepsilon$. The other inequalities follow from what we
have already proven, and that
$|z-w|\geq\max\\{d(z,\mathbb{{C}}\backslash\overline{S_{\gamma+\varepsilon}}),d(w,\overline{S_{\gamma}})\\}$
for all $z\in\overline{S_{\gamma}}$ and
$w\notin\overline{S_{\gamma+\varepsilon}}$. ∎

Lemma 3.4 will be used very frequently throughout the paper. We will not make
no further reference to it for the sake of brevity. In particular, as an
immediate application we have the following result.

######  Lemma 3.5.

Let $c\in\overline{S_{\gamma}}\backslash\\{0\\}$,
$\varepsilon\in(0,\pi-\gamma)$ and $f\in\mathcal{M}[\Omega_{A}]$ such that
$\mathcal{R}(f)\subset\overline{S_{\gamma}}\cup\\{\infty\\}$. Then,

$\displaystyle\left|R_{f}^{\lambda}(z)-\frac{\lambda}{\lambda-c}\right|$
$\displaystyle\lesssim\min\\{1,|f(z)-c|\\},$
$\displaystyle\left|R_{f}^{\lambda}(z)\right|$
$\displaystyle\lesssim\min\\{1,|\lambda||f(z)|^{-1}\\},$
$\displaystyle\left|R_{f}^{\lambda}(z)-1\right|$
$\displaystyle\lesssim\min\\{1,|\lambda|^{-1}|f(z)|\\},$

where all inequalities hold for all $z\in\mathcal{D}(f)$ and
$\lambda\notin\overline{S_{\gamma+\varepsilon}}$.

###### Proof.

It follows from Lemma 3.4 that all the above functions are uniformly bounded.

Now, let $c\in\overline{S_{\gamma}}$. Applying Lemma 3.4, one gets that

$\displaystyle\left|R_{f}^{\lambda}(z)-\frac{\lambda}{\lambda-c}\right|$
$\displaystyle=\left|\frac{\lambda}{\lambda-f(z)}\frac{f(z)-c}{\lambda-c}\right|\lesssim\left|\frac{f(z)-c}{\lambda-c}\right|\lesssim|f(z)-c|,$

for all $z\in\mathcal{D}(f),\,\lambda\notin\overline{S_{\gamma+\varepsilon}}$.
Likewise, one obtains that

$\displaystyle|R_{f}^{\lambda}(z)|=\left|\frac{\lambda}{\lambda-f(z)}\right|\lesssim|\lambda||f(z)|^{-1},\qquad|R_{f}^{\lambda}(z)-1|=\left|\frac{f(z)}{\lambda-f(z)}\right|\lesssim|\lambda|^{-1}|f(z)|,$

for all $z\in\mathcal{D}(f),\,\lambda\notin\overline{S_{\gamma+\varepsilon}}$.
The proof is finished. ∎

The following lemma gives an integral bound which will be very useful to prove
Proposition 3.7 below.

######  Lemma 3.6.

Let $I\subset\mathbb{{R}}^{+}$ be a measurable subset and let
$(f_{\nu})_{\nu}\mathrel{\mathop{\ordinarycolon}}I\to\mathbb{{C}}$ be a family
of complex-valued functions. Let
$F_{1},F_{2}\mathrel{\mathop{\ordinarycolon}}I\to\mathbb{{R}}^{+}$ be some
positive functions which are integrable with respect to the measure $dx/x$,
and let $r_{\nu}>0$ for all indices $\nu$, and $s_{n},t_{m}>0$ for
$n=1,...,N$, $m=1,...,M$ for some $N,M\in\mathbb{{N}}$. Assume that

$\displaystyle|f_{\nu}(x)|\lesssim\min\left\\{F_{1}(x)+\sum_{n\leq
N}(r_{\nu}x)^{s_{n}},F_{2}(x)+\sum_{m\leq M}(r_{\nu}x)^{-t_{m}}\right\\},$
(3.2)

for all $x\in I$ and all indices $\nu$. Then

$\displaystyle\sup_{\nu}\int_{I}|f_{\nu}(x)|\frac{dx}{x}<\infty.$

###### Proof.

By adding terms of the type $(r_{\nu}x)^{t_{m}}$ to the first expression
inside the minimum in (3.2), and terms of the type $(r_{\nu}x)^{-s_{n}}$, one
can assume that $N=M$ and $s_{n}=t_{n}$ for all $n=1,...,N$. It follows that

$\displaystyle\int_{I}|f_{\nu}(x)|\frac{dx}{x}$
$\displaystyle\lesssim\int_{I}\Big{(}F_{1}(x)+F_{2}(x)\Big{)}\frac{dx}{x}+\int_{I}\min\left\\{\sum_{n\leq
N}(r_{\nu}x)^{s_{n}},\sum_{n\leq N}(r_{\nu}x)^{-s_{n}}\right\\}\frac{dx}{x}.$

By the integrability condition on $F_{1},F_{2}$, it suffices to bound the
second term for all indices $\nu$. By bounding the integral on $I$ by the
integral on $(0,\infty)$, and a simple change of variable, one gets that

$\displaystyle\int_{I}\min\left\\{\sum_{n\leq N}(r_{\nu}x)^{s_{n}},\sum_{n\leq
N}(r_{\nu}x)^{-s_{n}}\right\\}\frac{dx}{x}=\int_{0}^{\infty}\min\left\\{\sum_{n\leq
N}x^{s_{n}},\sum_{n\leq N}x^{-s_{n}}\right\\}\frac{dx}{x}$
$\displaystyle\quad=\sum_{n\leq
N}\left(\int_{0}^{1}x^{s_{n}-1}\,dx+\int_{1}^{\infty}x^{-s_{n}-1}\,dx\right)<\infty.$

The proof is concluded. ∎

In order to prove the main result of this section, some integrals related to
resolvent operators need to be bounded. The techniques to bound them vary from
one case to another, as shows the proposition below.

######  Proposition 3.7.

For each point $d\in M$, there exists a family of functions
$(f^{\lambda})_{\lambda\notin\overline{S_{\gamma}}}$ that makes
$(R_{\widetilde{g}}^{\lambda})_{\lambda\notin\overline{S_{\gamma}}}$
$\varepsilon-$uniformly bounded at $d$ with respect to the NFC of $A$.

###### Proof.

We will proceed by examining all the possible cases. Throughout the proof
$\varepsilon$ is any appropriate number in $(0,\pi-\gamma)$ whenever it
appears. Also, $b>a$ for the rest of the proof. We proceed in several steps.

Step 1: Let $d=a$ and $c_{a}\in\overline{S_{\gamma}}\backslash\\{0,\infty\\}$.
We claim that the family of functions given by

$\displaystyle
f_{a,c_{a}}^{\lambda}(z)\mathrel{\mathop{\ordinarycolon}}=\frac{\lambda}{\lambda-
c_{a}}\frac{b^{2}-a^{2}}{2a}\frac{a+z}{b^{2}-z^{2}},\quad
z\in\mathcal{D}({\widetilde{g}}),$

makes $(R_{\widetilde{g}}^{\lambda})_{\lambda\notin\overline{S_{\gamma}}}$
$\varepsilon-$uniformly bounded at $a$. Indeed, it follows from Lemma 3.4 that
$|f_{a,c_{a}}^{\lambda}(z)|$ is uniformly bounded for all
$z\in\mathcal{D}(f_{a,c_{a}}^{\lambda})=\mathcal{D}({\widetilde{g}})$ and
$\lambda\notin\overline{S_{\gamma+\varepsilon}}$. Moreover,
$f_{a,c_{a}}^{\lambda}\in H(A)$ with

$\displaystyle f_{a,c_{a}}^{\lambda}(A)=\frac{\lambda}{\lambda-
c_{a}}\frac{b^{2}-a^{2}}{2a}(A+aI)R(b^{2},A^{2}),$

so $\|f_{a,c_{a}}^{\lambda}(A)\|_{\mathcal{L}(X)}$ is also uniformly bounded
for all $\lambda\notin\overline{S_{\gamma+\varepsilon}}$ (recall that
$\sigma(A^{2})=\sigma(A)^{2}$). It is clear that the integrability property
(c) in Definition 3.3 holds, since again $\lambda/(\lambda-c_{a})$ is bounded
and $(z+a)/(b^{2}-z^{2})$ is integrable with respect to
$\|R(z,A)\|_{\mathcal{L}(X)}|dz|$ at the neighborhoods of $-a$ and $\infty$.
Finally, we have that

$\displaystyle\left|R_{\widetilde{g}}^{\lambda}(z)-f_{a,c_{a}}^{\lambda}(z)\right|$
$\displaystyle\leq\left|R_{\widetilde{g}}^{\lambda}(z)-\frac{\lambda}{\lambda-
c_{a}}\right|+\left|\frac{\lambda}{\lambda-
c_{a}}\right|\left|\frac{b^{2}-a^{2}}{2a}\frac{z+a}{b^{2}-z^{2}}-1\right|\lesssim|{\widetilde{g}}(z)-c_{a}|+|z-a|,$

where the first estimate is obtained by an application of Lemma 3.5, and the
second one by using Lemma 3.4 and Taylor’s expansion of order $1$. Since
${\widetilde{g}}$ is regular at $a$ with limit $c_{a}$, it follows that
$|R_{\widetilde{g}}^{\lambda}(z)-f_{a,c_{a}}^{\lambda}(z)|$ satisfies the
integrability property (d) in Definition 3.3 and the claim is proven.

Step 2: Next, let $d=a$ and $c_{a}=0$. Since $a\in
M_{A}\cap{\widetilde{g}}^{-1}(\\{0,\infty\\})$, it follows that in a
neighborhood $\Omega_{a}$ containing $a$ and a real number $\alpha>0$, we have
that $|{\widetilde{g}}(z)|\sim|z-a|^{\alpha}$ for all
$z\in\Omega_{a}\cap\mathcal{D}({\widetilde{g}})$. We consider the family of
functions given by

$\displaystyle
f_{a,0}^{\lambda}(z)\mathrel{\mathop{\ordinarycolon}}=\frac{|\lambda|^{1/\alpha}}{|\lambda|^{1/\alpha}+a-z}\frac{b^{2}-a^{2}}{2a}\frac{a+z}{b^{2}-z^{2}},\quad
z\in\mathcal{D}({\widetilde{g}}).$

Let us show that
$(f_{a,0}^{\lambda})_{\lambda\notin\overline{S_{\gamma+\varepsilon}}}$
satisfies the desired properties. First of all, it is clear that
$|f_{a,0}^{\lambda}(z)|$ is uniformly bounded for all
$z\in\mathcal{D}(f_{a,0}^{\lambda})=\mathcal{D}({\widetilde{g}})$ and
$\lambda\notin\overline{S_{\gamma+\varepsilon}}$. Moreover,
$f_{a,0}^{\lambda}\in H(A)$ with

$\displaystyle
f_{a,0}^{\lambda}(A)=\frac{b^{2}-a^{2}}{2a}(A+aI)R(b^{2},A^{2})|\lambda|^{1/\alpha}R(|\lambda|^{1/\alpha},A-aI).$

Thus, it follows from the definition of bisectorial operators that
$\|f_{a,0}^{\lambda}(A)\|_{\mathcal{L}(X)}$ is uniformly bounded for all
$\lambda\notin\overline{S_{\gamma+\varepsilon}}$. It is readily seen that it
satisfies Property (c) in Definition 3.3, since
$|\lambda|^{1/\alpha}/(|\lambda|^{1/\alpha}+a-z)$ is uniformly bounded.

Next, we consider $|R_{\widetilde{g}}^{\lambda}(z)-f_{c_{a},0}^{\lambda}(z)|$
in $\Omega_{a}\cap\mathcal{D}({\widetilde{g}})$. On the one hand, by the
triangle inequality and various applications of Lemmas 3.4 and 3.5, we get

$\displaystyle|R_{\widetilde{g}}^{\lambda}(z)-f_{a,0}^{\lambda}(z)|$
$\displaystyle\leq\left|R_{\widetilde{g}}^{\lambda}(z)\right|+\frac{b^{2}-a^{2}}{2a}\left|\frac{a+z}{b^{2}-z^{2}}\right|\left|\frac{|\lambda|^{1/\alpha}}{|\lambda|^{1/\alpha}+a-z}\right|$
$\displaystyle\lesssim|\lambda||{\widetilde{g}}(z)|^{-1}+|\lambda|^{1/\alpha}|z-a|^{-1}\lesssim|\lambda^{-1/\alpha}(z-a)|^{-\alpha}+|\lambda^{-1/\alpha}(z-a)|^{-1},$

for all $z\in\Omega_{a}\cap\mathcal{D}({\widetilde{g}})$ and
$\lambda\notin\overline{S_{\gamma+\varepsilon}}$. On the other hand, it
follows that

$\displaystyle|R_{\widetilde{g}}^{\lambda}(z)-f_{a,0}^{\lambda}(z)|$
$\displaystyle\leq|R_{\widetilde{g}}^{\lambda}(z)-1|+|f_{a,0}^{\lambda}(z)-1|.$

Another application of Lemma 3.5 yields that
$|R_{\widetilde{g}}^{\lambda}(z)-1|\lesssim|\lambda|^{-1}|{\widetilde{g}}(z)|\lesssim|\lambda^{-1/\alpha}(z-a)|^{\alpha}$
for all $z\in\Omega_{a}\cap\mathcal{D}({\widetilde{g}})$ and
$\lambda\notin\overline{S_{\gamma+\varepsilon}}$. Moreover, one gets that

$\displaystyle|f_{a,0}^{\lambda}(z)-1|$
$\displaystyle\leq\left|f_{a,0}^{\lambda}(z)-\frac{b^{2}-a^{2}}{2a}\frac{a+z}{b^{2}-z^{2}}\right|+\left|\frac{b^{2}-a^{2}}{2a}\frac{a+z}{b^{2}-z^{2}}-1\right|$
$\displaystyle\lesssim\left|\frac{a-z}{|\lambda|^{1/\alpha}+a-z}\right|+|z-a|\lesssim|\lambda^{-1/\alpha}(z-a)|+|z-a|,$

for all $z\in\Omega_{a}\cap\mathcal{D}({\widetilde{g}})$ and
$\lambda\notin\overline{S_{\gamma+\varepsilon}}$. Summarizing, if we set
$U_{\lambda}(z)\mathrel{\mathop{\ordinarycolon}}=|\lambda|^{1/\alpha}|z-a|$,
then we obtain that

$\displaystyle|R_{\widetilde{g}}^{\lambda}(z)-f_{a,0}^{\lambda}(z)|\lesssim\min\left\\{\sum_{j\in\\{1,\alpha\\}}U_{\lambda}(z)^{-j},|z-a|+\sum_{j\in\\{1,\alpha\\}}U_{\lambda}(z)^{j}\right\\},$

for all $z\in\Omega_{a}\cap\mathcal{D}({\widetilde{g}})$ and
$\lambda\notin\overline{S_{\gamma+\varepsilon}}$. An application of Lemma 3.6
together with the bound of the resolvent of a bisectorial operator, yield that
$(f_{a,0}^{\lambda})_{\lambda\notin\overline{S_{\gamma}}}$ satisfies Property
(d) in Definition 3.3, so in fact
$(f_{a,0}^{\lambda})_{\lambda\notin\overline{S_{\gamma}}}$ makes
$(R_{\widetilde{g}}^{\lambda})_{\lambda\notin\overline{S_{\gamma}}}$
$\varepsilon-$uniformly bounded at $a$ with respect to the NFC of $A$.

Step 3: Next, let $d=a$ and $c_{a}=\infty$. By hypothesis, in a neighborhood
$\Omega_{a}$ containing $a$, and a real number $\alpha>0$, we have that
$|{\widetilde{g}}(z)|\sim|z-a|^{-\alpha}$ for all
$z\in\Omega_{a}\cap\mathcal{D}({\widetilde{g}})$. Set

$\displaystyle
f_{a,\infty}^{\lambda}(z)\mathrel{\mathop{\ordinarycolon}}=\frac{a-z}{|\lambda|^{-1/\alpha}+a-z}\frac{b^{2}-a^{2}}{2a}\frac{a+z}{b^{2}-z^{2}},\quad
z\in\mathcal{D}({\widetilde{g}}).$

Similar reasoning as in the above cases together with the observation that

$\displaystyle(aI-A)R(|\lambda|^{-1/\alpha},A-aI)=I-|\lambda|^{-1/\alpha}R(|\lambda|^{-1/\alpha},A-aI),$

leads to the fact that the family
$(f_{a,\infty}^{\lambda})_{\lambda\notin\overline{S_{\gamma}}}$ satisfies
Properties (a), (b) and (c) in Definition 3.3. By Lemma 3.4, it easily follows
that $|f_{a,\infty}^{\lambda}(z)|\lesssim|\lambda|^{1/\alpha}|z-a|$.
Therefore, the triangle inequality and an application of Lemma 3.5 yield

$\displaystyle|R_{\widetilde{g}}^{\lambda}(z)-f_{a,\infty}^{\lambda}(z)|$
$\displaystyle\leq|\lambda^{1/\alpha}(z-a)|^{\alpha}+|\lambda^{1/\alpha}(z-a)|,$

for all $z\in\Omega_{a}\cap\mathcal{D}({\widetilde{g}})$ and
$\lambda\notin\overline{S_{\gamma+\varepsilon}}$. This implies that

$\displaystyle|R_{\widetilde{g}}^{\lambda}(z)-f_{a,\infty}^{\lambda}(z)|$
$\displaystyle\leq\left|R_{\widetilde{g}}^{\lambda}(z)-\frac{a-z}{|\lambda|^{-1/\alpha}+a-z}\right|+\left|\frac{a-z}{|\lambda|^{-1/\alpha}+a-z}\right|\left|\frac{b^{2}-a^{2}}{2a}\frac{a+z}{b^{2}-z^{2}}-1\right|$
$\displaystyle\lesssim\left|R_{\widetilde{g}}^{\lambda}(z)-\frac{a-z}{|\lambda|^{-1/\alpha}+a-z}\right|+|z-a|,$

where we have used again the Taylor expansion of order $1$ and the fact that
$|(a-z)/(|\lambda|^{-1/\alpha}+a-z)|$ is uniformly bounded. In addition, it
follows that

$\displaystyle\left|R_{\widetilde{g}}^{\lambda}(z)-\frac{a-z}{|\lambda|^{-1/\alpha}+a-z}\right|=$
$\displaystyle\left|\frac{\lambda|\lambda|^{-1/\alpha}+{\widetilde{g}}(z)(a-z)}{(\lambda-{\widetilde{g}}(z))(|\lambda|^{-1/\alpha}+a-z)}\right|$
$\displaystyle\leq$
$\displaystyle\left|\frac{\lambda|\lambda|^{-1/\alpha}}{(\lambda-{\widetilde{g}}(z))(|\lambda|^{-1/\alpha}+a-z)}\right|+\left|\frac{{\widetilde{g}}(z)(a-z)}{(\lambda-{\widetilde{g}}(z))(|\lambda|^{-1/\alpha}+a-z)}\right|$
$\displaystyle\lesssim$
$\displaystyle|\lambda^{1/\alpha}(z-a)|^{-1}+|\lambda^{1/\alpha}(z-a)|^{-\alpha},$

for all $z\in\Omega_{a}\cap\mathcal{D}({\widetilde{g}})$ and
$\lambda\notin\overline{S_{\gamma+\varepsilon}}$, where we have used various
applications of Lemmas 3.4 and 3.5 in the last step. Finally, reasoning as in
the case before with Lemma 3.6, one obtains that
$(f_{a,\infty}^{\lambda})_{\lambda\notin\overline{S_{\gamma}}}$ satisfies also
Property (d) in Definition 3.3.

Step 4: Similar reasoning as in Step 1 shows that, if either $-a,\infty\in
M_{A}$ with
$c_{-a},c_{\infty}\in\overline{S_{\gamma}}\backslash\\{0,\infty\\}$, the
families of functions given by

$\displaystyle f_{-a,c_{-a}}^{\lambda}(z)$
$\displaystyle\mathrel{\mathop{\ordinarycolon}}=\frac{\lambda}{\lambda-
c_{-a}}\frac{b^{2}-a^{2}}{2a}\frac{a-z}{b^{2}-z^{2}},\quad
z\in\mathcal{D}({\widetilde{g}})$ $\displaystyle
f_{\infty,c_{\infty}}^{\lambda}(z)$
$\displaystyle\mathrel{\mathop{\ordinarycolon}}=\frac{\lambda}{\lambda-
c_{\infty}}\frac{a^{2}-z^{2}}{b^{2}-z^{2}},\quad\quad\quad\qquad
z\in\mathcal{D}({\widetilde{g}}),$

make $(R_{\widetilde{g}}^{\lambda})_{\lambda\notin\overline{S_{\gamma}}}$
$\varepsilon-$uniformly bounded at $-a$ and $\infty$, respectively, with
respect to the NFC of $A$.

Step 5: Let either $-a,\infty\in M_{A}$ with polynomial limits
$c_{a},c_{\infty}=0$ of exactly order $\alpha>0$, that is,
$|{\widetilde{g}}(z)|\sim|z+a|^{\alpha}$ near $z=-a$ and
$|{\widetilde{g}}(z)|\sim|z|^{-\alpha}$ near $z=\infty$, respectively.
Proceeding as in Step 2, one has that the families of functions given by

$\displaystyle f_{-a,0}^{\lambda}(z)$
$\displaystyle\mathrel{\mathop{\ordinarycolon}}=\frac{|\lambda|^{1/\alpha}}{|\lambda|^{1/\alpha}+a+z}\frac{b^{2}-a^{2}}{2a}\frac{a-z}{b^{2}-z^{2}},\quad
z\in\mathcal{D}({\widetilde{g}}),$ $\displaystyle f_{\infty,0}^{\lambda}(z)$
$\displaystyle\mathrel{\mathop{\ordinarycolon}}=\frac{b-z}{|\lambda|^{-1/\alpha}+b-z}\frac{a^{2}-z^{2}}{b^{2}-z^{2}},\quad\qquad\quad
z\in\mathcal{D}({\widetilde{g}}),$

satisfy the desired properties.

Step 6: Finally, assume that either $-a,\infty\in M_{A}$ with polynomial
limits $c_{a},c_{\infty}=\infty$ of exactly order $\alpha>0$, i.e.
$|{\widetilde{g}}(z)|\sim|z+a|^{-\alpha}$ near $z=-a$ and
$|{\widetilde{g}}(z)|\sim|z|^{\alpha}$ near $z=\infty$, respectively.
Analogous computations as in Step 3 yields to the fact that the families of
functions given by

$\displaystyle f_{-a,\infty}^{\lambda}(z)$
$\displaystyle\mathrel{\mathop{\ordinarycolon}}=\frac{a+z}{|\lambda|^{-1/\alpha}+a+z}\frac{b^{2}-a^{2}}{2a}\frac{a-z}{b^{2}-z^{2}},\quad
z\in\mathcal{D}({\widetilde{g}}),$ $\displaystyle
f_{\infty,\infty}^{\lambda}(z)$
$\displaystyle\mathrel{\mathop{\ordinarycolon}}=\frac{|\lambda|^{1/\alpha}}{|\lambda|^{1/\alpha}+b-z}\frac{a^{2}-z^{2}}{b^{2}-z^{2}},\quad\qquad\qquad
z\in\mathcal{D}({\widetilde{g}}),$

make $(R_{\widetilde{g}}^{\lambda})_{\lambda\notin\overline{S_{\gamma}}}$
$\varepsilon$-uniformly bounded at $-a$ and $\infty$ respectively with respect
to the NFC of $A$. The proof is complete. ∎

######  Remark 3.8.

If $M_{A}$ is a proper subset of $\\{-a,a,\infty\\}$ one can slightly simplify
the families of functions given in the proof of Proposition 3.7. More
precisely, for $d\in M_{A}$, one can eliminate the terms in $f_{d}^{\lambda}$
that make the functions uniformly bounded and integrable in the rest of the
points in $M_{A}$. For instance, if $M_{A}=\\{a,\infty\\}$ then the behavior
of the functions near $-a$ is irrelevant. Thus, if
$c_{a}\notin\\{0,\infty\\}$, then the family of functions given by

$\displaystyle\frac{\lambda}{\lambda-c_{a}}\frac{b-a}{b-z},\quad
z\in\mathcal{D}({\widetilde{g}}),$

also satisfies the desired properties.

######  Remark 3.9.

Let $(f_{d}^{\lambda})_{\lambda\notin\overline{S_{\gamma}}}$ be a family of
functions that makes
$(R_{\widetilde{g}}^{\lambda})_{\lambda\notin\overline{S_{\gamma}}}$
$\varepsilon-$uniformly bound at each $d\in M_{A}$ with respect to the NFC of
$A$. From the bounds appearing in the proof of Proposition 3.7, one obtains
that in fact $(R_{\widetilde{g}}^{\lambda}-\sum_{d\in
M_{A}}f_{d}^{\lambda})\in\mathcal{E}_{0}$ for all
$\lambda\notin\overline{S_{\gamma}}$.

We are now ready to state the main result of this section.

######  Theorem 3.10.

Let $(\omega,a)\in(0,\pi/2]\times[0,\infty)$ and $\beta\in[0,\pi)$. Let
$A\in\text{{BSect}}(\omega,a)$ in a Banach space $X$ and
$g\in\mathcal{M}_{A}$. Assume the following:

  1. (a)

For any $\gamma>\beta$, one can find $\varphi\in(0,\omega)$ such that
$g(BS_{\varphi,a})\subset\overline{S_{\gamma}}\cup\\{\infty\\}$.

  2. (b)

$g$ is quasi-regular at $M_{A}$.

  3. (c)

$g$ has exactly polynomial limits at $M_{A}\cap g^{-1}(\\{0,\infty\\})$.

Then, $g(A)$ is a sectorial operator of angle $\beta$.

###### Proof.

Our result will follow once we have proven that $g(A)$ is a sectorial operator
of angle $\gamma$ for all $\gamma>\beta$. Indeed, if that’s true, we will have
that

$\displaystyle\beta\geq\inf_{\gamma\in[0,\pi)}\\{\gamma\,\mathrel{\mathop{\ordinarycolon}}\,g(A)\text{
is sectorial of angle }\gamma\\},$

which implies that $g(A)$ is a sectorial operator of angle $\beta$, see the
beginning of [18, Section 2.1].

Indeed, let $\gamma>\beta$ and set
${\widetilde{g}}\mathrel{\mathop{\ordinarycolon}}=g|_{BS_{\varphi,a}}$, where
$\varphi\in(0,\omega)$ is chosen such that
$\mathcal{R}({\widetilde{g}})\subset\overline{S_{\gamma}}\cup\\{\infty\\}$.
Notice that ${\widetilde{g}}(A)=g(A)$. Now, the spectral inclusion
$\widetilde{\sigma}({\widetilde{g}}(A))\subset\overline{S_{\gamma}}\cup\\{\infty\\}$
holds by Proposition 2.13. It remains to prove the bound for the resolvent.
More precisely, we have to show that for all $\varepsilon>0$,

$\displaystyle\sup_{\lambda\notin\overline{S_{\gamma+\varepsilon}}}\|\lambda
R(\lambda,{\widetilde{g}}(A))\|_{\mathcal{L}(X)}<\infty.$

By Lemma 2.8, it follows that $\lambda
R(\lambda,{\widetilde{g}}(A))=R_{\widetilde{g}}^{\lambda}(A)$ for all
$\lambda\notin\overline{S_{\gamma}}$. Also, by Proposition 3.7, we have that
for each $d\in M_{A}$, there exist some families of functions
$(f_{d}^{\lambda})_{\lambda\notin\overline{S_{\gamma}}}$ which make
$(R_{\widetilde{g}}^{\lambda})_{\lambda\notin\overline{S_{\gamma}}}$
$\varepsilon$-uniformly bounded at $d$ with respect to the NFC of $A$. We have
that

$\displaystyle\|\lambda R(\lambda,{\widetilde{g}}(A))\|_{\mathcal{L}(X)}$
$\displaystyle\leq\left\|\left(R_{\widetilde{g}}^{\lambda}-\sum_{d\in
M_{A}}f_{d}^{\lambda}\right)(A)\right\|_{\mathcal{L}(X)}+\sum_{d\in
M_{A}}\left\|f_{d}^{\lambda}(A)\right\|_{\mathcal{L}(X)}.$ (3.3)

By Property (c) in Definition 3.3, one has that
$\sup_{\lambda\notin\overline{S_{\gamma+\varepsilon}}}\|f_{d}^{\lambda}(A)\|_{\mathcal{L}(X)}<\infty$
for each $d\in M_{A}$. It remains to uniformly bound the first term.

Let $\Gamma$ be an integration path for the primary functional calculus of
$A$, and $(\Omega_{d})_{d\in M_{A}}$ some appropriate open sets for which
$d\in\Omega_{d}$ and the uniform integral bounds of Definition 3.3 hold for
each $(f_{d}^{\lambda})_{\lambda\notin\overline{S_{\gamma}}}$. Since
$(R_{\widetilde{g}}^{\lambda}-\sum_{d\in
M_{A}}f_{d}^{\lambda})\in\mathcal{E}_{0}$ (see Remark 3.9), one has that

$\displaystyle\left\|\left(R_{\widetilde{g}}^{\lambda}-\sum_{d\in
M_{A}}f_{d}^{\lambda}\right)(A)\right\|_{\mathcal{L}(X)}\leq\int_{\Gamma}\left|R_{\widetilde{g}}^{\lambda}(z)-\sum_{d\in
M_{A}}f_{d}^{\lambda}(z)\right|\|R(z,A)\|_{\mathcal{L}(X)}\,|dz|.$ (3.4)

Next, we split the integral on $\Gamma$ to the sum of integrals on
$\Gamma\cap\Omega_{d}$ for each $d\in M_{A}$, and
$\Gamma\backslash\left(\cup_{d\in M_{A}}\Omega_{d}\right)$. Notice that by
Property (b) in Definition 3.3 and Lemma 3.5, $|f_{d}^{\lambda}(z)|$ and
$|R_{\widetilde{g}}^{\lambda}(z)|$ are uniformly bounded. Thus,

$\displaystyle\sup_{\lambda\notin\overline{S_{\gamma+\varepsilon}}}\int_{\Gamma\backslash\left(\cup_{d\in
M_{A}}\Omega_{d}\right)}\left|R_{\widetilde{g}}^{\lambda}(z)-\sum_{d^{\prime}\in
M_{A}}f_{d^{\prime}}^{\lambda}(z)\right|\|R(z,A)\|_{\mathcal{L}(X)}\,|dz|\lesssim\int_{\Gamma\backslash\left(\cup_{d\in
M}\Omega_{d}\right)}\|R(z,A)\|_{\mathcal{L}(X)}\,|dz|<\infty.$

Finally, for each $d\in M_{A}$, one has that

$\displaystyle\sup_{\lambda\notin\overline{S_{\gamma+\varepsilon}}}\int_{\Gamma\cap\Omega_{d}}\left|R_{\widetilde{g}}^{\lambda}(z)-\sum_{d^{\prime}\in
M_{A}}f_{d^{\prime}}^{\lambda}(z)\right|\|R(z,A)\|_{\mathcal{L}(X)}\,|dz|$
$\displaystyle\leq$
$\displaystyle\sup_{\lambda\notin\overline{S_{\gamma+\varepsilon}}}\sum_{d^{\prime}\in
M_{A}\backslash\\{d\\}}\int_{\Gamma\cap\Omega_{d}}|f_{d^{\prime}}^{\lambda}(z)|\|R(z,A)\|_{\mathcal{L}(X)}\,|dz|+\sup_{\lambda\notin\overline{S_{\gamma+\varepsilon}}}\int_{\Gamma\cap\Omega_{d}}\left|R_{\widetilde{g}}^{\lambda}(z)-f_{d}^{\lambda}(z)\right|\|R(z,A)\|_{\mathcal{L}(X)}\,|dz|.$

But, these two supremums of the integrals are finite by Properties (c) and (d)
in Definition 3.3, respectively. Combining these estimates with (3.3)-(3.4) we
get the resolvent bound, and as a consequence ${\widetilde{g}}(A)=g(A)$ is a
sectorial operator of angle $\gamma$ for all $\gamma>\beta$. The proof is
finished. ∎

The following corollaries are immediate consequences of Theorem 3.10. The
first one has been already stated in [7, Theorem 4.6] for the particular case
where $-A$ generates a bounded group.

######  Corollary 3.11.

Let $a\geq 0$, $A\in\text{{BSect}}(\pi/2,a)$, and let $\alpha>0$ with $\alpha$
not an odd number, so $\alpha\in(2n-1,2n+1)$ for a unique $n\in\mathbb{{N}}$.
Then, for any $\varepsilon>0$, there exists $\rho\geq 0$ such that $\rho
I+(-1)^{n}(A+aI)^{\alpha}$ is a sectorial operator of angle
$\pi\left|\frac{\alpha}{2}-n\right|+\varepsilon$. Moreover, if $a=0$ then we
can take $\rho=0$.

######  Corollary 3.12.

Let $0<\omega\leq\frac{\pi}{2}$ and $a\geq 0$. Let
$A\in\text{{BSect}}(\omega,a)$ in a Banach space $X$ and
$g\in\mathcal{M}_{A}$. Assume that there are $\beta\in[\pi/2,\pi)$ and $b\geq
0$ such that the following hold:

  1. (a)

For any $\gamma>\beta$, one can find $\varphi\in(0,\omega)$ for which
$g(BS_{\varphi,a})\subset\overline{BS}_{\gamma,b}$.

  2. (b)

$g$ is quasi-regular at $M_{A}$.

  3. (c)

$g$ has exactly polynomial limits at $M_{A}\cap g^{-1}(\\{-b,b,\infty\\})$.

Then, $g(A)$ is a bisectorial operator of angle $\pi-\beta$ and half-width
$b$.

###  3.2. The natural functional calculus for sectorial operators

Most proofs which we have presented in this text are generic, and as a
consequence, the results shown here will hold for functional calculus
analogous to that in Sections 2.2 and 2.3. In particular, it is
straightforward to adapt the preceding results to the case of NFC of sectorial
operators that can be found in [17], where the reader can find the definitions
of the appropriate versions of the function spaces
$\mathcal{E}(S_{\varphi}),\mathcal{E}_{0}(S_{\varphi}),\mathcal{M}(S_{\varphi})_{A}$,…
For instance, one obtains the following version of Theorem 3.10 adapted to
this NFC.

######  Theorem 3.13.

Let $0\leq\omega<\pi$, $\beta\in[0,\pi)$, $A$ a sectorial operator of angle
$\omega$ in a Banach space $X$, and $g\in\mathcal{M}_{A}$ in the sense of the
NFC for sectorial operators. Assume that the following hold:

  1. (a)

For any $\gamma>\beta$, one can find $\varphi\in(\omega,\pi)$ such that
$g(S_{\varphi})\subset\overline{S_{\gamma}}\cup\\{\infty\\}$.

  2. (b)

$g$ is quasi-regular at $\\{0,\infty\\}\cap\widetilde{\sigma}(A)$.

  3. (c)

$g$ has exactly polynomial limits at
$\\{0,\infty\\}\cap\widetilde{\sigma}(A)\cap g^{-1}(\\{0,\infty\\})$.

Then, $g(A)$ is sectorial of angle $\beta$.

##  4\. Some generation results of holomorphic semigroups and their properties

As a consequence of the bijection between generators of bounded holomorphic
semigroups and sectorial operators, the results obtained in Section 3
encourage us to study the properties related to the holomorphic semigroup
generated by $-g(A)$ whenever $A$ is a bisectorial operator, and $g$ a
meromorphic function satisfying the hypothesis of Theorem 3.10 with an angle
strictly smaller than $\frac{\pi}{2}$. We state explicitly this connection in
the following result.

######  Corollary 4.1.

Let $A,\beta,g$ be as in Theorem 3.10. In addition, assume that
$\beta\in[0,\pi/2)$. Then, $-g(A)$ generates a bounded holomorphic semigroup
$T_{g}$ of angle $\frac{\pi}{2}-\beta$.

###### Proof.

This is an immediate consequence of Theorem 3.10 and the fact that an operator
$B$ is sectorial of angle $\beta<\frac{\pi}{2}$ if and only if $-B$ is the
generator of a bounded holomorphic semigroup of angle $\frac{\pi}{2}-\beta$,
see for example [12, Theorem 4.6] or [18, Proposition 3.4.4]. ∎

The lemma below will be useful in the proof of our main results within the
case where either $aI-A$ or $aI+A$ is not injective. Its proof is analogous to
the related result for sectorial operators (see e.g. [16, Lemma 4.3]).

######  Lemma 4.2.

Let $A\in\text{{BSect}}(\omega,a),\,g\in\mathcal{M}_{A}$ with quasi-regular
limits at $M_{A}$, and assume that $d\in\sigma_{p}(A)$. Then, $g$ has a finite
limit $c_{d}$ at $d$.

###### Proof.

First, we have that the limit $c_{d}$ of $g$ at $d$ exists in
$\mathbb{{C}}_{\infty}$. Indeed, this is clear if $d\notin M_{A}$. And if
$d\in M_{A}$, by hypothesis $g$ is quasi-regular at $d\in M_{A}$, and
therefore has a limit $c_{d}\in\mathbb{{C}}_{\infty}$.

Second, let $e\in\mathcal{E}$ be a regularizer for $g$, so $eg\in\mathcal{E}$
and $e(A)$ is injective. One can easily check that $e(A)x=e(d)x$ for all
$x\in\mathcal{N}(dI-A)$. This implies that $e(d)\neq 0$. Since
$eg\in\mathcal{E}$, we have that $g$ has a finite limit
$c\mathrel{\mathop{\ordinarycolon}}=g(a)$ at $a$, and the assertion follows. ∎

Recall that the space of strong continuity $\mathbb{D}_{T}$ of a (holomorphic)
semigroup $T$ generated by $A$ is precisely $\overline{\mathcal{D}(A)}$. The
following result characterizes the space $\mathbb{D}_{T}$ in our setting. Let
us point out that the result holds even if the angle of sectoriality $\beta$
of $g(A)$ is greater or equal than $\frac{\pi}{2}$.

######  Proposition 4.3.

Let $A,g$ be as in Theorem 3.10. If $g^{-1}(\infty)\cap M_{A}=\emptyset$, then
$\overline{\mathcal{D}(g(A))}=X$. Otherwise,

$\displaystyle\overline{\mathcal{D}(g(A))}=\bigcap_{d\in g^{-1}(\infty)\cap
M_{A}}\overline{\mathcal{R}(dI-A)},$

where $\mathcal{R}(\infty
I-A)\mathrel{\mathop{\ordinarycolon}}=\mathcal{D}(A)$.

###### Proof.

First of all, notice that if $g^{-1}(\infty)\cap M_{A}=\emptyset$, then
$g^{-1}(\infty)=\emptyset$ (see Remark 3.2), so by the inclusion of the
spectra (Proposition 2.13), $g(A)\in\mathcal{L}(X)$ and $\mathcal{D}(g(A))=X$.

Let $d\in M_{A}$. Note that if $d\notin g^{-1}(\infty)$, then $g$ is regular
at $d$ with $g(d)\in\mathbb{{C}}$. For any $b>a$, consider

$f_{d}(z)\mathrel{\mathop{\ordinarycolon}}=\begin{cases}\displaystyle
g(a)\frac{b^{2}-a^{2}}{2a}\frac{a+z}{b^{2}-z^{2}},\quad&\mbox{ if }d=a,\\\
\displaystyle g(-a)\frac{b^{2}-a^{2}}{2a}\frac{a-z}{b^{2}-z^{2}},\quad&\mbox{
if }d=-a,\\\ \displaystyle
g(\infty)\frac{a^{2}-z^{2}}{b^{2}-z^{2}},\quad&\mbox{ if
}d=\infty.\end{cases}$

Then, $g-f_{d}$ is regular at $d$ with limit $0$, and the behavior of
$g-f_{d}$ at $M_{A}\backslash\\{d\\}$ remains the same as the behavior of $g$
at those points. Moreover, since $f_{d}(A)\in\mathcal{L}(X)$, it follows that

$\displaystyle\mathcal{D}(g(A))=\mathcal{D}(g_{\bullet}(A))\;\mbox{ where
}\;g_{\bullet}(z)\mathrel{\mathop{\ordinarycolon}}=g(z)-\sum_{d\notin
g^{-1}(\infty)\cap M_{A}}f_{d}(z).$

Thus, we can assume that $g$ has regular limits equal to $0$ at
$M_{A}\backslash g^{-1}(\infty)$.

Now, we proceed by showing both inclusions $\subset,\supset$ of the statement,
starting with the latter one. For all $t>0$ small enough (for which
$b\notin\sigma(tA)$), set

$\displaystyle
h_{t}(z)\mathrel{\mathop{\ordinarycolon}}=\frac{(a-z)^{n_{a}}(a+z)^{n_{-a}}b^{n_{\infty}}}{(t+a-z)^{n_{a}}(t+a+z)^{n_{-a}}(b+tz)^{n_{\infty}}},\quad
z\in\mathcal{D}(g),$ (4.1)

with $n_{d}=0$ if $d\notin g^{-1}(\infty)\cap M_{A}$ and the rest
$n_{d}\in\mathbb{{N}}$ large enough so that $h_{t}g\in\mathcal{E}$. Then
$h_{t}g(A)\in\mathcal{L}(X)$ with $\mathcal{D}((h_{t}g)(A))=X$, and note that
$h_{t}^{-1}\in\mathcal{M}_{A}$ since $\sigma_{p}(A)\cap
g^{-1}(\infty)=\emptyset$ (see Lemma 4.2). Therefore,
$g(A)\supset(h_{t}g)(A)h_{t}^{-1}(A)$, which implies that
$\mathcal{D}(g(A))\supset\mathcal{D}(h_{t}^{-1}(A))=\mathcal{R}(h_{t}(A))$ for
all $t>0$ small enough. In addition, since both $aI+A$ and $aI-A$ are
sectorial operators, we have that

$\displaystyle\lim_{t\to 0}h_{t}(A)x=x,\quad\text{for all }x\in\bigcap_{d\in
g^{-1}(\infty)\cap M_{A}}\overline{\mathcal{R}(dI-A)},$

see [18, Proposition 2.1.1 (c)], which yields that

$\bigcap_{d\in g^{-1}(\infty)\cap
M_{A}}\overline{\mathcal{R}(dI-A)}\subset\overline{\mathcal{R}(h_{t}(A))}\subset\overline{\mathcal{D}(g(A))},$

and the inclusion $\supset$ of the assertion follows.

Let us prove the reverse inclusion $\subset$. Assume that $\infty\in
g^{-1}(\infty)\cap M_{A}$. Then, $|g(z)|\sim|z|^{\alpha}$ as $z\to\infty$ for
some $\alpha>0$. It follows that $(1+g(z))^{-1}$ regularizes
$(z+a)^{\alpha^{\prime}}$ for any $\alpha^{\prime}\in(0,\alpha)$, which
implies that $\mathcal{D}(g(A))\subset\mathcal{D}((A+aI)^{\alpha^{\prime}})$.
Reasoning similarly with $-a,a$, one obtains that for any $\alpha^{\prime}>0$
small enough,

$\displaystyle\mathcal{D}(g(A))\subset\bigcap_{d\in g^{-1}(\infty)\cap
M_{A}}\mathcal{R}((dI-A)^{\alpha^{\prime}}),$

where $\mathcal{R}((\infty
I-A)^{\alpha^{\prime}})\mathrel{\mathop{\ordinarycolon}}=\mathcal{D}((aI+A)^{\alpha^{\prime}})$.
Then, our proof will finish if we show that
$\overline{\mathcal{R}((dI-A)^{\alpha^{\prime}})}\subset\overline{\mathcal{R}(dI-A)}$.
Assume that $d=\infty$. It follows from Theorem 3.10 that
$(aI+A)^{\alpha^{\prime}}$ is a sectorial operator for a small enough
$\alpha^{\prime}>0$. Moreover, $aI+A$ is also a sectorial operator, and
$(aI+A)^{\alpha^{\prime}}=f_{\alpha^{\prime}}(aI+A)$ (where
$f(z)=z^{\alpha^{\prime}}$) by using the NFC of sectorial operators (see [18,
Section 2.3]). Then, by the composition rule for sectorial operators (see e.g.
[18, Theorem 2.4.2]), one has that
$f_{1/\alpha^{\prime}}((aI+A)^{\alpha^{\prime}})=aI+A$. Reasoning analogously
as in the proof of the inclusion $\supset$, one gets that

$\displaystyle\overline{\mathcal{D}(aI+A)}=\overline{\mathcal{D}((f_{1/\alpha^{\prime}})((aI+A)^{\alpha^{\prime}}))}\supset\overline{\mathcal{D}((aI+A)^{\alpha^{\prime}})},$

as we wanted to prove. The cases $d\in\\{-a,a\\}$ are solved in an analogous
way, by using the operators
$(aI+A)^{-\alpha^{\prime}},(aI-A)^{-\alpha^{\prime}}$, respectively. The proof
is finished. ∎

######  Corollary 4.4.

If $X$ is reflexive, then $\overline{\mathcal{D}(g(A))}=X$.

###### Proof.

By [18, Proposition 2.1.1 (h)], one has that

$X=\overline{\mathcal{D}(A)}=\mathcal{N}(aI-A)\oplus\overline{\mathcal{R}(aI-A)}=\mathcal{N}(aI+A)\oplus\overline{\mathcal{R}(aI+A)}$

if $X$ is reflexive. Since $\sigma_{p}(A)\cap g^{-1}(\infty)=\emptyset$ (see
Lemma 4.2), the statement follows by an application of Proposition 4.3. ∎

Recall that $\exp_{-w}(z)\mathrel{\mathop{\ordinarycolon}}=\exp(-wz)$ for all
$z,\,w\in\mathbb{{C}}$. Since $T_{g}(w)=\exp_{-w}(g(A))$, it is natural to
conjecture that $T_{g}(w)=(\exp_{-w}\circ g)(A)$. The theorem below answers
this question positively. Its proof is inspired by the composition rule for
sectorial operators given in [16], but carefully adapted to cover all our
cases. Indeed, one could easily generalize the result below to a composition
rule from bisectorial to sectorial operators, addressing a larger class of
functions. However, this would require to introduce several new definitions
and additional cumbersome notations. Thus, in order for the paper to be
accessible for a broad class of mathematicians, we will limit to our specific
cases.

######  Theorem 4.5.

Let $\beta,A,g$ be as in Corollary 4.1, so that $-g(A)$ generates a
holomorphic semigroup $T_{g}$ of angle $\frac{\pi}{2}-\beta$. Then, for any
$w\in S_{\pi/2-\beta}$, we have that $(\exp_{-w}\circ g)\in\mathcal{M}_{A}$
and

$\displaystyle T_{g}(w)=(\exp_{-w}\circ g)(A).$ (4.2)

###### Proof.

First of all, the claim is trivial if $g=0$, so we will assume that $g\neq 0$.
Fix $w\in S_{\pi/2-\beta}$. Then, it is straightforward to check that
$(\exp_{-w}\circ g)$ is regular at $M_{A}$, so Lemma 2.12 yields that
$(\exp_{-w}\circ g)\in\mathcal{M}(A)$.

Now set $f_{w}(z)\mathrel{\mathop{\ordinarycolon}}=\exp_{-w}(z)-(1+z)^{-1}$.
Then, $f_{w}\in\mathcal{E}_{0}[S_{\beta}]$. As
$-1\notin\sigma(g(A))\subset\overline{S_{\beta}}$, an application of Lemma 2.8
(f) yields that $(f_{w}\circ g)\in\mathcal{M}_{A}$ and
$(I+g)^{-1}(A)=(I+g(A))^{-1}$. Therefore, our statement will follow if we
prove that $(f_{w}\circ g)(A)=f_{w}(g(A))$.

Recall that, for $d\in\mathbb{{C}}$, we denote by $c_{d}$ the limit of $g(z)$
as $z\to d$ whenever it exists. In particular, it exists if
$d\in\sigma_{p}(A)$, see Lemma 4.2. Let $b>a$, and for any
$\lambda\notin\overline{S_{\beta}}$, set

$\displaystyle
G_{\lambda}(z)\mathrel{\mathop{\ordinarycolon}}=\frac{1}{\lambda-g(z)}-\sum_{d\in\sigma_{p}(A)\cap\\{-a,a\\}}\frac{1}{\lambda-
c_{d}}\frac{z+d}{b-z}\frac{b-d}{2d},\quad z\in\mathcal{D}(g).$

Since $\lambda\in\rho(A)$, one has that $G_{\lambda}\in\mathcal{M}_{A}$.
Moreover, $G_{\lambda}\in H^{\infty}(\mathcal{D}(g))$ with $G_{\lambda}(d)=0$
for all $d\in\sigma_{p}(A)\cap\\{-a,a\\}$. Furthermore, it is readily seen
that one can find a regularizer $e\in\mathcal{E}$ independent of $\lambda$,
for which $eG_{\lambda}\in\mathcal{E}_{0}$. Indeed, to check the regularity of
$eG_{\lambda}$ at the points $d^{\prime}\in M_{A}$, one can add to $e$ powers
of the function $(z-d^{\prime})/(z-b)^{2}$ if $d^{\prime}\notin\sigma_{p}(A)$.
Otherwise, the regularity is obtained by the bounds in Lemma 3.5 (recall that
in this case, $c_{d^{\prime}}\neq\infty$ by Lemma 4.2).

Then, let $\Gamma^{\prime}$ be an appropriate path for the NFC of the
sectorial operator $g(A)$ and the function $f_{w}$. It follows that

$\displaystyle f_{w}(g(A))$
$\displaystyle=e(A)^{-1}e(A)f_{w}(g(A))=e(A)^{-1}\frac{1}{2\pi
i}\int_{\Gamma^{\prime}}f_{w}(\lambda)e(A)g(A)\,d\lambda$
$\displaystyle=e(A)^{-1}\frac{1}{2\pi
i}\int_{\Gamma^{\prime}}f_{w}(\lambda)(e(z)G_{\lambda}(z))(A)\,d\lambda$
$\displaystyle\quad+\sum_{d\in\sigma_{p}(A)\cap\\{-a,a\\}}\frac{b-d}{2d}(dI+A)R(b,A)\;\frac{1}{2\pi
i}\int_{\Gamma^{\prime}}\frac{f_{w}(\lambda)}{\lambda-c_{d}}\,d\lambda.$

By Cauchy’s integral theorem, one has that the last term is precisely

$\displaystyle\sum_{d\in\sigma_{p}(A)\cap\\{-a,a\\}}\frac{b-d}{2d}f(c_{d})(dI+A)R(b,A).$

Now, let us compute the first term. Let $\Gamma$ be an appropriate path of the
NFC of the bisectorial operator $A$. Since $eG_{\lambda}\in\mathcal{E}_{0}$,
one has that

$\displaystyle e(A)^{-1}\frac{1}{2\pi
i}\int_{\Gamma^{\prime}}f_{w}(\lambda)(e(z)G_{\lambda}(z))(A)\,d\lambda$
$\displaystyle=e(A)^{-1}\frac{1}{(2\pi
i)^{2}}\int_{\Gamma^{\prime}}f_{w}(\lambda)\int_{\Gamma}e(z)G_{\lambda}(z)R(z,A)\,dzd\lambda$
$\displaystyle=e(A)^{-1}\frac{1}{(2\pi
i)^{2}}\int_{\Gamma}e(z)R(z,A)\int_{\Gamma^{\prime}}f_{w}(\lambda)G_{\lambda}(z)\,d\lambda
dz.$ (4.3)

Let us go on with the proof before checking the hypothesis for Fubini’s
theorem that we have applied in the last equality in (4). By Cauchy’s theorem,
it easily follows that

$\displaystyle\frac{1}{2\pi
i}\int_{\Gamma^{\prime}}f_{w}(\lambda)G_{\lambda}(z)\,d\lambda=f_{w}(g(z))-\sum_{d\in\sigma_{p}(A)\cap\\{-a,a\\}}f(c_{d})\frac{z+d}{b-z}\frac{b-d}{2d}.$

From this, we can conclude that in fact

$\displaystyle e(A)^{-1}\frac{1}{(2\pi
i)^{2}}\int_{\Gamma}e(z)R(z,A)\int_{\Gamma}f_{w}(\lambda)G_{\lambda}(z)\,d\lambda
dz$ $\displaystyle\quad=(f_{w}\circ
g)(A)-\sum_{d\in\sigma_{p}(A)\cap\\{-a,a\\}}\frac{b-d}{2d}f(c_{d})(dI+A)R(b,A),$

and our assertion follows.

Let us check now that indeed Fubini’s theorem can be applied. For that, we
have to check the integrability of the function

$\displaystyle
F(\lambda,z)\mathrel{\mathop{\ordinarycolon}}=\frac{f_{w}(\lambda)}{\lambda}\lambda
G_{\lambda}(z)\frac{e(z)}{\min\\{|z-a|,|z+a|\\}},$

on $\Gamma^{\prime}\times\Gamma$. First, $f_{w}(\lambda)/\lambda$ is clearly
integrable on $\Gamma^{\prime}$ and, by Lemma 3.5, $\lambda G_{\lambda}(z)$ is
uniformly bounded on $\Gamma^{\prime}\times\Gamma$. Now, one can assume that
$\frac{e(z)}{\min\\{|z-a|,|z+a|\\}}$ is integrable on $\Gamma$ if and only if
$\\{-a,a\\}\cap\sigma_{p}(A)=\emptyset$. Otherwise, one has to check a uniform
bound for the integral of $F(\lambda,z)$ on the intersection of $\Gamma$ with
a neighborhood of $d\in\sigma_{p}(A)\cap\\{-a,a\\}$.

So let $d\in\sigma_{p}(A)\cap\\{-a,a\\}$. Recall that in this case,
$c_{d}\in\overline{S_{\beta}}$ with $c_{d}\neq\infty$. If $c_{d}\neq 0$, then
$\lambda G_{\lambda}$ is of the same type as the function appearing in Step 1
in the proof of Proposition 3.7, and proceeding as there, one can easily check
the integrability condition.

Thus, we can assume that $c_{d}=0$. So, one has that

$\displaystyle|\lambda
G_{\lambda}(z)|\lesssim\frac{|g(z)|}{|\lambda-g(z)|}+|z-d|,\qquad\mbox{ as
}\;z\to d,$

where the $|z-d|$ term is the result of applying a Taylor expansion of order
$1$ in a similar way as in Step 2 in the proof of Proposition 3.7. It is
readily seen that the $|z-d|$ term does not entangle the bound of
$F(\lambda,z)$. Moreover, for any $\delta\in(0,1)$, one has that

$\displaystyle\left|\frac{f_{w}(\lambda)}{\lambda}\frac{g(z)}{\lambda-g(z)}\frac{e(z)}{z-d}\right|=\left|\frac{f_{w}(\lambda)}{\lambda^{1+\delta}}\right|\left|\frac{\lambda^{\delta}g(z)^{1-\delta}}{\lambda-g(z)}\right|\left|\frac{(eg^{\delta})(z)}{z-d}\right|.$

It is easy to see that $f_{w}(\lambda)/\lambda^{1+\delta}$ is still integrable
on $\Gamma^{\prime}$, and that the middle term is uniformly bounded. Moreover,
since $c_{d}=0$, we have by hypothesis that $|g(z)|\sim|z-d|^{\alpha}$ as
$z\to d$ for some $\alpha>0$. Thus
$g^{\delta}(z)\lesssim|z-d|^{\alpha\delta}$, so the last term is integrable in
$\Gamma$. The proof is finished. ∎

##  5\. Generalized Black-Scholes equations on interpolation spaces

In this section, we apply the theory developed in the previous sections to
introduce and study generalized Black–Scholes equations on
$(L^{1}-L^{\infty})$-interpolation spaces. Throughout the following, without
any mention, $E$ will denote a $(L^{1},L^{\infty})$-interpolation space on
$(0,\infty)$. We recall that

$(G_{E}(t)f)(x)\mathrel{\mathop{\ordinarycolon}}=f(e^{-t}x),\qquad
x>0,\,t\in\mathbb{{R}},\,f\in E,$

defines a group of bounded operators $G_{E}=(G_{E}(t))_{t\in\mathbb{{R}}}$ on
$E$ with $\|G_{E}(t)\|_{\mathcal{L}(E)}\leq\max\\{1,e^{t}\\}$ for
$t\in\mathbb{{R}}$, and which is strongly continuous if and only $E$ has order
continuous norm. Then, the lower and upper Boyd indices
$\underline{\eta}_{E},\overline{\eta}_{E}$ are defined by

$\displaystyle\underline{\eta}_{E}\mathrel{\mathop{\ordinarycolon}}=-\lim_{t\to\infty}\frac{\log\|G_{E}(-t)\|_{\mathcal{L}(E)}}{t},\quad\overline{\eta}_{E}\mathrel{\mathop{\ordinarycolon}}=\lim_{t\to\infty}\frac{\log\|G_{E}(t)\|_{\mathcal{L}(E)}}{t},$

and they satisfy $0\leq\underline{\eta}_{E}\leq\overline{\eta}_{E}\leq 1$.

The generator $J_{E}$ of the group $G_{E}$ is given by

$\begin{cases}\mathcal{D}(J_{E})=\Big{\\{}f\in
E\,\mathrel{\mathop{\ordinarycolon}}f\in AC_{\rm loc}(0,\infty)\text{ and
}-xf^{\prime}(x)\in E\Big{\\}},\\\
(J_{E}f)(x)\mathrel{\mathop{\ordinarycolon}}=-xf^{\prime}(x),\quad
x>0,\,f\in\mathcal{D}(J_{E}),\end{cases}$ (5.1)

and its spectrum is given by

$\sigma(J_{E})=\\{\lambda\in\mathbb{{C}}\,\mathrel{\mathop{\ordinarycolon}}\,\underline{\eta}_{E}\leq\Re\lambda\leq\overline{\eta}_{E}\\},$

see [2] for more details about the operator $J_{E}$ and the classical
Black–Scholes equation on exact $(L^{1}-L^{\infty})$-interpolation spaces.
However, as the authors indicate in [2], every
$(L^{1}-L^{\infty})$-interpolation space can be equivalently renormed so that
it becomes exact (e.g. [8, Proposition III.1.13]). Thus, we may apply the
results in [2] to arbitrary $(L^{1}-L^{\infty})$-interpolation spaces, without
requiring them to be exact.

As a consequence of the above properties, for any
$\underline{\varepsilon},\overline{\varepsilon}>0$, both
$\left(\underline{\eta}_{E}+\underline{\varepsilon}\right)I+J_{E}$ and
$\left(\overline{\eta}_{E}+\overline{\varepsilon}\right)I-J_{E}$ are sectorial
operators of angle $\frac{\pi}{2}$ (see e.g. [18, Section 2.1.1]). Therefore,
$J_{E}-\frac{\overline{\eta}_{E}+\underline{\eta}_{E}+\overline{\varepsilon}-\underline{\varepsilon}}{2}I$
is a bisectorial operator of angle $\pi/2$ and half-width
$\frac{\overline{\eta}_{E}-\underline{\eta}_{E}+\overline{\varepsilon}+\underline{\varepsilon}}{2}$.
However, to avoid cumbersome notations we will write $f(J_{E})$ to refer to
$f_{k}(J_{E}-k)$ for
$k=\frac{\overline{\eta}_{E}+\underline{\eta}_{E}+\overline{\varepsilon}-\underline{\varepsilon}}{2}$
and $f_{k}(z)=f(z+k)$. Notice that one may take
$\underline{\varepsilon}=\overline{\varepsilon}=0$ if $\underline{\eta}_{E}=0$
and $\overline{\eta}_{E}=1$, respectively, or if $E=L^{p}$ with $1\leq
p\leq\infty$.

In [2], the authors make use of the operator $J_{E}$ to study the classical
Black-Scholes partial differential equation in
$(L^{1},L^{\infty})$-interpolation spaces. Recall that the classical Black-
Scholes equation is the degenerate parabolic equation given by

$\displaystyle u_{t}=x^{2}u_{xx}+xu_{x},\quad x,t>0.$ (5.2)

In fact, we can rewrite (5.2) as $u_{t}=J_{E}^{2}u$, where $J_{E}$ is the
operator defined in (5.1).

Next, we introduce the fractional operators that generalize the Black–Scholes
equation (5.2). On the one hand, we will consider fractional powers of the
operator $J_{E}$. If $\alpha\in(0,n)$, $n\in\mathbb{N}$, it follows that
$\mathcal{D}(J_{E}^{n})\subset\mathcal{D}(J_{E}^{\alpha})$ (see [18,
Proposition 3.1.1 ]). If in addition $0<\alpha<1$, an application of Fubini’s
theorem to the Balakrishnan representation of $J_{E}^{\alpha}f$ together with
the resolvent identity yields that, whenever $\underline{\eta}_{E}>0$,

$\displaystyle(J_{E}^{\alpha}f)(x)=\frac{-1}{\Gamma(1-\alpha)}\int_{x}^{\infty}\left(\log\frac{s}{x}\right)^{-\alpha}f^{\prime}(s)\,ds,\quad
f\in\mathcal{D}(J_{E}),\,x>0.$

If $\underline{\eta}_{E}=0$, then one cannot apply Fubini’s theorem to obtain
the above expression. However, one can use the fact that $(J_{E}+\varepsilon
I)^{\alpha}f\to J_{E}^{\alpha}f$ in $E$ as $\varepsilon\downarrow 0$ (see [18,
Proposition 3.1.9]), together with

$\displaystyle((J_{E}+\varepsilon
I)^{\alpha}f)(x)=\frac{-1}{\Gamma(1-\alpha)}\int_{x}^{\infty}\left(\log\frac{s}{x}\right)^{-\alpha}\left(\frac{x}{s}\right)^{\varepsilon}f^{\prime}(s)\,ds,$
(5.3)

for any $f\in\mathcal{D}(J_{E})$ and $x,\varepsilon>0$.

Next, let $\alpha>0$ be a real number and recall that we denote by
$D^{-\alpha}$ the Riemann-Liouville fractional integral of order $\alpha$, and
by $W^{-\alpha}$ the Weyl fractional integral of order $\alpha$, see (2.4) and
(2.5), respectively. Similarly, $D^{\alpha}$ denotes the Riemann-Liouville
fractional derivative of order $\alpha$, and $W^{\alpha}$ the Weyl fractional
derivative of order $\alpha$, defined in (2.6) and (2.7), respectively. Also,
if $m^{s}$ is the multiplication operator by $x^{s}$ for any
$s\in\mathbb{{R}}$, we have that the generalized Cesàro operator
$\mathcal{C}_{\alpha}$ of order $\alpha$, and its adjoint
$\mathcal{C}_{\alpha}^{\ast}$, are given, respectively, by

$\displaystyle\mathcal{C}_{\alpha}=\Gamma(\alpha+1)m^{-\alpha}I^{-\alpha}\;\mbox{
and }\quad$
$\displaystyle\mathcal{D}^{\alpha}=(\mathcal{D}^{-\alpha})^{-1},\mathcal{C}_{\alpha}^{\ast}=\Gamma(\alpha+1)W^{-\alpha}m^{-\alpha},$

see (1.6) and (1.6). Note that these operators are injective due to the fact
the operators $D^{-\alpha},W^{-\alpha}$ and $m^{-\alpha}$ are injective.
Moreover, by equality (1.6) again, one has that $\mathcal{C}_{\alpha}$ defines
a bounded operator $\mathcal{C}_{\alpha,E}$ when restricted to our
$(L^{1}-L^{\infty})$-interpolation space $E$ with $\overline{\eta}_{E}<1$,
since

$\displaystyle\mathcal{C}_{\alpha,E}=\alpha\int_{0}^{\infty}e^{-s}(1-e^{-s})^{\alpha-1}G_{E}(s)\,ds=\alpha\mathbb{B}(I-J_{E},\alpha),$

where we have applied Proposition A.3, and $\mathbb{B}$ denotes the usual Beta
function.

Similarly, $\mathcal{C}_{\alpha}^{\ast}$ defines a bounded operator
$\mathcal{C}_{\alpha,E}^{\ast}$ on any $(L^{1}-L^{\infty})$-interpolation
space $E$ with $\underline{\eta}_{E}>0$, satisfying
$\mathcal{C}_{\alpha,E}^{\ast}=\alpha\mathbb{B}(J_{E},\alpha)$. As a
consequence, one obtains that

$\mathcal{D}_{E}^{\alpha}\mathrel{\mathop{\ordinarycolon}}=(\mathcal{C}_{\alpha,E})^{-1}=\left(\alpha\mathbb{B}(I-J_{E},\alpha)\right)^{-1}\mbox{
and
}\mathcal{W}_{E}^{\alpha}\mathrel{\mathop{\ordinarycolon}}=(\mathcal{C}_{\alpha,E}^{\ast})^{-1}=\left(\alpha\mathbb{B}(J_{E},\alpha)\right)^{-1},$

are closed operators on $E$ whenever $\overline{\eta}_{E}<1$ and
$\underline{\eta}_{E}>0$, respectively. The above identities appear in [21]
for a family of Sobolev spaces on $(0,\infty)$.

###  5.1. Generation results of holomorphic semigroups of fractional powers
operators

The identity $J_{E}=\mathcal{W}_{E}^{1}=I-\mathcal{D}_{E}^{1}$ holds whenever
the operators are well defined on $E$ (see e.g. [2]). In particular, we have
that

$\displaystyle(J_{E})^{2}=(I-\mathcal{D}_{E}^{1})^{2}=(\mathcal{W}_{E}^{1})^{2}=\mathcal{W}_{E}^{1}(I-\mathcal{D}_{E}^{1}).$
(5.4)

This motivates us to study different fractional versions of the Black–Scholes
equation (5.2). In this section, we make use of the theory we developed in the
preceding sections to obtain that the operators

$\displaystyle(J_{E})^{2\alpha},\;\;(I-\mathcal{D}_{E}^{\alpha})^{2},\;\;(\mathcal{W}_{E}^{\alpha})^{2},\;\;\mathcal{W}_{E}^{\alpha}(I-\mathcal{D}_{E}^{\alpha}),$
(5.5)

are indeed generators of exponentially bounded holomorphic semigroups on $E$
for suitable values of $\alpha$.

We start with the operator $(J_{E})^{2\alpha}$.

######  Proposition 5.1.

Let $E$ be a $(L^{1}-L^{\infty})$-interpolation space, $n\in\mathbb{{N}}$ and
$\alpha\in\left(n-\frac{1}{2},n+\frac{1}{2}\right)$. Then, the operator
$(-1)^{n+1}(J_{E})^{2\alpha}$ generates an exponentially bounded holomorphic
semigroup $T_{(-1)^{n+1}(J_{E})^{2\alpha}}$ of angle
$\pi\left(\frac{1}{2}-\left|\alpha-n\right|\right)$, which is given by

$\displaystyle\left(T_{(-1)^{n+1}(J_{E})^{2\alpha}}(w)f\right)(x)=\frac{1}{2\pi}\int_{0}^{\infty}\frac{f(s)}{s}\int_{-\infty}^{\infty}\left(\frac{s}{x}\right)^{iu}\exp((-1)^{n+1}wu^{2\alpha})\,duds,\quad
x>0,$

for any $w\in S_{\pi\left(\frac{1}{2}-|\alpha-n|\right)}$ and $f\in E$. In
addition,
$\overline{\mathcal{D}((J_{E})^{2\alpha})}=\overline{\mathcal{D}(J_{E})}$.

###### Proof.

That the operator $(-1)^{n+1}(J_{E})^{2\alpha}$ generates an exponentially
bounded holomorphic semigroup with the given angle follows from Corollary
3.11. The expression given for $T_{(-1)^{n+1}(J_{E})^{2\alpha}}$ is an
immediate consequence of Theorem 4.5 and Proposition A.3. The assertion about
$\overline{\mathcal{D}((J_{E})^{2\alpha})}$ follows from Proposition 4.3. ∎

Next, we have the following result for the operator
$(I-\mathcal{D}_{E}^{\alpha})^{2}$.

######  Proposition 5.2.

Let $E$ be a $(L^{1}-L^{\infty})$-interpolation space with
$\overline{\eta}_{E}<1$, $n\in\mathbb{{N}}$ and
$\alpha\in\left(n-\frac{1}{2},n+\frac{1}{2}\right)$. Then, the operator
$(-1)^{n+1}(I-\mathcal{D}_{E}^{\alpha})^{2}$ generates an exponentially
bounded holomorphic semigroup $T_{(-1)^{n+1}(I-\mathcal{D}_{E}^{\alpha})^{2}}$
of angle $\pi\left(\frac{1}{2}-|\alpha-n|\right)$, which is given by

$\displaystyle\left(T_{(-1)^{n+1}(I-\mathcal{D}_{E}^{\alpha})^{2}}(w)f\right)(x)$
$\displaystyle=\frac{1}{2\pi}\int_{0}^{\infty}\frac{f(s)}{s}\int_{-\infty}^{\infty}\left(\frac{s}{x}\right)^{iu}\exp\left((-1)^{n+1}w\left(1-\frac{1}{\alpha\mathbb{B}(1-iu,\alpha)}\right)^{2}\right)\,duds,$

for $x>0,\,w\in S_{\pi\left(\frac{1}{2}-|\alpha-n|\right)}$ and $f\in E$. In
addition,
$\overline{\mathcal{D}((I-\mathcal{D}_{E}^{\alpha})^{2})}=\overline{\mathcal{D}(J_{E})}$.

###### Proof.

First, recall that
$\mathcal{D}_{E}^{\alpha}=(\alpha\mathbb{B}(I-J_{E},\alpha))^{-1}$, so
$(I-\mathcal{D}_{E}^{\alpha})^{2}=(I-\alpha\mathbb{B}(I-J_{E},\alpha))^{-1}$.
It follows that

$\left(1-\frac{1}{\alpha\mathbb{B}(1-z,\alpha)}\right)^{2}=\left(1-\frac{1}{\Gamma(\alpha+1)}\frac{\Gamma(1+\alpha-z)}{\Gamma(1-z)}\right)^{2},$

which is holomorphic in $\mathbb{{C}}\backslash\\{1,2,3,...\\}$. In addition,
for $\lambda,z\in\mathbb{{C}}$, one has that

$\frac{\Gamma(z+\lambda)}{\Gamma(z)}=z^{\lambda}\left(1+O(|z|^{-1})\right),\quad\mbox{
as }|z|\to\infty,$ (5.6)

whenever $z\neq 0,-1,-2,...$ and $z\neq-\lambda,-\lambda-1,-\lambda-2...$,
(see e.g. [26] for more details). As a consequence, one gets that

$\displaystyle\left(1-\frac{1}{\alpha\mathbb{B}(1-z,\alpha)}\right)^{2}=\frac{(-z)^{2\alpha}}{\alpha}\left(1+O(|z|^{-1})\right),\quad\mbox{
as }|z|\to\infty.$

Thus, for any $\beta\in\left(0,\pi\left(\frac{1}{2}-|\alpha-n|\right)\right)$,
one can find a $\rho>0$ large enough such that the function
$\rho+(-1)^{n+1}\left(1-\frac{1}{\alpha\mathbb{B}(1-z,\alpha)}\right)^{2}$
satisfies the hypothesis of Corollary 4.1, i.e.
$(-1)^{n+1}(I-\mathcal{D}_{E}^{\alpha})^{2}$ generates an exponentially
bounded holomorphic semigroup of angle
$\pi\left(\frac{1}{2}-|\alpha-n|\right)$. The rest of the statement follows by
a similar reasoning as in the proof of Proposition 5.1. ∎

We have the following generation result for the operator
$(\mathcal{W}_{E}^{\alpha})^{2}$.

######  Proposition 5.3.

Let $E$ be a $(L^{1}-L^{\infty})$-interpolation space with
$\underline{\eta}_{E}>0$, $n\in\mathbb{{N}}$ and
$\alpha\in\left(n-\frac{1}{2},n+\frac{1}{2}\right)$. Then, the operator
$(-1)^{n+1}(\mathcal{W}_{E}^{\alpha})^{2}$ generates an exponentially bounded
holomorphic semigroup $T_{(-1)^{n+1}(\mathcal{W}_{E}^{\alpha})^{2}}$ of angle
$\pi\left(\frac{1}{2}-|\alpha-n|\right)$, which is given by

$\displaystyle\left(T_{(-1)^{n+1}(\mathcal{W}_{E}^{\alpha})^{2}}(w)f\right)(x)$
$\displaystyle=\frac{1}{2\pi}\int_{0}^{\infty}\frac{f(s)}{s}\int_{-\infty}^{\infty}\left(\frac{s}{x}\right)^{iu+\delta}\exp\left((-1)^{n+1}w\left(\alpha\mathbb{B}(iu+\delta,\alpha)\right)^{-2}\right)\,duds,$

for $x>0$, $w\in S_{\pi\left(\frac{1}{2}-|\alpha-n|\right)}$, and $f\in E$,
where $\delta$ is any number $\delta>0$. In addition,
$\overline{\mathcal{D}((\mathcal{W}_{E}^{\alpha})^{2})}=\overline{\mathcal{D}(J_{E})}$.

###### Proof.

The proof is analogous to the proof of Proposition 5.2, using that
$\mathcal{W}_{E}^{\alpha}=(\alpha\mathbb{B}(J_{E},\alpha))^{-1}$. The only
difference comes out that one cannot apply Cauchy’s Theorem and translate the
inner integral path in $u$ to make $\delta=0$ since the Euler-Beta function
$\mathbb{B}(0,\alpha)$ has an essential singularity for any non natural number
$\alpha$. ∎

Finally, we have the following generation result for the operator
$\mathcal{W}_{E}^{\alpha}(I-\mathcal{D}_{E}^{\alpha})$.

######  Proposition 5.4.

Let $E$ be a $(L^{1}-L^{\infty})$-interpolation space with
$\underline{\eta}_{E}>0$ and $\overline{\eta}_{E}<1$, and let $\alpha>0$.
Then, $\mathcal{W}_{E}^{\alpha}(I-\mathcal{D}_{E}^{\alpha})$ generates an
exponentially bounded holomorphic semigroup
$T_{\mathcal{W}_{E}^{\alpha}(I-\mathcal{D}_{E}^{\alpha})}$ of angle
$\frac{\pi}{2}$, which is given by

$\displaystyle\left(T_{\mathcal{W}_{E}^{\alpha}(I-\mathcal{D}_{E}^{\alpha})}(w)f\right)(x)$
$\displaystyle=$
$\displaystyle\frac{1}{2\pi}\int_{0}^{\infty}\frac{f(s)}{s}\int_{-\infty}^{\infty}\left(\frac{s}{x}\right)^{iu+\delta}\exp\left(\frac{w}{\alpha\mathbb{B}(\delta+iu,\alpha)}\left(1-\frac{1}{\alpha\mathbb{B}(1-\delta-
iu,\alpha)}\right)\right)\,duds,$

for $x>0,\,w\in S_{\frac{\pi}{2}}$, and $f\in E$, where $\delta\in(0,1)$ is
any number. In addition,
$\overline{\mathcal{D}(\mathcal{W}_{E}^{\alpha}(I-\mathcal{D}_{E}^{\alpha}))}=\overline{\mathcal{D}(J_{E})}$.

###### Proof.

The proof is analogous to the proof of Propositions 5.2 and 5.3. Here, the
statement is valid for any $\alpha>0$ since, by (5.6), we have that

$\displaystyle\frac{1}{\alpha\mathbb{B}(z,\alpha)}\left(1-\frac{1}{\alpha\mathbb{B}(1-z,\alpha)}\right)=\frac{z^{\alpha}(-z)^{\alpha}}{2\alpha}(1+O(|z|^{-1})),\quad\mbox{
as }|z|\to\infty.$

The proof is finished. ∎

###  5.2. Generalized Black-Scholes partial differential equations

Let $B_{E}$ be a closed linear operator on a Banach space $E$, and consider
the following abstract Cauchy problem:

$\displaystyle\begin{cases*}\displaystyle u\in C^{1}((0,\infty);E),\quad
u(t)\in\mathcal{D}(B_{E}),\quad t>0,\\\ \displaystyle
u^{\prime}(t)=B_{E}u(t),\qquad\,\,\,t>0,\\\ \displaystyle\lim_{t\downarrow
0}u(t)=f\in E.\end{cases*}$ ($ACP_{0}$)

We say that the Cauchy problem ($ACP_{0}$) is well-posed, if for for any $f\in
E$, there exists a unique solution $u$.

We are ready to state the following result concerning the well-posedness of
the fractional Black-Scholes equation. Before that, let us state explicitly
how these equations look like. Let $n\in\mathbb{{N}}$, $\alpha>0$, and recall
that $D^{\alpha}$ and $W^{\alpha}$ denote, respectively, the Riemann-Liouville
and Weyl fractional derivatives of order $\alpha$ acting on the spatial
domain.

  1. (1)

In the case $B_{E}=(-1)^{n+1}(J_{E})^{2\alpha}$ we have the following
situation:

     * •

If $\underline{\eta}_{E}>0$, one can use the Balakrishnan representation, to
obtain

$(-1)^{n+1}u_{t}(x)=\frac{-1}{\Gamma(1-\alpha)}\int_{x}^{\infty}\left(\log\frac{s}{x}\right)^{-2\alpha+n}U_{n}^{\prime}(s)\,ds,\quad
t,x>0.$

     * •

If $\underline{\eta}_{E}=0$, one has to proceed as in (5.3) to obtain

$(-1)^{n+1}u_{t}(x)=\lim_{\varepsilon\downarrow
0}\frac{-1}{\Gamma(1-\alpha)}\int_{x}^{\infty}\left(\log\frac{s}{x}\right)^{-2\alpha+n}\left(\frac{x}{s}\right)^{\varepsilon}U_{n}^{\prime}(s)\,ds,\quad
t,x>0.$

In both cases, $n\in\mathbb{{N}}$ is the whole part of $2\alpha$ and
$U_{n}\mathrel{\mathop{\ordinarycolon}}=(J_{E})^{n}U$.

  2. (2)

If $B_{E}=(-1)^{n+1}(I-\mathcal{D}_{E}^{\alpha})^{2}$, one obtains the
equation

$(-1)^{n+1}u_{t}=\frac{1}{\Gamma(\alpha+1)^{2}}D^{\alpha}(x^{\alpha}D^{\alpha}(x^{\alpha}u))-\frac{2}{\Gamma(\alpha+1)}D^{\alpha}(x^{\alpha}u)+u,\quad
t,x>0.$

  3. (3)

If $B_{E}=(-1)^{n+1}(\mathcal{W}_{E}^{\alpha})^{2}$, one gets the equation

$(-1)^{n+1}u_{t}=\frac{1}{\Gamma(\alpha+1)^{2}}x^{\alpha}W^{\alpha}(x^{\alpha}W^{\alpha}u),\quad
t,x>0.$

  4. (4)

The case $B_{E}=\mathcal{W}_{E}^{\alpha}(I-\mathcal{D}_{E}^{\alpha})$ leads to
the equation

$u_{t}=\frac{1}{\Gamma(\alpha+1)}x^{\alpha}W^{\alpha}u-\frac{1}{\Gamma(\alpha+1)^{2}}D^{\alpha}(x^{2\alpha}W^{\alpha}u),\quad
t,x>0.$

We have the following result.

######  Theorem 5.5.

Let $E$ be a $(L^{1}-L^{\infty})$-interpolation space with order continuous
norm, $n\in\mathbb{{N}}$, and $\alpha>0$. Then, the following assertions hold.

  1. (a)

If $\alpha\in\left(n-\frac{1}{2},n+\frac{1}{2}\right)$, then ($ACP_{0}$) is
well-posed with $B_{E}=(-1)^{n+1}(J_{E})^{2\alpha}$.

  2. (b)

If $\overline{\eta}_{E}<1$ and
$\alpha\in\left(n-\frac{1}{2},n+\frac{1}{2}\right)$, then ($ACP_{0}$) is well-
posed with $B_{E}=(-1)^{n+1}(I-\mathcal{D}_{E}^{\alpha})^{2}$.

  3. (c)

If $\underline{\eta}_{E}>0$ and
$\alpha\in\left(n-\frac{1}{2},n+\frac{1}{2}\right)$, then ($ACP_{0}$) is well-
posed with $B_{E}=(-1)^{n+1}(\mathcal{W}_{E}^{\alpha})^{2}$.

  4. (d)

If $\overline{\eta}_{E}<1$ and $\underline{\eta}_{E}>0$, then ($ACP_{0}$) is
well-posed with $B_{E}=\mathcal{W}_{E}^{\alpha}(I-\mathcal{D}_{E}^{\alpha})$.

In any case, the solution $u$ of ($ACP_{0}$) is given by $u(t)=T_{B_{E}}(t)f$
for $t>0$. In addition, identifying $u(t,x)=u(t)(x)$, we obtain that $u\in
C^{\infty}((0,\infty)\times(0,\infty))$.

###### Proof.

In all cases, $B_{E}$ is the generator of a holomorphic semigroup with
$\overline{\mathcal{D}(B_{E})}=\overline{\mathcal{D}(J_{E})}$ by Propositions
5.1, 5.2, 5.3, and 5.4. Moreover, $T_{B_{E}}$ is strongly continuous since one
has that $\mathcal{D}(J_{E})$ is dense in $E$ if and only if $E$ has order
continuous norm (see e.g. [2, Remark 4.2]). Then, the assertions follow
immediately by the relation between the well-posedness of a Cauchy problem,
and the fact that $B_{E}$ generates a strongly continuous semigroup (see for
example [1, Proposition 3.1.2 and Theorem 3.1.12]).

Regarding the regularity result, one has that $u(t)$ is $E$-holomorphic in $t$
in $(0,\infty)$ since $T_{B_{E}}$ is a holomorphic semigroup. Even more, it
satisfies $u(t)=T_{E}(t)f$, $u^{(k)}(t)=(B_{E})^{k}u(t)$, and that
$u^{(k)}(t)\in\mathcal{D}((B_{E})^{n})$ for all $k,n\in\mathbb{{N}}$ and $t>0$
(see [1, Chapter 3]). Now, reasoning as in the proof of Proposition 4.3 with
any of the operators $B_{E}$ yields that
$\mathcal{D}(B_{E})\subset\mathcal{D}((J_{E})^{\varepsilon})$ for sufficiently
small $\varepsilon>0$. In addition, since $\mathcal{D}(J_{E})\subset AC_{\rm
loc}(0,\infty)$, we have that $\mathcal{D}((J_{E})^{j+1})\subset
C^{j}(0,\infty)$. As
$u^{(k)}(t)\in\mathcal{D}((B_{E})^{n})\subset\mathcal{D}((J_{E})^{n\varepsilon})$
for all $k,n\in\mathbb{{N}}$, one obtains that $u^{(k)}(t)\in
C^{\infty}(0,\infty)$ for all $k\in\mathbb{{N}}$ and $t>0$. The proof is
finished. ∎

######  Remark 5.6.

As stated in the above proof, $T_{B_{E}}$ is strongly continuous at $0$ if and
only if $E$ has order continuous norm. Hence, Theorem 5.5 does not hold for a
general $(L^{1}-L^{\infty})$-interpolation space. To address all interpolation
spaces, we follow the ideas given in [2] and consider the Köthe dual
$E^{\star}$ of $E$, given by

$\displaystyle
E^{\star}\mathrel{\mathop{\ordinarycolon}}=\left\\{g\mathrel{\mathop{\ordinarycolon}}(0,\infty)\to\mathbb{{C}}\mbox{
measurable and }\;\int_{0}^{\infty}|f(x)g(x)|\;dx<\infty\quad\text{for all
}f\in E\right\\}.$

Every $g\in E^{\star}$ defines a bounded (order continuous) linear functional
$\varphi_{g}$ on $E$, given by

$\langle
f,\varphi_{g}\rangle_{E,E^{\star}}\mathrel{\mathop{\ordinarycolon}}=\int_{0}^{\infty}f(x)g(x)\;dx\;\mbox{
for all }\;f\in E.$

In this way we can identify $E^{\star}$ with a subspace of the dual
$E^{\prime}$ (and under the present assumptions on $E$, this subspace is
norming for $E$). It is known that, when equipped with the norm
$\|g\|_{E^{\star}}=\|\varphi_{g}\|_{E^{\prime}}$, then $E^{\star}$ is a
$(L^{1},L^{\infty})$-interpolation space on $(0,\infty)$.

Next, we consider the following abstract Cauchy problem:

$\displaystyle\begin{cases*}\displaystyle u\in C^{1}((0,\infty);E),\quad
u(t)\in\mathcal{D}(B_{E}),\quad t>0,\\\ \displaystyle
u^{\prime}(t)=B_{E}u(t),\qquad\,\,\,t>0,\\\ \displaystyle\lim_{t\downarrow
0}\langle u(t),\varphi\rangle_{E,E^{\star}}=\langle
f,\varphi\rangle_{E,E^{\star}},\quad f\in E\text{ and for all }\varphi\in
E^{\star}.\end{cases*}$ ($ACP_{1}$)

Again, we say that ($ACP_{1}$) is well-posed if, for any $f\in E$, there
exists a unique $u$ which is a solution of ($ACP_{1}$).

We have the following result.

######  Theorem 5.7.

Let $E$ be a $(L^{1}-L^{\infty})$-interpolation space, $n\in\mathbb{{N}}$, and
$\alpha>0$. Then, the following assertions hold.

  1. (a)

If $\alpha\in\left(n-\frac{1}{2},n+\frac{1}{2}\right)$, then ($ACP_{1}$) is
well-posed with $B_{E}=(-1)^{n+1}(J_{E})^{2\alpha}$.

  2. (b)

If $\overline{\eta}_{E}<1$ and if
$\alpha\in\left(n-\frac{1}{2},n+\frac{1}{2}\right)$, then ($ACP_{1}$) is well-
posed with $B_{E}=(-1)^{n+1}(I-\mathcal{D}_{E}^{\alpha})^{2}$.

  3. (c)

If $\underline{\eta}_{E}>0$ and if
$\alpha\in\left(n-\frac{1}{2},n+\frac{1}{2}\right)$, then ($ACP_{1}$) is well-
posed with $B_{E}=(-1)^{n+1}(\mathcal{W}_{E}^{\alpha})^{2}$.

  4. (d)

If $\overline{\eta}_{E}<1$ and $\underline{\eta}_{E}>0$, then ($ACP_{1}$) is
well-posed with $B_{E}=\mathcal{W}_{E}^{\alpha}(I-\mathcal{D}_{E}^{\alpha})$.

In any case, the solution $u$ of ($ACP_{1}$) is given by $u(t)=T_{B_{E}}(t)f$
for $t>0$. In addition, identifying $u(t,x)=u(t)(x)$, we obtain that $u\in
C^{\infty}((0,\infty)\times(0,\infty))$.

To prove the theorem, we need the following lemma.

######  Lemma 5.8.

Let $a\geq 0$ and let $A\in\text{{BSect}}(\pi/2,a)$ on $E$ be such that $A$
generates an exponentially bounded group $(G(t))_{t\in\mathbb{{R}}}$ for which
$\|G(t)\|\lesssim e^{a|t|}$ for $t\in\mathbb{{R}}$. Let $g\in\mathcal{M}_{A}$
satisfy all the hypothesis in Corollary 4.1. Assume furthermore that the
following hold:

  1. (a)

$g$ is quasi-regular in $\\{-a,a,\infty\\}$ with $g(a),g(-a)\neq\infty$.

  2. (b)

The group $(G(t))_{t\in\mathbb{{R}}}$ is $\sigma(E,E^{\star})$-continuous,
that is, $\lim_{t\to 0}\langle G(t)f,\varphi\rangle_{E,E^{\star}}=\langle
f,\varphi\rangle_{E,E^{\star}}$ for all $f\in E$ and $\varphi\in E^{\star}$.

Then, the semigroup $(T_{g}(t))_{t\geq 0}$ generated by the operator $-g(A)$
is also $\sigma(E,E^{\star})$-continuous, i.e.,

$\lim_{t\downarrow 0}\langle T_{g}(t)f,\varphi\rangle_{E,E^{\star}}=\langle
f,\varphi\rangle_{E,E^{\star}}\quad\text{for all }f\in E\text{ and }\varphi\in
E^{\star}.$

###### Proof.

We ask for the regularity conditions at $\\{-a,a,\infty\\}$ instead of just
$M_{A}$ in order to apply the results given in the appendix of this paper.
Now, Proposition A.3 yields that
$T_{g}(t)=\int_{-\infty}^{\infty}G(s)\,\mu_{h_{t}}(ds)$, where
$h_{t}(z)\mathrel{\mathop{\ordinarycolon}}=\exp(-tg(z))$, and $\mu_{h_{t}}\in
M_{a}(\mathbb{{R}})$ is given in Lemma A.1. By Lemma A.1 again, one obtains
that

$\int_{-\infty}^{\infty}\,\mu_{t}(ds)=\int_{-\infty}^{\infty}e^{i0s}\,\mu_{t}(ds)=h_{t}(0)=\exp(-tg(0)).$

Then, for any $f\in E$ and $\varphi\in E^{\star}$, we have that

$\displaystyle\langle T_{g}(t)f,\varphi\rangle_{E,E^{\star}}-\langle
f,\varphi\rangle_{E,E^{\star}}=\int_{-\infty}^{\infty}\langle
G(s)f-f,\varphi\rangle_{E,E^{\star}}\mu_{h_{t}}(s)\,ds+(e^{-tg(0)}-1)\langle
f,\varphi\rangle_{E,E^{\star}}.$ (5.7)

We have to prove that the integral term in (5.7) tends to $0$ as $t\downarrow
0$. Since by assumption $(G(t))_{t\in\mathbb{{R}}}$ is
$\sigma(E,E^{\star})$-continuous, we have that $\lim_{t\downarrow 0}\langle
G(t)f,\varphi\rangle_{E,E^{\star}}=\langle f,\varphi\rangle_{E,E^{\star}}$.
Thus, for any $\varepsilon>0$, there exists $\delta>0$ such that $|\langle
G(s)f-e^{irs}f,\varphi\rangle_{E,E^{\star}}|<\varepsilon$ for all
$|s|<\delta$. Hence, for some $C>0$ independent of $t$ and $\varepsilon$, we
have that

$\displaystyle\limsup_{t\downarrow 0}\left|\int_{-\infty}^{\infty}\langle
G(s)f-f,\varphi\rangle_{E,E^{\star}}\mu_{h_{t}}(s)\,ds\right|\leq
C\varepsilon+\limsup_{t\downarrow 0}\left|\int_{|s|>\delta}\langle
G(s)f-f,\varphi\rangle_{E,E^{\star}}\mu_{h_{t}}(s)\,ds\right|.$

Let us work with the above integral when $s>\delta$, leaving the case
$s<-\delta$, which is completely analogous. By Lemma A.1, one gets that

$\displaystyle\int_{s>\delta}\langle
G(s)f-f,\varphi\rangle_{E,E^{\star}}\,\mu_{h_{t}}(s)\,ds=\int_{s>\delta}\langle
G(s)f-f,\varphi\rangle_{E,E^{\star}}\,\frac{1}{2\pi
i}\int_{\Gamma_{+}}e^{-zs}e^{-tg(z)}\,dzds$ $\displaystyle=$
$\displaystyle\int_{s>\delta}\langle
G(s)f-f,\varphi\rangle_{E,E^{\star}}\frac{1}{2\pi
i}\int_{\Gamma_{+}}e^{-zs}\left(e^{-tg(z)}-e^{-tg(a)}\frac{b+a}{b+z}\right)\,dzds,$

where we have used Cauchy’s theorem in the last equality and the fact that for
$b>a$, we have that

$e^{-tg(a)}\int_{\Gamma_{+}}e^{-zs}\frac{b+a}{b+z}\,dz=0,\qquad\text{for all
}s,t>0.$

Now, applying the Lebesgue Dominated Convergence Theorem we obtain that

$\displaystyle\limsup_{t\downarrow 0}\left|\int_{s>\delta}\langle
G(s)f-f,\varphi\rangle_{E,E^{\star}}\,\mu_{h_{t}}(s)\,ds\right|=\left|\int_{s>\delta}\langle
G(s)f-f,\varphi\rangle_{E,E^{\star}}\frac{1}{2\pi
i}\int_{\Gamma_{+}}e^{-zs}\frac{z-a}{b+z}\,dzds\right|=0,$

where we have used again Cauchy’s theorem in the last equality. To check the
hypothesis of the Dominated Convergence Theorem, one has to bound the
following expression:

$\displaystyle F_{t}(s,z)\mathrel{\mathop{\ordinarycolon}}=e^{-s(\Re
z-a)}\left|e^{-tg(z)}-e^{-tg(a)}\frac{b+a}{b+z}\right|,\quad
s>\delta,\,z\in\Gamma_{+},$

by an integrable function for all $t\in(0,\varepsilon^{\prime})$, where
$\varepsilon^{\prime}$ is any number $\varepsilon^{\prime}>0$. Since $\Re
g(z)\geq 0$ implies that $\sup_{t>0,z\in\Gamma_{+}}|e^{-tg(z)}|<\infty$, an
easy bound of the term between $|(\cdot)|$ leads to

$\displaystyle F_{t}(s,z)\lesssim e^{-s(\Re
z-a)}\min\left\\{1,\frac{|z-a|+|g(z)-g(a)|}{|b+z|}\right\\},$

which is easily seen to be integrable by integrating first on $s$ and then in
$z$ (recall that the function $g$ is regular at $a$). The proof is finished. ∎

###### Proof of Theorem 5.7.

Once we have proven that $T_{B_{E}}(t)f$ is $\sigma(E,E^{\star})$-continuous
on $t$ as $t\downarrow 0$ for all $f\in E$, the assertions follow by a similar
reasoning as in the proofs of Theorem 5.5 and [2, Theorem 5.8]. Then, for the
operators we are considering, we only have to check the exponentially bound
condition of Lemma 5.8. But, except for the case $\underline{\eta}_{E}=0$ and
$B_{E}=(J_{E})^{2\alpha}$, we can always assume that they are satisfied since
the functions $g_{B_{E}}$, for which $B_{E}=g_{B_{E}}(J_{E})$, are holomorphic
in strictly wider bisectors than the ones with singular points in
$\underline{\eta}_{E},\overline{\eta}_{E}$. If this is the case, given any
$\varepsilon>0$, we have that
$\|G_{E}(t)\|_{\mathcal{L}(E)}\lesssim\max\\{e^{(\underline{\eta}_{E}-\varepsilon)t},e^{(\overline{\eta}_{E}+\varepsilon)t}\\}$
for all $t\in\mathbb{{R}}$. And regarding the case $\underline{\eta}_{E}=0$,
one still has that $\|G_{E}(t)\|_{\mathcal{L}(E)}\lesssim 1$ for $t\leq 0$
(see e.g. [2]). Then, we can apply Lemma 5.8 to obtain that $T_{B_{E}}(t)f$ is
$\sigma(E,E^{\star})$-continuous. The proof is finished. ∎

It is easy to check that when $\alpha=1$, all the different generalized
Black–Scholes equations presented above yield the classical Black–Scholes
equation given by (BS). In this case, the above results retrieve the ones
obtained in [2, Section 5]. In particular, one gets the formula for the
semigroup $T_{B_{E}}$, given by

$\displaystyle\left(T_{B_{E}}(w)f\right)(x)$
$\displaystyle=\frac{1}{2\pi}\int_{0}^{\infty}\frac{f(s)}{s}\int_{-\infty}^{\infty}\left(\frac{s}{x}\right)^{iu}\exp\left(-wu^{2}\right)\,duds$
$\displaystyle=\frac{1}{\sqrt{4\pi w}}\int_{0}^{\infty}\exp\left(-\frac{(\log
x-\log s)^{2}}{4w}\right)\frac{f(s)}{s}\,ds,\qquad x>0,\,\Re w>0,$

where in the last equality we have made use of the integral identity [15,
Formula 3.233(2)].

######  Remark 5.9.

The above results do not cover (in general) the case $\alpha=1/2,3/2,5/2,...$.
This is closely related to the odd powers of a generator of a group (see
Corollary 3.11 and [7, Theorem 4.6]). Indeed, one can prove that when
$\alpha=1/2,3/2,5/2,...$, the considered operators for $B_{E}$, except the
last one $\mathcal{W}_{E}^{\alpha}(I-\mathcal{D}_{E}^{\alpha})$, are
bisectorial operators of angle $\frac{\pi}{2}$. Unfortunately, this is a
necessary but not sufficient condition to determine that they generate
semigroups.

##  Appendix A Functional calculus of generators of exponentially bounded
groups

In this appendix, we give some auxiliary results in the case where $A$ is the
generator of an exponentially bounded group $(G(t))_{t\in\mathbb{{R}}}$ on a
Banach space $X$ satisfying $\|G(t)\|_{\mathcal{L}(X)}\lesssim\exp(a|t|)$ for
all $t\in\mathbb{{R}}$ and some $a\geq 0$. It is well known that in this case,
$A\in\text{{BSect}}(\pi/2,a)$, see for example [18, Section 2.1.1]. The
following results are completely analogous to the ones given in [6, Theorem
5.2] for the primary functional calculus of strip operators; or in [18,
Section 3.3] for the NFC of sectorial operators.

It should be mentioned that through this appendix, we will only work with the
NFC for bisectorial rate operators, not including the different NFCs presented
in Section 2.3. The reason for this is that, in order to successfully apply
some identities, we will need that the integration paths of the NFC leave the
spectrum of $A$ completely on one side. This is enough to cover all the
results that use the appendix.

First, recall the integral representation for the resolvent of the generator
$A$ of an exponentially bounded group $(G(t))_{t\in\mathbb{{R}}}$ given by

$\displaystyle R(z,A)=\int_{0}^{\infty}e^{-zt}G(t)\,dt,\text{ if }\Re
z>a,\quad R(z,A)$ $\displaystyle=-\int_{-\infty}^{0}e^{-zt}G(t)\,dt,\text{ if
}\Re z<-a.$ (A.1)

Next, for any $a\geq 0$, let $M_{a}(\mathbb{{R}})$ be the set of Borel
measures $\mu$ on $\mathbb{{R}}$ for which $e^{a|t|}$ is $\mu$-integrable. It
is readily seen that $M_{a}(\mathbb{{R}})$ is closed under translation and
convolution. Moreover, for any $\mu\in M_{a}(\mathbb{{R}})$, one can define
its Fourier transform $\mathcal{F}$ given by

$\displaystyle(\mathcal{F}\mu)(z)=\int_{-\infty}^{\infty}e^{-zt}\,\mu(dt),\qquad\text{for
all }z\in BS_{\pi/2,a}.$

######  Lemma A.1.

Let $a\geq 0$ and $f\in\mathcal{E}[BS_{\pi/2,a}]\oplus\mathbb{{C}}\mathbf{1}$.
Then, there exists a (unique) measure $\mu_{f}\in M_{a}(\mathbb{{R}})$ such
that $f(z)=\mathcal{F}\mu_{f}(-z)$ for all $z\in BS_{\pi/2,a}$, which is given
by $\mu_{f}(dt)=\psi_{f}(t)dt+c\delta_{0}(dt)$, where $c=f(\infty)$ and

$\displaystyle\psi_{f}(t)\mathrel{\mathop{\ordinarycolon}}=\begin{cases*}\displaystyle\frac{-1}{2\pi
i}\int_{\Gamma_{-}}e^{-zt}f(z)\,dz,\quad t<0,\\\ \displaystyle\frac{1}{2\pi
i}\int_{\Gamma_{+}}e^{-zt}f(z)\,dz,\quad t>0,\end{cases*}$ (A.2)

and where $\Gamma$ is any path of integration for the NFC of bisectorial
operators, $\Gamma_{-}\mathrel{\mathop{\ordinarycolon}}=\Gamma\cap\Re z<-a$
and $\Gamma_{+}\mathrel{\mathop{\ordinarycolon}}=\Gamma\cap\Re z>a$.

###### Proof.

The proof is the same as in the case of sectorial operators (see [18, Lemma
3.3.1]). We omit the details for the sake of brevity. ∎

######  Remark A.2.

Let $f$ be as above, and assume furthermore that
$|f(z)|\lesssim|z|^{-(1+\varepsilon)}$ as $z\to\infty$ for some
$\varepsilon>0$. An easy application of Cauchy’s theorem to (A.2) yields that

$\displaystyle\psi_{f}(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{-itu}f(iu)\,du,\quad
t\in\mathbb{{R}}.$

######  Proposition A.3.

Let $A$ be the generator of an exponentially bounded group
$(G(t))_{t\in\mathbb{{R}}}$ on $X$ satisfying
$\|G(t)\|_{\mathcal{L}(X)}\lesssim e^{a|t|}$ for some $a\geq 0$, so that
$A\in\text{{BSect}}(\pi/2,a)$. Let $\mu\in M_{a}(\mathbb{{R}})$ be such that
$f(z)\mathrel{\mathop{\ordinarycolon}}=\mathcal{F}\mu(-z)\in\mathcal{M}[BS_{\pi/2,a}]_{A}$.
Then,

$f(A)=\int_{-\infty}^{\infty}G(t)\,\mu(dt).$

###### Proof.

The proof follows as in the case of sectorial operators (see [18, Proposition
3.3.2]). We omit the details for the sake of brevity. ∎

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