# An introduction to hyperholomorphic spectral theories and fractional powers
of vector operators

Fabrizio Colombo  (FC) Politecnico di Milano
Dipartimento di Matematica
Via E. Bonardi, 9
20133 Milano, Italy<EMAIL_ADDRESS>,  Jonathan Gantner  (JG) (was
PhD student at) Politecnico di Milano
Dipartimento di Matematica
Via E. Bonardi, 9
20133 Milano, Italy<EMAIL_ADDRESS>and  Stefano Pinton  (SP)
Politecnico di Milano
Dipartimento di Matematica
Via E. Bonardi, 9
20133 Milano, Italy<EMAIL_ADDRESS>
###### Abstract.

The aim of this paper is to give an overview of the spectral theories
associated with the notions of holomorphicity in dimension greater than one. A
first natural extension is the theory of several complex variables whose
Cauchy formula is used to define the holomorphic functional calculus for
$n$-tuples of operators $(A_{1},...,A_{n})$. A second way is to consider
hyperholomorphic functions of quaternionic or paravector variables. In this
case, by the Fueter-Sce-Qian mapping theorem, we have two different notions of
hyperholomorphic functions that are called slice hyperholomorphic functions
and monogenic functions. Slice hyperholomorphic functions generate the
spectral theory based on the $S$-spectrum while monogenic functions induce the
spectral theory based on the monogenic spectrum. There is also an interesting
relation between the two hyperholomorphic spectral theories via the
$F$-functional calculus. The two hyperholomorphic spectral theories have
different and complementary applications. Here we also discuss how to define
the fractional Fourier’s law for nonhomogeneous materials, such definition is
based on the spectral theory on the $S$-spectrum.

AMS Classification 47A10, 47A60.

Keywords: Spectral theory, $S$-spectrum, monogenic spectrum, hyperholomorphic
spectral theories, fractional powers of vector operators.

today

##  1\. Introduction

The problem to define functions of an operator $A$ or of an $n$-tuple of
operators $(A_{1},...,A_{n})$ is very important both in mathematics and in
physics and has been investigated with different methods starting from the
beginning of the last century. The spectral theorem is one of the most
important tools to define functions of normal operators on a Hilbert space and
it is of crucial importance in quantum mechanic as well as the Weyl functional
calculus.

The theory of holomorphic functions plays a central role in operator theory.
In fact, the Cauchy formula allows to define the holomorphic functional
calculus (often called Riesz-Dunford functional calculus) in Banach spaces
[14], and this calculus can be extended to unbounded operators. For sectorial
operators the $H^{\infty}$-functional calculus, introduced by A. McIntosh in
[67], turned out to be the most important extension.

Holomorphic functions of one complex variable
$f:\Omega\subseteq\mathbb{C}\to\mathbb{C}$ (denoted by $\mathcal{O}(\Omega)$)
have the following extensions:

  * (E1)

Systems of Cauchy-Riemann equations, for functions
$f:\Pi\subseteq\mathbb{C}^{n}\to\mathbb{C}$, give the theory of several
complex variables.

  * (E2)

Holomorphicity of vector fields is connected with quaternionic-valued
functions or more in general with Clifford algebra-valued functions. There are
two different extensions that are obtain by the Fueter-Sce-Qian theorem, also
called the Fueter-Sce-Qian construction, and gives two different notions of
hyperholomorphic functions, see for more details [30].

Consider functions defined on an open set $U$ in the quaternions $\mathbb{H}$
or in $\mathbb{R}^{n+1}$ for Clifford algebra-valued functions, then the
Fueter-Sce-Qian extension consists of two steps.

  * Step (I) gives the class of slice hyperholomorphic functions (denoted by $SH(U)$), these functions are also called slice monogenic for Clifford algebra-valued functions and slice regular in the quaternionic case.

  * Step (II) gives the monogenic functions (denoted by $M(U)$) and Fueter regular functions in the case of the quaternions.

Both classes of hyperholomorphic functions have a Cauchy formula that can be
used to define functions of quaternionic operators or of $n$-tuples of
operators that do not necessarily commute.

  * (S)

The Cauchy formula of slice hyperholomorphic functions generates the
$S$-functional calculus for quaternionic linear operators or for $n$-tuples of
not necessarily commuting operators, this calculus is based on the the notion
of $S$-spectrum. The spectral theorem for quaternionic operators is also based
on the $S$-spectrum.

  * (M)

The Cauchy formula of monogenic functions generates the monogenic functional
calculus that is based on the monogenic spectrum.

The hyperholomorphic functional calculi coincide with the Riesz-Dunford
functional calculus when they are applied to a single complex operator.

If we denote by the symbol ${FSQ}$ the Fueter-Sce-Qian construction then the
following diagram illustrates the possible extensions:

$\begin{CD}{\color[rgb]{0,0,0}\mathcal{O}(\Omega)}@>{{FSQ}-construction}>{}>{\color[rgb]{0,0,0}Hyperholomorphic\
functions}\\\ @V{}V{}V@V{}V{}V\\\ {\color[rgb]{0,0,0}Several\ complex\
variables}{\color[rgb]{0,0,0}S-spectrum\ and\ monogenic\ spectrum}\\\
@V{}V{}V@V{}V{}V\\\ {\color[rgb]{0,0,0}Taylor\ joint\ spectrum}\
{\color[rgb]{0,0,0}Hyperholomorphic\ spectral\ theories\ (HST)}\\\
@V{}V{}V@V{}V{}V\\\ {\color[rgb]{0,0,0}Complex\ spectral\ theory}\
{\color[rgb]{0,0,0}Connections\ between\ (HST)}\end{CD}$

The first mathematicians who understood the importance of hypercomplex
analysis to define functions of noncommuting operators on Banach spaces were
A. McIntosh and his collaborators, staring from preliminary results in [70].
Using the theory of monogenic functions they developed the monogenic
functional calculus and several of its applications, see [69]. The
$S$-functional calculus, and in general the spectral theory on the
$S$-spectrum, started its development only in 2006 when F. Colombo and I.
Sabadini discovered the $S$-spectrum. The discovery of the $S$-spectrum and of
the $S$-functional calculus is well explained in the introduction of the book
[24] with a complete list of the references and it is also described how
hypercomplex analysis methods were used to identify the appropriate notion of
quaternionic spectrum whose existence was suggested by quaternionic quantum
mechanics.

If we denote by $\mathcal{B}(V)$ the Banach space of all bounded right linear
operators acting on a two sided quaternionic Banach space $V$ then the
appropriate notion quaternionic spectrum, that is called the $S$-spectrum, is
defined in a very counterintuitive way because it involves the square of the
quaternionic linear operator $T$ and it is define as:

$\sigma_{S}(T)=\\{s\in\mathbb{H}\ |\ T^{2}-2s_{0}T+|s|^{2}\mathcal{I}\ \ {\rm
is\ not\ invertible\ in}\ \ B(V)\\}.$

The $S$-spectrum for quaternionic operators can be naturally defined also for
paravector operators when we work in a Clifford algebra, see [43] and the book
[46].

The problem of the definition of the quaternionic spectrum for the
quaternionic spectral theorem has been an open problem for long time even
though several attempts have been done by several authors in the past decades,
see e.g. [81, 82], however the correct definition of spectrum was not
specified. Finally in [2] the spectral theorem on the $S$-spectrum was proved
for both bounded and unbounded normal operators on a quaternionic Hilbert
space.

The main problems with the quaternionic notion of spectrum can be easily
described with the following considerations related to bounded linear
operators just for the sake of simplicity. Let $T:V\to V$ be a right linear
bounded quaternionic operator acting on a two sided quaternionic Banach space
$V$. If we readapt the notion of spectrum for a complex linear operator to the
quaternionic setting we obtain two different notions of spectra because of the
noncommutativity of the quaternions. The left spectrum $\sigma_{L}(T)$ of $T$
is defined as

$\sigma_{L}(T)=\\{s\in\mathbb{H}\ \ |\ \ s\mathcal{I}-T\ \ \ {\rm is\ not\
invertible\ in\ }\mathcal{B}(V)\\},$

where the notation $s\mathcal{I}$ in $\mathcal{B}(V)$ means that
$(s\mathcal{I})(v)=sv$. The right spectrum $\sigma_{R}(T)$ of $T$ is
associated with the right eigenvalue problem, i.e., the search of those
quaternions $s$ such that there exists a nonzero vector $v\in V$ satisfying

$T(v)=vs.$

In both spectral problems it is unclear how to associate to the spectrum a
resolvent operator with the property of being an hyperholomorphic function
operator-valued. In fact, for the left spectrum $\sigma_{L}(T)$ it is not
clear what notion of hyperholomorphicity is associated to the map
$s\to(s\mathcal{I}-T)^{-1}$, for $s\in\mathbb{H}\setminus\sigma_{L}(T)$ and
for the right spectrum it is even more weird because the operator
$\mathcal{I}s-T$ (where $\mathcal{I}s$ means $(\mathcal{I}s)(v)=vs$) is not
linear, so it is not clear which operator is the candidate to be the resolvent
operator.

######  Remark 1.1.

One of the main motivations that suggested the existence of the $S$-spectrum
is the paper [13] by G. Birkhoff and J. von Neumann, where they showed that
quantum mechanics can be formulated also on quaternionic numbers. Since that
time, several papers and books treated this topic, however it is interesting,
and somewhat surprising, that an appropriate notion of spectrum for
quaternionic linear operators was not present in the literature. Moreover, in
quaternionic quantum mechanics the right spectrum $\sigma_{R}(T)$ is the most
useful notion of spectrum to study the bounded states of a quantum systems.
Before 2006 only in one case the quaternionic spectral theorem was proved
specifying the spectrum and it is the case of quaternionic normal matrices,
see [51], where the right spectrum $\sigma_{R}(T)$ has been used.

Now we recall some research directions and applications of the
hyperholomorphic function theories and related spectral theories.

The first step of FSQ-construction generates slice hyperholomorphic functions
and the spectral theory of the $S$-spectrum, we have:

  * •

The foundation of the quaternionic spectral theory on the $S$-spectrum are
organized in the books [23, 24], and for paravector operators see [46].

  * •

The mathematical tools for quaternionic quantum mechanics is the spectral
theorem based on the $S$-spectrum [2, 55].

  * •

Quaternionic evolution operators, Phillips functional calculus,
$H^{\infty}$-functional calculus, see [23].

  * •

Quaternionic approximation [54].

  * •

The characteristic operator functions and applications to linear system theory
[5].

  * •

Quaternionic spectral operators [56].

  * •

Quaternionic perturbation theory and invariant subspaces [16].

  * •

Schur analysis in the slice hyperholomorphic setting [4].

  * •

The theory of function spaces of slice hyperholomorphic functions [6].

  * •

New classes of fractional diffusion problems based on fractional powers of
quaternionic linear operators, see the book [23] and the more recent
contributions [17, 18, 19, 28, 29].

In the last section of this paper we explain how to treat fractional diffusion
problems using the quaternionic spectral theory on the $S$-spectrum and we
show some of the recent results on fractional Fourier’s law for nonhomogeneous
materials recently obtained. An example of problems that we can treat is the
following.

We warn the reader that in this paper, with an abuse of notations, we use the
symbol $\underline{x}$ for both the coordinates of a point
$(x_{1},x_{2},\ldots,x_{n})\in\mathbb{R}^{n}$ or for the vector part of a
quaternion or for the imaginary part of a paravector in a Clifford algebra.

Let $\Omega$ be a bounded or an unbounded domain in $\mathbb{R}^{3}$ and let
$\tau>0$ and denote by $v$ the temperature of the material contained in
$\Omega$. Let $\underline{x}=(x_{1},x_{2},x_{3})\in\Omega$ and consider the
evolution problem

$\left\\{\begin{split}&\partial_{t}v(\underline{x},t)+{\rm
div}\,T(\underline{x})v(\underline{x},t)=0,\ \ \ \
(\underline{x},t)\in\Omega\times(0,\tau]\\\
&v(\underline{x},0)=f(\underline{x}),\ \ \ \underline{x}\in\Omega\\\
&v(\underline{x},t)=0,\ \ \underline{x}\in\partial\Omega\ \ \ \
t\in[0,\tau],\end{split}\right.$ (1)

where $f$ is a given datum and the heat flux for the nonhomogeneous material
contained in $\Omega$, is given by the vector differential operator:

$T(\underline{x})=a(\underline{x})\partial_{x_{1}}e_{1}+b(\underline{x})\partial_{x_{2}}e_{2}+c(\underline{x})\partial_{x_{3}}e_{3},\
\ \ \underline{x}\in\Omega.$ (2)

We determine the conditions on the coefficients $a$, $b$,
$c:\Omega\to\mathbb{R}$ under which the operator $T(\underline{x})$ generates
the fractional powers $P_{\alpha}(T(\underline{x}))$ of $T(\underline{x})$,
for $\alpha\in(0,1)$. The vector part of $P_{\alpha}(T(\underline{x}))$ of
$T(\underline{x})$ is defined to be the nonlocal Fourier’s law associated with
$T(\underline{x})$.

The second step of FSQ-construction generates Fueter or monogenic functions
and the spectral theory on the monogenic spectrum. We highlight some
references for the research directions in this area:

  * •

Monogenic spectral theory and applications [69]. Here one can also find the
relations of the monogenic functional calculus with the Taylor functional
calculus and the Weyl functional calculus see also some of the original
contributions [64, 65, 66, 68, 75].

  * •

Harmonic analysis in higher dimension, singular integrals and Fourier
transform see the recent book [77].

  * •

Algebraic Analysis of Dirac systems [42].

  * •

The theory of spinor valued function [50].

  * •

Boundary value problems treated with quaternionic techniques [60].

  * •

The extension of Schur analysis in the Fueter setting and related topics [8,
9, 10].

  * •

Dirac operator on manifolds and spectral theory [52, 61].

We conclude by saying that before the recent works on slice hyperholomorphic
functions, this function theory was simply seen as an intermediate step in the
Fueter-Sce-Qian’s construction. The literature on hyperholomorphic function
theories and related spectral theories is nowadays very large. For the
function theory of slice hyperholomorphic functions the main books are [6, 44,
46, 54, 58], while for the spectral theory on the $S$-spectrum we mention the
books [5, 23, 24, 46]. For the Fueter and monogenic function theory and
related topics see the books [15, 42, 50, 61, 62, 69, 78].

##  2\. Spectral theory in the complex setting

In this section we discuss what is a functional calculus of a single operator
on a Banach space and also for the case of several operators. When we consider
a closed linear operator $A$ with domain $\mathcal{D}(A)\subset X$, where $X$
is a Banach space, the resolvent set $\rho(A)$ of $A$ is defined as

$\rho(A)=\\{\lambda\in\mathbb{C}\ |\ (\lambda\mathcal{I}-A)^{-1}\in B(X)\\}$

where $B(X)$ in the space of all bounded linear operators on $X$ and the
spectrum of $A$ is the set $\sigma(A)=\mathbb{C}\setminus\rho(A)$ and for
$\lambda\in\rho(A)$ the map $\lambda\to(\lambda\mathcal{I}-A)^{-1}$ is called
the resolvent operator. We start with the following intuitive definition of
what is a functional calculus.

A functional calculus for a closed linear operator $A$ on complex Banach space
$X$ is a mathematical technique that allows to construct in a meaningful way
an operator $f(A)$ for any function $f$ in a certain class of functions
$\mathfrak{F}$ defined on sets that contain the spectrum $\sigma(A)$ of $A$.

The formulation in a meaningful way usually means that the functional calculus
is compatible with formally plugging $A$ into the function $f$, whenever this
is possible. That is, whenever $f(z)$ can be expressed by a formula so that
formally replacing $z$ by the operator $A$ yields an expression that is
meaningful, then $f(A)$ should correspond to this expression. Some examples
should clarify this idea:

  1. (a)

For any polynomial $p(z)=a_{n}z^{n}+\ldots+a_{1}z+a_{0}\in\mathfrak{F}$ with
$a_{\ell}\in\mathbb{C}$, the operator $p(A)$ should be given by
$p(A)=a_{n}A^{n}+\ldots+a_{1}A+a_{0}\mathcal{I}$.

  2. (b)

For any $\lambda\in\rho(A)$ with
$R_{\lambda}(z)=(\lambda-z)^{-1}\in\mathfrak{F}$, the functional calculus is
compatible with the resolvent operator at $\lambda$. That is, we have
$R_{\lambda}(A)=(\lambda\mathcal{I}-A)^{-1}$.

  3. (c)

For any rational function $r(z)=p(z)/q(z)\in\mathfrak{F}$ with polynomials
$p(z)=a_{n}z^{n}+\ldots+a_{1}z+a_{0}$ and
$q(z)=b_{n}z^{n}+\ldots+b_{1}z+b_{0}$ with $a_{\ell},b_{\ell}\in\mathbb{C}$,
the operator $r(A)$ should be given by $r(A)=p(A)q(A)^{-1}$, where
$p(A)=a_{n}A^{n}+\ldots+a_{1}A+a_{0}\mathcal{I}$ and
$q(A)=b_{m}A^{m}+\ldots+b_{1}A+b_{0}\mathcal{I}$.

  4. (d)

If $A$ is the infinitesimal generator of a strongly continuous group
$U_{A}(t),t\geq 0$ and $\exp(tz)\in\mathfrak{F}$, then $\exp(tA)=U_{A}(t)$.

Of course in the case of unbounded operators one has to pay attention to the
domain of the operators. Usually the class $\mathfrak{F}$ constitutes an
algebra, often even a Banach algebra, and the meaningfulness of the functional
calculus as described above follows from the compatibility of the functional
calculus with the algebraic operation. Precisely, a functional calculus
usually satisfies several (or all) of the following statements:

  1. (I)

The functional calculus is an algebra homomorphism that is
$(af+g)(T)=af(T)+g(T)$ for all $f,g\in\mathfrak{F}$ and all $a\in\mathbb{C}$.

  2. (II)

For $f$ and $g\in\mathfrak{F}$ such that $fg\in\mathfrak{F}$ we expect
$(fg)(A)=f(A)g(A)$.

  3. (III)

For $f(z)=1$, we have $f(A)=1(A)=\mathcal{I}$.

  4. (VI)

For the identity $f(z)=z$, we have $f(A)=z(A)=A$.

  5. (V)

If $X$ is a Hilbert space and both $f$ and $\overline{f}(z)=\overline{f(z)}$
belong to $\mathfrak{F}$, then $\overline{f}(A)=f(A)^{*}$.

  6. (VI)

If $\mathfrak{F}$ is normed, then the functional calculus defines a continuous
mapping into the space of bounded linear operators $\mathcal{B}(X)$ on $X$,
that is $\|f(A)\|_{\mathcal{B}(X)}\leq C\|f\|_{\mathfrak{F}}$.

We restrict ourselves to the case $\mathfrak{F}$ consists of functions that
are at least continuous and to the case that the topology on $\mathfrak{F}$ is
coarses than the topology of locally uniform convergence. In particular,
convergence in $\mathfrak{F}$ implies pointwise convergence.

For the measurable functional calculus the main statements of this section
hold true, but it needs different arguments to show them.

There are two main methods for defining a functional calculus, often both of
them can be applied in order to construct a specific functional calculus.

######  Method 2.1.

One considers a subalgebra $\mathfrak{F}_{0}$ of $\mathfrak{F}$ that is dense
in $\mathfrak{F}$ such that $f(A)$ can be defined easily for any
$f\in\mathfrak{F}_{0}$ (for instance the set of polynomials or the set of
rational functions in $\mathfrak{F}$). If $f\in\mathfrak{F}$ is arbitrary, one
chooses an approximating sequence $(f_{n})_{n\in\mathbb{N}}$ in
$\mathfrak{F}_{0}$ for $f$ and defines

$f(A):=\lim_{n\to+\infty}f_{n}(A).$

######  Method 2.2.

If any $f\in\mathfrak{F}$ admits an integral representation

$f(z)=\int K(\xi,z)f(\xi)\,d\mu(\xi),$

and formally replacing $z$ by $A$ in $K(\xi,z)$ yields a meaningful operator
$K(\xi,A)$, then one may define

$f(A):=\int K(\xi,A)f(\xi)\,d\mu(\xi).$

An example for 2.1 is the continuous functional calculus. With 2.2 we define
for example the Riesz-Dunford-functional calculus or the Philips functional
calculus.

There is also another concept behind the notion of functional calculus: the
operator $f(A)$ should be defined by letting $f$ act on the spectral values of
$A$. In particular, this means that

$Ax=\lambda x\qquad\Longrightarrow\qquad f(A)x=f(\lambda)x,\ \ \ {\rm for}\ \
x\in X.$ (3)

This idea is usually not to much emphasized in the complex setting when one
introduces and explains the concept of a functional calculus and we shall see
here in the sequel the reason way this happens. However, it is this relation
that explains why functional calculi are the fundamental techniques for
investigating linear operators. If it doesn’t hold, then a functional calculus
does not provide any information about the operator even though it generates
functions of operators.

Interestingly enough a deep difference between the theory of complex and the
theory of quaternionic linear operators (or more in general for
hyperholomorphic spectral theories) is revealed here so that in the latter,
the relation (3) needs to be addressed explicitly.

Let us start our considerations by justifying the importance of the relation
(3). We therefore recall the easiest result that is shown by an application of
a functional calculus.

######  Theorem 2.3.

Let $A\in\mathbb{C}^{m\times m}$. Then $A$ has an eigenvalue and for any
polynomial $p\in\mathbb{C}[n]$ with $p(A)=0$, the set of eigenvalues of $A$ is
contained in the set of roots of $p$.

It is obvious that even for the above, very easy, and fundamental result, the
fact that the polynomial functional calculus satisfies the relation (3) is
crucial.

The fact that the relation (3) trivially holds true for any known functional
calculus in the complex setting is shown in the next two results, and this is
the reason for which it is not explicitly mentioned.

######  Theorem 2.4.

Let $\Phi:\mathfrak{F}\to\mathcal{B}(X)$ be a functional calculus for an
operator $A$ defined via method 2.1. If (3) holds true for any function in
$\mathfrak{F}_{0}$, then the functional calculus $\Phi$ is compatible with
(3). This is in particular the case if $\mathfrak{F}_{0}$ consists of
polynomials or rational functions.

######  Theorem 2.5.

Let $\Phi:\mathfrak{F}\to\mathcal{B}(X)$ be a functional calculus for an
operator $A$ defined via method 2.2. If (3) holds true for $K(\lambda,\cdot)$
for any $\lambda$, then the functional calculus $\Phi$ is compatible with (3).

We now recalling that the spectral theorem works as a functional calculus. In
the finite dimensional case when we pick an $n\times n$ matrix $A=(a_{i,j})$
for $i,j=1,\ldots n$ of complex numbers such that $\overline{a_{i,j}}=a_{j,i}$
for all $i,j=1,\ldots,n$. The spectrum $\sigma(A)$ consists of eigenvalues of
$A$, that is, complex numbers $\lambda$ for which the equation $Av=\lambda v$
has a nonzero vector $v\in\mathbb{C}^{n}$ as a solution. The hermitian matrix
$A$ has a unique decomposition as a finite sum

$A=\sum_{j=1}^{n}\lambda_{j}E_{\lambda_{j},A}$

where $E_{\lambda_{j},A}$ is the orthogonal projection onto the eigenspace of
the eigenvalue $\lambda_{j}$. In the case of bounded selfadjoint (or more in
general normal operators) operators $A$ acting in Hilbert space the spectral
theorem is the most important tool for the complete description of such
operators, in fact we have

$A=\int_{\sigma(A)}\lambda dE_{\lambda;A}$

with respect to a spectral measure $E_{\lambda;A}$ associated with $A$. From
the spectral theorem we can define $f(A)$ by

$f(A)=\int_{\sigma(A)}f(\lambda)dE_{\lambda;A}$

for any continuous (but also bounded Borel measurable) function
$f:\sigma(A)\to\mathbb{C}$. The mapping $f\to f(A)$ is an algebra homomorphism
into the space of bounded linear operators. The spectral theorem can be
generalized to the case of $n$-tuple of commuting bounded selfadjoint
operators $(A_{1},\ldots,A_{n})$, as

$f(A_{1},\ldots,A_{n})=\int_{\sigma(A_{1},\ldots,A_{n})}f(\lambda)dE_{\lambda;A_{1},\ldots,A_{n}}$

is valid for the joint spectral measure $E_{\lambda;A_{1},\ldots,A_{n}}$
associated with $A_{1},\ldots,A_{n}$. The joint spectrum of
$A_{1},\ldots,A_{n}$ in $\mathbb{R}^{n}$ is the support of
$E_{\lambda;A_{1},\ldots,A_{n}}$ and
$f:\sigma(A_{1},\ldots,A_{n})\to\mathbb{C}$ is any bounded Borel measurable
function. The theorem holds more in general for unbounded normal operators but
one has to pay attention to the definitions of commutativity in this case, see
the book [80].

In the case we work in a Banach space the most natural way to define functions
of bounded (and also of unbounded) operators is the Riesz-Dunford functional
calculus

$f(A)=\frac{1}{2\pi i}\int_{C}(\lambda\mathcal{I}-A)^{-1}f(\lambda)d\lambda$

which holds for all holomorphic functions $f$ defined in a neighborhood of
$\sigma(A)$ in the complex plane. The simple closed contour $C$ surrounds
$\sigma(A)$ and is contained in the domain of the function $f$. There is a
natural generalization of Riesz-Dunford functional calculus for $n$-tuples of
bounded operators $A_{1},\ldots,A_{n}$ as

$f(A_{1},\ldots,A_{n})=\frac{1}{(2\pi
i)^{n}}\int_{C_{1}}\ldots\int_{C_{n}}(\lambda_{1}\mathcal{I}-A_{1}1)^{-1}\ldots(\lambda_{n}\mathcal{I}-A_{n})^{-1}f(\lambda_{1},\ldots,\lambda_{n})d\lambda$

where $d\lambda=d\lambda_{1}\cdots d\lambda_{n}$ and $f$ is any holomorphic
function in a neighborhood of $\sigma(A_{1})\times\ldots\times\sigma(A_{n})$
in $\mathbb{C}^{n}$. For each $j=1,\ldots,n$ the simple closed contour $C_{j}$
surrounds $\sigma(A_{j})$ and $C_{1}\times\ldots\times\mathbb{C}_{n}$ is
contained in the domain of $f$ in $\mathbb{C}^{n}$. Also when the operators
$A_{1},\ldots,A_{n}$ do not commute with each other, the functional calculus
makes sense with any change in the operator ordering of the function

$(\lambda_{1},\ldots,\lambda_{n})\mapsto(\lambda_{1}\mathcal{I}-A_{1}1)^{-1}\ldots(\lambda_{n}\mathcal{I}-A_{n})^{-1}\
\ \ {\rm in}\ \ \
\mathbb{C}^{n}\setminus(\sigma(A_{1})\times\ldots\sigma(A_{n})).$

For non commuting operators things are in general more complicated and we will
not enter into the details here.

The material discussed in this section can be found is several classical
books.

Some remarks in view of the hyperholomorphic spectral theories.

(I) In operator theory on the $S$-spectrum, the statements of Theorems 2.4 and
2.5 still hold true but just for a subclass of functions. The conditions that
the functions in the dense subspace $\mathfrak{F}_{0}$ resp. the kernel
$K(\lambda,A)$ satisfy (3) is not true in this setting. It is neither
satisfied by the $S$-resolvent operator, nor by the $F$-resolvent operator,
nor by slice hyperholomorphic rational functions with non-real coefficients.
In particular it is not satisfied, whenever the left-linear structure of the
space has a prominent role.

(II) In general the definitions of hyperholomorphic functional calculi
($S$-functional calculus, the $F$-functional calculus, the monogenic
functional calculus) the product rule, the composition rule or the spectral
mapping theorem do not hold. One needs additional arguments to show that
functional calculi based on the $S$-spectrum satisfy (3) at least for a
subclass of functions namely, the class of intrinsic functions and for these
functions, the problems mentioned before do not occur.

##  3\. Spectral theories in the hyperholomorphic setting

At the beginning of the last century several authors started the study of
hyperholomorphic functions and the most popular class of functions are
nowadays called Fueter (or Cauchy-Fueter) regular functions in the case of the
quaternions and monogenic functions (or functions in the kernel of the Dirac)
for Clifford algebra setting. The second class of hyperholomorphic functions
have been developed more recently, just at the beginning of this century, and
different definitions are possible even though they are not totally
equivalent.

In the following we will discuss mainly the implications of the Fueter-Sce-
Qian construction in the Clifford setting, the quaternionic setting is similar
and we will use it for the fractional powers of vector operators in the last
section of this paper.

Let $\mathbb{R}_{n}$ be the real Clifford algebra over $n$ imaginary units
$e_{1},\ldots,e_{n}$ satisfying the relations $e_{\ell}e_{m}+e_{m}e_{\ell}=0$,
$\ell\not=m$, $e_{\ell}^{2}=-1.$ An element in the Clifford algebra will be
denoted by $\sum_{A}e_{A}x_{A}$ where
$A=\\{\ell_{1}\ldots\ell_{r}\\}\in\mathcal{P}\\{1,2,\ldots,n\\},\ \
\ell_{1}<\ldots<\ell_{r}$ is a multi-index and
$e_{A}=e_{\ell_{1}}e_{\ell_{2}}\ldots e_{\ell_{r}}$, $e_{\emptyset}=1$. A
point $(x_{0},x_{1},\ldots,x_{n})\in\mathbb{R}^{n+1}$ will be identified with
the element
$x=x_{0}+\underline{x}=x_{0}+\sum_{j=1}^{n}x_{j}e_{j}\in\mathbb{R}_{n}$ called
paravector and the real part $x_{0}$ of $x$ will also be denoted by
$\mathrm{Re}(x)$. The imaginary part of $x$ is defined by ${\rm
Im}(x)=x_{1}e_{1}+\ldots+x_{n}e_{n}$ and for the sake of simplicity we also
use the notation $\underline{x}$ for ${\rm Im}(x)$. The conjugate of $x$ is
denoted by $\overline{x}=x_{0}-{\rm Im}(x)$ and the Euclidean modulus of $x$
is given by $|x|^{2}=x_{0}^{2}+\ldots+x_{n}^{2}$. The sphere of purely
imaginary paravectors with modulus $1$, is defined by

$\mathbb{S}=\\{x=e_{1}x_{1}+\ldots+e_{n}x_{n}\ |\
x_{1}^{2}+\ldots+x_{n}^{2}=1\\}.$

The element $I\in\mathbb{S}$ are such that $I^{2}=-1$ so we will denote the
complex place with imaginary unit $I$ by $\mathbb{C}_{I}$. For this reason the
elements of $\mathbb{S}$ are also called imaginary units. Given a non-real
paravector $x=x_{0}+{\rm Im}(x)=x_{0}+J_{x}|{\rm Im}(x)|$, $J_{x}:={\rm
Im}(x)/|{\rm Im}(x)|\in\mathbb{S}$, we can associate to it the sphere defined
by

$[x]=\\{x_{0}+J|{\rm Im}(x)|\ \ |\ \ J\in\mathbb{S}\\}.$

The set of quaternions will be denoted by $\mathbb{H}$ and the above
definitions adapts in this setting in a natural way.

######  Definition 3.1.

Let $U\subseteq\mathbb{R}^{n+1}$ (or $U\subseteq\mathbb{H}$ ). We say that $U$
is axially symmetric if, for every $u+Iv\in U$, all the elements $u+Jv$ for
$J\in\mathbb{S}$ are contained in $U$.

For operator theory the most appropriate definition of slice hyperholomorphic
functions is the one that comes from the Fueter-Sce-Qian mapping theorem
because it allows to define functions on axially symmetric open sets.

######  Definition 3.2.

Let $U\subseteq\mathbb{R}^{n+1}$ be an axially symmetric open set and let
$\mathcal{U}\subseteq\mathbb{R}\times\mathbb{R}$ be such that $x=u+Jv\in U$
for all $(u,v)\in\mathcal{U}$. We say that a function $f:U\to\mathbb{R}_{n}$
of the form

$f(x)=f_{0}(u,v)+Jf_{1}(u,v)$

is left slice hyperholomorphic if $f_{0}$, $f_{1}$ are $\mathbb{R}_{n}$-valued
differentiable functions such that

$f_{0}(u,v)=f_{0}(u,-v),\ \ \ f_{1}(u,v)=-f_{1}(u,-v)\ \ \ {\rm for\ all}\ \
(u,v)\in\mathcal{U}$

and if $f_{0}$ and $f_{1}$ satisfy the Cauchy-Riemann system

$\partial_{u}f_{0}-\partial_{v}f_{1}=0,\ \ \ \ \
\partial_{v}f_{0}+\partial_{u}f_{1}=0.$

The above definition adapt naturally to the quaternionic setting. Since we
will restrict just to left slice hyperholomorphic function on $U$ we introduce
the symbol ${SH}_{L}(U)$ to denote them. The subset of intrinsic functions
consist of those slice hyperholomophic functions such that $f_{0}$, $f_{1}$
are real-valued and is denoted by $N(U)$. We recall that right slice
hyperholomorphic functions are of the form

$f(x)=f_{0}(u,v)+f_{1}(u,v)J$

where $f_{0}$, $f_{1}$ satisfy the above conditions.

######  Definition 3.3 (Monogenic functions).

Let $f:U\to\mathbb{R}_{n}$ be a continuously differentiable function defined
on an open subset $U\subseteq\mathbb{R}^{n+1}$. We say that $f$ is (left)
monogenic on $U$, if

$Df(x)=0$

where $D$ is the Dirac operator defined by

$D=\partial_{x_{0}}+\sum_{j=1}^{n}e_{j}\partial_{x_{j}}.$

The definition of slice hyperholomorphic functions and of monogenic functions
can be seen as to two steps in the Fueter-Sce- Qian constructions to extend
holomorphic functions to dimension greater than one for the vector-valued
functions (quaternionic or Clifford valued-functions).

In fact, starting from holomorphic functions, R. Fueter in 1935, see [53],
showed an interesting way to generate Cauchy-Fueter regular functions. More
then 20 years later in 1957 M. Sce, see [79], extended this result in a very
pioneering and general way that includes Clifford algebras, see the English
translation of his works in hypercomplex analysis with commentaries collected
in the recent book [30].

In the original construction of R. Fueter the holomorphic functions are
defined on open sets of the upper half complex plane. This condition can be
relaxed by taking function

$g(z)=g_{0}(u,v)+ig_{1}(u,v),\ \ \ z=x+iy$

defined in a set $D\subseteq\mathbb{C}$, symmetric with respect to the real
axis such that

$g_{0}(u,-v)=g_{0}(u,v)\ \ \ {\rm and}\ \ \ g_{1}(u,-v)=-g_{1}(u,v)$

namely if $g_{0}$ and $g_{1}$ are, respectively, even and odd functions in the
variable $v$. Additionally the pair $(g_{0},g_{1})$ satisfies the Cauchy-
Riemann system. The above remark holds also for M. Sce’s theorem that we state
in the following for Clifford algebras.

######  Theorem 3.4 (Sce [79]).

Consider the Euclidean space $\mathbb{R}^{n+1}$ whose elements are identified
with paravectors $x=x_{0}+\underline{x}$. Let
$\tilde{f}(z)=f_{0}(u,v)+if_{1}(u,v)$ be a holomorphic function defined in a
domain (open and connected) $D$ in the upper-half complex plane and let

$\Omega_{D}=\\{x=x_{0}+\underline{x}\ \ |\ \ (x_{0},|\underline{x}|)\in D\\}$

be the open set induced by $D$ in $\mathbb{R}^{n+1}$. The following map

$f(x)=T_{FS1}(\tilde{f}):={\color[rgb]{0,0,0}f_{0}(x_{0},|\underline{x}|)+\frac{\underline{x}}{|\underline{x}|}f_{1}(x_{0},|\underline{x}|)}$

takes the holomorphic functions $\tilde{f}(z)$ and induces the Clifford-valued
function $f(x)$. Then the function

$\breve{f}(x):={\color[rgb]{0,0,0}T_{FS2}}\Big{(}{\color[rgb]{0,0,0}f_{0}(x_{0},|\underline{x}|)+\frac{\underline{x}}{|\underline{x}|}f_{1}(x_{0},|\underline{x}|)}\Big{)},$

where $T_{FS2}:=\Delta_{n+1}^{\frac{n-1}{2}}$ and $\Delta_{n+1}$ is the
laplacian in $n+1$ dimensions, is in the kernel of the Dirac operator, i.e.,

$D\breve{f}(x)=0\ \ \ {\rm on}\ \ \Omega_{D}.$

The case in which the operator $\Delta_{n+1}^{\frac{n-1}{2}}$ has a fractional
index has been treated by T. Qian in [74]. Observe that for the Fueter’s
theorem the operator $T_{FS2}$ is equal to the laplacian $\Delta$ in $4$
dimensions. Further developments can be found in [71, 72, 73] see also the
survey [76]. We can summarize the Fueter-Sce contractions as follows. Denoting
by $\mathcal{O}(D)$ the set of holomorphic functions on $D$, by
${N(\Omega_{D})}$ the set of induced functions on $\Omega_{D}$ (which turn out
to be intrinsic slice hyperholomorphic functions) and by $AM(\Omega_{D})$ the
set of axially monogenic functions on $\Omega_{D}$ the Fueter-Sce construction
can be visualized by the diagram:

$\begin{CD}{\color[rgb]{0,0,0}\mathcal{O}(D)}@>{T_{FS1}}>{}>{\color[rgb]{0,0,0}N(\Omega_{D})}@>{\
\ T_{FS2}=\Delta^{(n-1)/2)}}>{}>{\color[rgb]{0,0,0}AM(\Omega_{D})},\end{CD}$

where $T_{FS1}$ denotes the first linear operator of the Fueter-Sce
construction and $T_{FS2}$ the second one. The Fueter-Sce mapping theorem
induces two spectral theories according to the two classes of hyperholomorphic
functions it generate.

Recently also the problem of construction the inversion of the maps that
appear in the Fueter-Sce-Qian extension has been treated, we mention the
papers [11, 12, 39, 40, 41, 31], while a different method to connect slise
monogenic and monogenic functions is via the Radon and dual Radon transform,
see [27].

######  Remark 3.5.

The theory of slice hyperholomorphic functions was somewhat abandoned until
2006 when G. Gentili and D. C. Struppa (inspired by C. G. Cullen [49])
introduced in [57] the notion of slice regular functions for the quaternions.
Further developments of the theory of slice regular functions were discussed
also in [25] and the above definition was extended by F. Colombo, I. Sabadini
and D.C. Struppa, in [45], (see also [47, 48, 32]) to the Clifford algebra
setting. Slice regular functions as defined in [57] and their generalization
to the Clifford algebra as in [45], called slice monogenic functions, possess
good properties on specific open sets that are called axially symmetric slice
domains. When it is not necessary to distinguish between the quaternionic case
and the Clifford algebra case we call these functions slice hyperholomorphic.
The extension slice hyperholomorphic functions on real alternative algebras
can be found in [59].

######  Remark 3.6.

It is also possible to define slice hyperholomorphic functions, as functions
in the kernel of the first order linear differential operator (introduced in
[26])

$Gf=\Big{(}|\underline{x}|^{2}\frac{\partial}{\partial
x_{0}}+\underline{x}\sum_{j=1}^{n}x_{j}\frac{\partial}{\partial
x_{j}}\Big{)}f=0,$

where $\underline{x}=x_{1}e_{1}+\ldots+x_{n}e_{n}$ . While, a forth way to
introduce slice hyperholomorphicity, done in 1998 by G. Laville and I.
Ramadanoff in the paper [63], is inspired by the Fueter-Sce-Qian mapping
theorem. They introduce the so called Holomorphic Cliffordian functions
defined by the differential equation $D\Delta^{m}f=0$ over
$\mathbb{R}^{2m+1}$, where $D$ is the Dirac operator. Observe that the
definition via the global operator $G$ requires less regularity of the
functions with respect to the definition in [63].

We now recall the hyperholomorphic Cauchy formulas that are the heart of the
hyperholomorphic spectral theories. It is important to remark that the
hypotheses of the following Cauchy formula are related to the Definition 3.2
of slice hyperholomorphic functions.

######  Theorem 3.7 (Cauchy formula for slice hyperholomorphic functions).

Let $U\subseteq\mathbb{R}^{n+1}$ be an axially symmetric open set such that
$\partial(U\cap\mathbb{C}_{I})$ is union of a finite number of continuously
differentiable Jordan curves, for every $I\in\mathbb{S}$. Let $f$ be an
$\mathbb{R}_{n}$-valued slice hyperholomorphic function on an open set
containing $\overline{U}$ and, for any $I\in\mathbb{S}$, we set $ds_{I}=-Ids$.
Then, for every $x\in U$, we have:

$f(x)=\frac{1}{2\pi}\int_{\partial(U\cap\mathbb{C}_{I})}S_{L}^{-1}(s,x)ds_{I}f(s),$
(4)

where the slice hyperholomorphic Cauchy kernel is given by

$S_{L}^{-1}(s,x)=-(x^{2}-2{\rm Re}(s)x+|s|^{2})^{-1}(x-\overline{s})$ (5)

and the value of the integral (4) depends neither on $U$ nor on the imaginary
unit $I\in\mathbb{S}$.

######  Remark 3.8.

In the paper [57] was introduced the notion of slice regularity, besides the
definition, the authors treated power series centered at the origin and some
consequences. Without any tools the Cauchy formula with slice hyperholomorphic
kernel and the representation formula were originally determine with the
following elementary considerations. To determine the slice hyperholomorphic
Cauchy kernel $S_{L}^{-1}(s,q)$ we observe that from the definition of slice
regularity its expansion

$S_{L}^{-1}(s,q):=\sum_{m=0}^{\infty}q^{m}s^{-1-m},\ \ |q|<|s|$

is true when $q$ and $s$ belong to the same complex plane $\mathbb{C}_{I}$,
for $I\in\mathbb{S}$. Then we ask ourself what is the closed form of the
series in the case $q$ and $s$ do not belong to the same complex plane
$\mathbb{C}_{I}$ observing that

$\Big{(}\sum_{m=0}^{\infty}q^{m}s^{-1-m}\Big{)}s-q\Big{(}\sum_{m=0}^{\infty}q^{m}s^{-1-m}\Big{)}=1$
(6)

is true also when $s$ and $q$ do not belong to the same complex plane
$\mathbb{C}_{I}$. In the quaternionic case it was observed that the inverse
$S$ of $S_{L}^{-1}(s,q)$ is the non trivial solution of the quaternionic
equation

$S^{2}+Sq-sS=0,$

which easily follows from (6). The unknown $S$ was determined using the
Niven’s Algorithm as it is shown in the historical Note 4.18.3 in the book
[46] and it gives

$S(s,q)=(q-\overline{s})^{-1}s(q-\overline{s})-q.$

Taking the inverse of $S(s,q)$ we have the Cauchy kernel defined in (5) for
the quaternions. This strategy to determine the Cauchy kernel shows that in
the Clifford setting the Cauchy kernel remains the same if we consider

$S_{L}^{-1}(s,x):=\sum_{m=0}^{\infty}x^{m}s^{-1-m},\ \ |x|<|s|$

where $x=x_{0}+x_{1}e_{1}+...+x_{n}e_{n}$ and
$s=x_{0}+s_{1}e_{1}+...+s_{n}e_{n}$ are paravectors. Moreover, observe that a
direct computation of the integral

$\frac{1}{2\pi}\int_{\partial(U\cap\mathbb{C}_{I})}S_{L}^{-1}(s,x)ds_{I}f(s)$

by computing the residues of the singularities of the kernel $S_{L}^{-1}(s,x)$
in the complex plane $\mathbb{C}_{I}$, gives:

$\begin{split}\frac{1}{2\pi}\int_{\partial(U\cap\mathbb{C}_{I})}S_{L}^{-1}(s,x)ds_{I}f(s)=\frac{1}{2}\Big{[}f(u+Jv)+f(u-Jv)\Big{]}+I\frac{1}{2}\Big{[}J[f(u-Jv)-f(u+Jv)]\Big{]},\end{split}$

choosing any $J\in\mathbb{S}$ for all $x=u+Iv\in U$. From here one can clearly
see the existence of the structure formula (or representation formula) for
slice monogenic functions, see for example [32] (or [33]):

$f(u+Iv)=\frac{1}{2}\Big{[}f(u+Jv)+f(u-Jv)\Big{]}+I\frac{1}{2}\Big{[}J[f(u-Jv)-f(u+Jv)]\Big{]}.$

The quaternionic setting is just a particular case and from these observations
started a full development of the theory of slice hyperholomorphic functions.

The second ingredient for our discussion in the following is the Cauchy
formula for monogenic functions.

######  Theorem 3.9 (Cauchy formula for monogenic functions).

Let $U\subset\mathbb{R}^{n+1}$ be an open set with smooth boundary $\partial
U$ and let $\eta(\omega)$ be the outer unit normal to $\partial U$ and
$dS(\omega)$ be the scalar element of surface area on $\partial U$. Let $f$ be
a monogenic function on an open set that contains $\overline{U}$ then

$f(x)=\int_{\partial U}G_{\omega}(x)\eta(\omega)f(\omega)dS(\omega)$

for every $x$ in $U$, where the monogenic Cauchy kernel is given by

$G_{\omega}(x):=\frac{1}{\sigma_{n}}\frac{\overline{\omega-x}}{|\omega-x|^{n+1}},\
\ \ x,\ \omega\in\mathbb{R}^{n+1},\ \ x\not=\omega$

and $\sigma_{n}:=2\pi^{\frac{n+1}{2}}/\Gamma\Big{(}\frac{n+1}{2}\Big{)}$ is
the volume of unit n-sphere in $\mathbb{R}^{n+1}$.

Before to introduce the basic fact on the hyperholomorphic spectral theories
we need some important considerations.

(I) Holomorphic functions of one complex variable and harmonic analysis are
strongly connected since the Cauchy-Riemann operator factorizes the Laplace
operator. The holomorphic functional calculus and the spectral theorem are
based on the same notion of spectrum.

(II) In order to restore the analogy with the holomorphic functional calculus
and the spectral theorem in the quaternionic setting we have to replace the
classical spectrum with the $S$-spectrum. In, fact the $S$-functional calculus
and the quaternionic spectral theorem are both based on the $S$-spectrum. The
Dirac operator factorizes the Laplace operator the monogenic functional
calculus, based on the monogenic spectrum, has applications in harmonic
analysis and in other related fields.

Let us consider a Banach space $V$ over $\mathbb{R}$ with norm $\|\cdot\|$. It
is possible to endow $V$ with an operation of multiplication by elements of
$\mathbb{R}_{n}$ which gives a two-sided module over $\mathbb{R}_{n}$ and by
$V_{n}$ we indicate the two-sided Banach module over $\mathbb{R}_{n}$ given by
$V\otimes\mathbb{R}_{n}$.

We start with the definition of a functional calculus for $(n+1)$-tuples of
not necessarily commuting operators using slice hyperholomorphic functions. So
we consider the paravector operator

$T=T_{0}+\sum_{j=1}^{n}e_{j}T_{j},$

where $T_{\mu}\in\mathcal{B}(V)$ for $\mu=0,1,...,n$, and where
$\mathcal{B}(V)$ is the space of all bounded $\mathbb{R}$-linear operators
acting on $V$. The notion of $S$-spectrum follows from the Cauchy formula of
slice hyperholomorphic functions and from some not trivial considerations on
the fact that we can replace in the Cauchy kernel $S^{-1}_{L}(s,x)$ the
paravector $x$ by the paravector operator $T$ also in the case the components
$(T_{0},T_{1},...,T_{n})$ of $T$ do not commute among themselves.

######  Remark 3.10.

We make a crucial observation which justifies the definition of $S$-spectrum
and of $S$-resolvent operator. With the procedure of Remark 3.8 it is natural
to replace the paravector $x$ by the paravector operator
$T=T_{0}+T_{1}e_{1}+...+T_{n}e_{n}$ with bounded not necessarily commuting
components $T_{\ell}$, $\ell=0,...,n$ in the Cauchy kernel series. We obtain

$\sum_{m=0}^{\infty}T^{m}s^{-1-m}=-(T^{2}-2{\rm
Re}(s)T+|s|^{2}\mathcal{I})^{-1}(T-\overline{s}\mathcal{I}),\ \ \|T\|<|s|.$

even though the components of $T$ do not commute. From this relation we
justify the definition of the $S$-resolvent operator and of the $S$-spectrum.
The quaternionic setting is just a particular case.

We have the following definition.

######  Definition 3.11 ($S$-spectrum).

Let $T\in\mathcal{B}(V_{n})$ be a paravector operator. We define the
$S$-spectrum $\sigma_{S}(T)$ of $T$ as:

$\sigma_{S}(T)=\\{s\in\mathbb{R}^{n+1}\ \ :\ \ T^{2}-2\,{\rm
Re}\,(s)T+|s|^{2}{\mathcal{I}}\ \ \ {\rm is\ not\ invertible\ in\
}\mathcal{B}(V_{n})\\}$

where $\mathcal{I}$ denotes the identity operator. The $S$-resolvent set of
$T$ is defined as

$\rho_{S}(T)=\mathbb{H}\setminus\sigma_{S}(T).$

######  Definition 3.12.

Let $T\in\mathcal{B}(V_{n})$ be a paravector operator and $s\in\rho_{S}(T)$.
We define the left $S$-resolvent operator as

$S_{L}^{-1}(s,T):=-(T^{2}-2{\rm
Re}\,(s)T+|s|^{2}{\mathcal{I}})^{-1}(T-\overline{s}{\mathcal{I}}).$ (7)

A similar definition can be given for the right resolvent operator.

######  Definition 3.13.

We denote by ${SH}^{L}_{\sigma_{S}(T)}$ the set of slice hyperholomorphic
functions defined on the axially symmetric set $U$ that contains the
$S$-spectrum of $T$.

A crucial result for the definition of the $S$-functional calculus is that
integral

${\frac{1}{2\pi}}\int_{\partial(U\cap\mathbb{C}_{I})}S_{L}^{-1}(s,T)\ ds_{I}\
f(s),\ \ \ \ \ {\rm for}\ \ \ \ \ f\in{SH}^{L}_{\sigma_{S}(T)}$ (8)

depends neither on $U$ nor on the imaginary unit $I\in\mathbb{S}$, so the
$S$-functional calculus turns out to be well defined.

######  Definition 3.14 ($S$-functional calculus).

Let $T\in{B}({V}_{n})$ and let $U\subset\mathbb{H}$ be as above. We set
$ds_{I}=-Ids$ and we define the $S$-functional calculus as

$f(T):={{1}\over{2\pi}}\int_{\partial(U\cap\mathbb{C}_{I})}S_{L}^{-1}(s,T)\
ds_{I}\ f(s),\ \ \ \ \ {\rm for}\ \ \ \ \ f\in{SH}^{L}_{\sigma_{S}(T)}.$ (9)

Observe that the definition of the $S$-functional calculus is very natural for
non commuting operators in noncommutative spectral theory. The heart of the
general version of the $S$-functional calculus can be found in the original
papers [1, 33, 36, 37] and its commutative version [35].

Warning. In the monogenic setting the natural functional calculus is for
vector operators that is when we set $T_{0}=0$ in the paravector operator
$T=T_{0}+\sum_{j=1}^{n}e_{j}T_{j}$. The reason will be clear in the sequel,
but to point out this fact we use the symbol $A=(A_{1},\ldots,A_{n})$ or
$A=\sum_{j=1}^{n}e_{j}A_{j}$ instead of $(T_{1},\ldots,T_{n})$ or
$T=\sum_{j=1}^{n}e_{j}T_{j}$.

Using the Cauchy integral formula for monogenic functions, we establish the
monogenic functional calculus for the $n$-tuple $A=(A_{1},\ldots,A_{n})$ of
bounded linear operators on a Banach space $X$ by substituting the $n$-tuple
$A$ for the vector $x\in\mathbb{R}^{n}$.

In the following for the monogenic functional calculus we limit ourselves to
the most simple case when $n$ is odd and the $n$-tuple
$A=(A_{1},\ldots,A_{n})$ of bounded linear operators commute among themselves.
Such restrictions can be removed but one needs to do further considerations.

######  Remark 3.15.

If $n$ is odd, A is a commutative $n$-tuple, that is, $A_{j}A_{k}=A_{k}A_{j}$
for $j,k=1,\cdots,n$, and each operator $A_{j}$ has real spectrum
$\sigma(A_{j})\subset\mathbb{R}$ for $j=1,\cdots,n$, then for suitable
$\omega\in\mathbb{R}^{n+1}$, the expression

$G_{\omega}(A):=\frac{1}{\sigma_{n}}\frac{\overline{\omega\mathcal{I}-A}}{|\omega\mathcal{I}-A|^{n+1}}$
(10)

makes sense as an element of $\mathcal{B}(V_{n})$ and it is called the
monogenic resolvent.

######  Remark 3.16.

For an even integer $m$ we have

$|\omega\mathcal{I}-A|^{-m}=\Big{(}\,\Big{(}\omega_{0}^{2}\mathcal{I}+\sum_{j=1}^{n}(\omega_{j}\mathcal{I}-A_{j})^{2}\Big{)}^{-1}\,\Big{)}^{m/2}$

and

$\overline{\omega\mathcal{I}-A}=\omega_{0}I-\sum_{j=1}^{n}(\omega_{j}\mathcal{I}-A_{j})e_{j}$

for $\omega=\omega_{0}+\sum_{j=1}^{n}\omega_{j}$. Observe that the operator

$\omega_{0}^{2}\mathcal{I}+\sum_{j=1}^{n}(\omega_{j}\mathcal{I}-A_{j})^{2}$

is invertible in $\mathcal{B}(V)$ for each $\omega_{0}\not=0$.

######  Definition 3.17 (Monogenic spectrum).

The function

$\omega\mapsto G_{\omega}(A),$

is defined on the set $\mathbb{R}^{n+1}\setminus(\\{0\\}\times\gamma(A))$
where

$\gamma(A)=\\{(\omega_{1},\ldots,\omega_{n})\ |\
\sum_{j=1}^{n}(\omega_{j}\mathcal{I}-A_{j})^{2}\ {\rm is\ not\ invertible\
in}\ \mathcal{B}(V)\\}$

is called the monogenic spectrum.

######  Definition 3.18 (The monogenic functional calculus).

Let $n$ be an odd number and let us assume that $A=(A_{1},\ldots,A_{n})$ is a
commutative $n$-tuple of bounded linear operators (that is
$A_{j}A_{k}=A_{k}A_{j}$ for $j,k=1,\cdots,n$), and each operator $A_{j}$ has
real spectrum $\sigma(A_{j})\subset\mathbb{R}$ for $j=1,\cdots,n$. If $f$ is a
monogenic function on an open set that contains
$\overline{U}\subset\mathbb{R}^{n+1}$ with $\gamma(A)\subset U$. Then we
define the monogenic functional calculus as

$f(A)=\int_{\partial U}G_{\omega}(A)\eta(\omega)f(\omega)dS(\omega)$

where $G_{\omega}(A)$ is the monogenic resolvent operator (10), $\eta(\omega)$
is the outer unit normal to $\partial U$ and $dS(\omega)$ is the scalar
element of surface area on $\partial U$.

In the case $m=2,4,6,...$ the operator $|\omega\mathcal{I}-A|^{-m}$ needs to
be defined in a suitable way. The direct formulation employs Taylor’s
functional calculus, but by using the plane wave decomposition of the Cauchy
kernel, the case of even n and noncommuting operators can be treated
simultaneously. For an $n$-tuple $(A_{1},....,A_{n})$ of commuting bounded
linear operators on a Banach space $V$ with real spectra, the nonempty compact
subset $\gamma(A)$ of $\mathbb{R}^{n}$ coincides with Taylor’s joint spectrum
defined in terms of the Koszul complex.

The Cauchy formula of slice hyperholomorphic functions allows to define the
notion of $S$-spectrum, while the Cauchy formula for monogenic functions
induces the notion of monogenic spectrum, as illustrated by the diagram:

$\begin{CD}{\color[rgb]{0,0,0}SH(U)}@>{T_{FS2}}>{}>{\color[rgb]{0,0,0}M(U)}\\\
@V{}V{}V@V{}V{}V\\\ {\color[rgb]{0,0,0}Slice\ Cauchy\
Formula}{\color[rgb]{0,0,0}Monogenic\ Cauchy\ Formula}\\\ @V{}V{}V@V{}V{}V\\\
{\color[rgb]{0,0,0}S-Spectrum}\ {\color[rgb]{0,0,0}Monogenic\ Spectrum}\\\
@V{}V{}V@V{}V{}V\\\ {\color[rgb]{0,0,0}S-Functional\ calculus}\
{\color[rgb]{0,0,0}Monogenic\ Functional\ Calculus}\end{CD}$

In the above diagram we have replaced the set of intrinsic functions $N$ by
the larger set of slice hyperholomorphic functions $SH$. This is clearly
possible because the map $T_{FS2}$ is the Laplace operator or its powers.

We finally recall that the quaternionic spectral theorem is based on the
$S$-spectrum and not on the monogenic spectrum. In 2015 (and published in
2016) the quaternionic spectral theorem for quaternionic normal operators was
finally proved, see [2] (see also [3]). Later on perturbation results of
quaternionic normal operators were proved in [16]. Beyond the spectral theorem
there are more recent developments in the direction of the characteristic
operator functions, see [5] and the theory of quaternionic spectral operators
was developed in [56].

Finally, we wish to give an idea of the structure of the quaternionic spectral
theorem. For a complete treatment see [24]. If $T\in\mathcal{B}(\mathcal{H})$
is a bounded normal quaternionic linear operator, on a quaternionic Hilbert
space $\mathcal{H}$, then there exist three quaternionic linear operators $A$,
$\mathfrak{J}$, $B$ such that $T=A+\mathfrak{J}B$, where $A$ is self-adjoint
and $B$ is positive, $\mathfrak{J}$ is an anti self-adjoint partial isometry
(called imaginary operator). Moreover, $A$, $B$ and $\mathfrak{J}$ mutually
commute.
There exists a unique spectral measure $E_{I}$ on
$\sigma_{S}(T)\cap\mathbb{C}_{I}^{+}$ so that for any slice continuous
intrinsic function $f=f_{0}+f_{1}I$ we have:

$\langle
f(T)x,y\rangle=\int_{\sigma_{S}(T)\cap\mathbb{C}_{I}^{+}}f_{0}(q)\,d\langle
E_{I}(q)x,y\rangle+\int_{\sigma_{S}(T)\cap\mathbb{C}_{I}^{+}}f_{1}(q)\,d\langle\mathfrak{J}E_{I}(q)x,y\rangle,\quad
x,y\in\mathcal{H}.$ (11)

This theorem extends to the case of unbounded operators as well and holds true
for a larger class of functions that are not necessarily continuous.

##  4\. Interaction of the hyperholomorphic spectral theories

Now we formulate the Fueter-Sce-Qian theorem in integral form and we use it to
define the $F$-functional calculus. This gives a version of the monogenic
functional calculus for $n$-tuples of commuting operators but it is based on
the $S$-spectrum instead of the monogenic spectrum. This calculus was
introduced in [38] and further investigated in [21, 34].

It is important to recall that the monogenic functional calculus is defined
for $n$-tuples of operators $A_{j}$, $j=1,...,n$ that have real spectrum
considered as operators $A_{j}:V\to V$ on the real Banach space $V$. The
$F$-functional calculus has advantages and disadvantages with respect to the
monogenic functional calculus. Precisely, the $F$-functional calculus allows
to consider a much larger class of operators because it does not require that
the spectrum of the operators $A_{j}$, $j=1,...,n$ has to be real. Moreover,
this calculus allows to consider paravector operators and not only vector
operators as the monogenic functional calculus imposes.

On the other hand, from the hyperholomorphic functions point of view the
$F$-functional calculus is less general with respect to the monogenic
functional calculus because it works for the subset of monogenic function
given by

$\breve{M}=\\{\breve{f}\ |\ \breve{f}(x)=\Delta^{\frac{n-1}{2}}f(x)\ {\rm
for}\ f\in SH(U)\\}.$

We now show how the Fueter-Sce mapping theorem provides an alternative way to
define the functional calculus for monogenic functions. The main idea is to
apply the Fueter-Sce operator $T_{FS2}$ to the slice hyperholomorphic Cauchy
kernel as illustrated by the diagram:

$\begin{CD}{SH(U)}{AM(U)}\\\ @V{}V{}V\\\ {Slice\ Cauchy\
Formula}@>{T_{FS2}}>{}>{Fueter-Sce\ theorem\ in\ integral\ from}\\\
@V{}V{}V@V{}V{}V\\\ {S-Functional\ calculus}{F-functional\ calculus}\end{CD}$

This method generates an integral transform, called the Fueter-Sce mapping
theorem in integral form, that allows to define the so called $F$-functional
calculus. This calculus uses slice hyperholomorphic functions and the
commutative version of the $S$-spectrum and now we show how it works. We point
out that the operator $T_{FS2}$ has a kernel and one has to pay attention to
this fact with the definition of the $F$-functional calculus, more details are
given in [24].

Now observe that one can apply the powers of the Laplace operators to both
sides of (4) so that we have

$\Delta^{h}f(x)=\frac{1}{2\pi}\int_{\partial(U\cap\mathbb{C}_{I})}\Delta^{h}S_{L}^{-1}(s,x)ds_{I}f(s).$

In general, it is not easy to compute $\Delta^{h}f$ and when we apply
$\Delta^{h}$ to the Cauchy kernel written in the form (5), we do not get a
simple formula. However, $S_{L}^{-1}(s,x)$ can be written in two equivalent
ways as follows.

######  Proposition 4.1.

Let $x$, $s\in\mathbb{R}^{n+1}$ (or in $\mathbb{H}$ in the quaternionic case)
be such that $x^{2}-2x{\rm Re}(s)+|s|^{2}\not=0$. Then the following identity
holds:

$\begin{split}S_{L}^{-1}(s,x)&=-(x^{2}-2x{\rm
Re}(s)+|s|^{2})^{-1}(x-\overline{s})=(s-\bar{x})(s^{2}-2{\rm
Re}(x)s+|x|^{2})^{-1}.\end{split}$ (12)

If we use the second expression for the Cauchy kernel we find a very simple
expression for $\Delta^{h}S_{L}^{-1}(s,x)$. In fact, we have:

######  Theorem 4.2.

Let $x$, $s\in\mathbb{R}^{n+1}$ be such that $x^{2}-2x{\rm
Re}(s)+|s|^{2}\not=0$. Let

$S_{L}^{-1}(s,x)=(s-\bar{x})(s^{2}-2{\rm Re}(x)s+|x|^{2})^{-1}$

be the slice monogenic Cauchy kernel and let
$\Delta=\sum_{i=0}^{n}\frac{\partial^{2}}{\partial x_{i}^{2}}$ be the Laplace
operator in the variables $(x_{0},x_{1},...,x_{n})$. Then, for $h\geq 1$, we
have:

$\Delta^{h}S_{L}^{-1}(s,x)=C_{n,h}\ (s-\bar{x})(s^{2}-2{\rm
Re}(x)s+|x|^{2})^{-(h+1)},$ (13)

where

$C_{n,h}:=(-1)^{h}\prod_{\ell=1}^{h}(2\ell)\prod_{\ell=1}^{h}(n-(2\ell-1)).$

The function $\Delta^{h}S^{-1}(s,x)$ is slice hyperholomorphic in $s$ for any
$h\in\mathbb{N}$ but is monogenic in $x$ if and only if $h=(n+1)/2$, namely if
and only if $h$ equals the Sce’s exponent. We define the kernel

$\begin{split}\mathcal{F}_{L}(s,x)&:=\Delta^{\frac{n-1}{2}}S_{L}^{-1}(s,x)=\gamma_{n}(s-\bar{x})(s^{2}-2{\rm
Re}(x)s+|x|^{2})^{-\frac{n+1}{2}},\end{split}$

where

$\gamma_{n}:=(-1)^{(n-1)/2}2^{(n-1)/2}(n-1)!\Big{(}\frac{n-1}{2}\Big{)}!$ (14)

which can be used to obtain the Fueter-Sce mapping theorem in integral form.

######  Theorem 4.3.

Let $n$ be an odd number. Let $f$ be a slice hyperholomorphic function defined
in an open set that contains $\overline{U}$, where $U$ is a bounded axially
symmetric open set. Suppose that the boundary of $U\cap\mathbb{C}_{I}$
consists of a finite number of rectifiable Jordan curves for any
$I\in\mathbb{S}$. Then, if $x\in U$, the function $\breve{f}(x)$, given by

$\breve{f}(x)=\Delta^{\frac{n-1}{2}}f(x)$

is monogenic and it admits the integral representation

$\breve{f}(x)=\frac{1}{2\pi}\int_{\partial(U\cap\mathbb{C}_{I})}\mathcal{F}_{L}(s,x)ds_{I}f(s),\
\ \ ds_{I}=ds/I,$ (15)

where the integral depends neither on $U$ nor on the imaginary unit
$I\in\mathbb{S}$.

In the sequel, we will consider bounded paravector operators $T$, with
commuting components $T_{\ell}\in\mathcal{B}(V)$ for $\ell=0,1,\ldots,n$. Such
subset of ${\mathcal{B}(V_{n})}$ will be denoted by $\mathcal{BC}^{\small
0,1}(V_{n})$. The $F$-functional calculus is based on the commutative version
of the $S$-spectrum given by

$\sigma_{S}(T)=\\{s\in\mathbb{R}^{n+1}\ \ :\ \
s^{2}\mathcal{I}-(T+\overline{T})s+T\overline{T}\ \ \ {\rm is\ not\
invertible\ in\ }\mathcal{B}(V_{n})\\}$

where the operator $\overline{T}$ is defined by

$\overline{T}=T_{0}-T_{1}e_{1}-\dots-T_{n}e_{n}.$

We observe that for historical reasons the commutative version of the
$S$-spectrum is sometimes called $F$-spectrum because it is used for the
$F$-functional calculus. So we define the $F$-resolvent operators.

######  Definition 4.4 ($F$-resolvent operators).

Let $n$ be an odd number and let $T\in\mathcal{BC}^{\small 0,1}(V_{n})$. For
$s\in\rho_{S}(T)$ we define the left $F$-resolvent operator by

$F_{L}(s,T):=\gamma_{n}(s{I}-\overline{T})(s^{2}\mathcal{I}-(T+\overline{T})s+T\overline{T})^{-\frac{n+1}{2}},$
(16)

and the constants $\gamma_{n}$ are given in (14).

######  Definition 4.5 (The $F$-functional calculus for bounded operators).

Let $n$ be an odd number, let
$T=T_{0}+T_{1}e_{1}+\dots+T_{n}e_{n}\in\mathcal{BC}^{\small 0,1}(V_{n})$ and
set $ds_{I}=ds/I$, for $I\in\mathbb{S}$. Let ${SH}^{L}_{\sigma_{S}(T)}$ and
$U$ be as in Definition 3.13. We define

$\breve{f}(T):=\frac{1}{2\pi}\int_{\partial(U\cap\mathbb{C}_{I})}F_{L}(s,T)\,ds_{I}\,f(s).$
(17)

The definition of the $F$-functional calculus is well posed since the
integrals in (17) depends neither on $U$ and nor on the imaginary unit
$I\in\mathbb{S}$.

We conclude this section with some considerations on the hyperholomorphic
functional calculi to show the difference with respect the the complex case.

(I) The product rule holds for the $S$-functional calculus but just in the
case one of the two functions is intrinsic function. For the monogenic
functional calculus the product rule does not hold. This is due to the fact
the that product of two monogenic functions is not monogenic. For the $F$\-
functional calculus the product rule does not hold.

(II) Regarding the compatibility with polynomials we have: that the
$S$-functional calculus and the monogenic functional calculus are compatible
with slice hyperholomorphic polynomials and with monogenic polynomials,
respectively. For the $F$-functional calculus the compatibility with
polynomials holds if we consider

$\breve{P}(q)=\Delta P(q)$

where $\breve{P}(q)$ is a monogenic (or Fueter) and $P$ is a slice monogenic
polynomials

$q\to T\Rightarrow\breve{P}(q)\Rightarrow\breve{P}(T)$

(III) The spectral properties of the operator $T$ can be deduced by the
$S$-functional calculus and the quaternionic spectral theorem for which

$Tx=\lambda x\qquad\Longrightarrow\qquad f(T)x=f(\lambda)x$ (18)

when we use intrinsic functions.

##  5\. The $S$-spectrum approach to fractional diffusion problems

An important extension of the $S$-functional calculus to unbounded sectorial
operators is the $H^{\infty}$-functional calculus which is one of the ways to
define functions of unbounded operators. The $H^{\infty}$-functional calculus
has been used to define fractional powers of paravector operators and of
quaternionic linear operators that define fractional Fourier laws for
nonhomogeneous material in the theory of heat propagation. For the original
contributions on fractional powers of vector operators and of quaternionic
operators and of the $H^{\infty}$-functional calculus based on the
$S$-spectrum see [7, 20, 22]. For a systematic and recent treatment of
quaternionic spectral theory on the $S$-spectrum and the fractional diffusion
problems based on techniques on the $S$-spectrum see the books [23, 24]
published in 2019. Moreover, in the monograph [46], published 2011, one can
find also the foundations of the spectral theory on the $S$-spectrum for
$n$-tuples of noncommuting operators.

The theory on the fractional powers of quaternionic operators has been
recently applied to physical problems and in particular to generate the
fractional Fourier law for the heat equation that is collected in the papers
[19, 28, 29, 17, 18], here we give an overview of some of our results.

We denote by $\underline{x}:=(x_{1},x_{2},x_{3})$ a generic point in
$\mathbb{R}^{3}$ (we warn the reader that the symbol $\underline{x}$ is also
used to denote the imaginary part of a quaternion. It will be clear from the
context which is the meaning of the symbol $\underline{x}$ otherwise it will
be specified). Let $\Omega\subset\mathbb{R}^{3}$ bounded or unbounded domain
(with $\mathcal{C}^{1}$ boundary), the heat equation for nonhomogeneous
materials with the associated initial-boundary conditions descibes the
evolution of the heat. Precisely, we determine
$v:\Omega\times(0,\tau]\to\mathbb{R}$ (for $\tau>0$) such that

$\left\\{\begin{split}&\partial_{t}v(\underline{x},t)+{\rm
div}\,T(x)v(\underline{x},t)=0,\ \ \ \
(\underline{x},t)\in\Omega\times(0,\tau]\\\
&v(\underline{x},0)=f(\underline{x}),\ \ \ \underline{x}\in\Omega\\\
&v(\underline{x},t)=0,\ \ \underline{x}\in\partial\Omega\ \ \ \
t\in[0,\tau],\end{split}\right.$ (19)

where $f$ is a given datum and

$T(\underline{x})=\left(\begin{split}a_{1}(\underline{x})\partial_{x_{1}}\\\
a_{2}(\underline{x})\partial_{x_{2}}\\\
a_{3}(\underline{x})\partial_{x_{3}}\end{split}\right)$ (20)

where we suppose that the coefficients $a_{1}$, $a_{2}$,
$a_{3}:\overline{\Omega}\subset\mathbb{R}^{3}\to\mathbb{R}$ of $T$ belong to
$\mathcal{C}^{1}(\overline{\Omega})$ and they are not necessarily constant. We
also consider the heat equation for nonhomogeneous materials with Robin
boundary conditions, that consists in finding
$v:\Omega\times(0,\tau]\to\mathbb{R}$ (for $\tau>0$) such that

$\begin{cases}&\partial_{t}v(\underline{x},t)+{\rm
div}\,T(x)v(\underline{x},t)=0,\ \ \ \
(\underline{x},t)\in\Omega\times(0,\tau]\\\
&v(\underline{x},0)=f(\underline{x}),\ \ \ \underline{x}\in\Omega\\\
&b(\underline{x})v(\underline{x},t)+\sum_{\ell=1}^{3}a_{\ell}(\underline{x})n_{\ell}(\underline{x})\partial_{x_{\ell}}v(\underline{x},t)=0,\
\ (\underline{x},t)\in\partial\Omega\times(0,\tau],\end{cases}$ (21)

where $n=(n_{1},n_{2},n_{3})$ is the outward unit normal vector to
$\partial\Omega$, and $b:\partial\Omega\to\mathbb{R}$ is a given continuous
function. From the physical point of view, if we call $q(\underline{x},t)$ the
flux of the quantity described by $v(\underline{x},t)$ at the instant $t$, the
Fourier’s law states that $q(\underline{x},t)=T(x)(v(x,t))$.

The simpler case is when we consider $\Omega=\mathbb{R}^{3}$ and the
homogeneous diffusion problem is the consequence of Fourier’s law

$q(\underline{x},t)=-\nabla v(\underline{x},t)$

and of the conservation of the energy

$\partial_{t}v(\underline{x},t)+\operatorname{div}(q(\underline{x},t))=0.$

In this case $T$ is reduced to the negative gradient operator $T=-\nabla$ and
observing that $\operatorname{div}\circ\nabla=\Delta$ the fractional diffusion
model is obtained by replacing in the heat equation the Laplace operator by
its fractional powers

$(-\Delta)^{\alpha}u(\underline{x}):=\int_{\mathbb{R}^{3}}\frac{u(\underline{x})-u(\underline{y})}{|\underline{x}-\underline{y}|^{3+2\alpha}}\,dV(\underline{y}),\
\ \ {\rm for}\ \ \ \alpha\in(0,1).$

The fractional versions of the evolution equation in $\mathbb{R}^{3}$ is given
by

$\partial_{t}v(\underline{x},t)+(-\Delta)^{\alpha}v(\underline{x},t)=0.$ (22)

We observe that the fractional diffusion problem modifies both the Fourier’s
law and the conservation of the energy. Using the quaternionic functional
calculus we are able to define the fractional powers of vector operators, such
as $\nabla$ or $T$, in a bounded or unbounded domain $\Omega$ of
$\mathbb{R}^{3}$. Denoted, just for the moment, by $T^{\alpha}$ or
$\nabla^{\alpha}$ these fractional operators, we can define the fractional
diffusion problem (22) in the divergence form

$\partial_{t}v(\underline{x},t)+\operatorname{div}T^{\alpha}(x)v(\underline{x},t)=0,\
\ \ \ (\underline{x},t)\in\Omega\times(0,\tau].$ (23)

The boundary conditions one has to associate with the fractional evolution
problem are a very delicate issue and will not discussed here. We just mention
that the most natural boundary conditions are $v=0$ at infinity in the case
$\Omega=\mathbb{R}^{3}$.

######  Remark 5.1.

The boundary condition that we have to assume to generate the fractional
powers of $T$ are given by

$a(\underline{x})v(\underline{x},t)+\sum_{\ell=1}^{3}a^{2}_{\ell}(\underline{x})n_{\ell}(\underline{x})\partial_{x_{\ell}}v(\underline{x},t)=0,\
\ (\underline{x},t)\in\partial\Omega\times(0,\tau],$ (24)

where $a:\partial\Omega\to\mathbb{R}$ is a given continuous function. These
Robin-like boundary condition differs from the boundary condition in (21) by a
power two on the coefficients $a_{\ell}$’s. This is due to the fact that in
(21) the boundary condition rise from a physical condition on the flux through
the boundary, instead the boundary condition in (24) naturally comes from the
definition of $T^{\alpha}$. In any case, the two boundary conditions are
related when $T$ has coefficients that become constant on the boundary
$\partial\Omega$. Indeed, suppose that there exists a constant $\mu$ such that
the functions $a_{1}$, $a_{2}$, $a_{3}$ satisfy the conditions

$a_{1}(\underline{x})=a_{2}(\underline{x})=a_{3}(\underline{x})=\mu\ \ {\rm
for\ all}\ \ x\in\partial\Omega$ (25)

and the coefficients $a$ and $b$ are such that

$a(\underline{x})=\mu b(\underline{x})\ \ {\rm for\ all}\ \
\underline{x}\in\partial\Omega.$ (26)

Then the relation

$\sum_{\ell=1}^{3}a_{\ell}(\underline{x})n_{\ell}(\underline{x})\partial_{x_{\ell}}v(\underline{x})+b(\underline{x})v(\underline{x})=0$

is equivalent to

$\sum_{\ell=1}^{3}a^{2}_{\ell}(\underline{x})n_{\ell}(\underline{x})\partial_{x_{\ell}}v(\underline{x})+a(x)v(\underline{x})=0$

when $x\in\partial\Omega.$ For, using (25) and (26), we have

$\begin{split}\sum_{\ell=1}^{3}a^{2}_{\ell}(\underline{x})n_{\ell}(\underline{x})\partial_{x_{\ell}}+a(\underline{x})I&=\mu^{2}\sum_{\ell=1}^{3}n_{\ell}(\underline{x})\partial_{x_{\ell}}+\mu
b(\underline{x})I=\mu\Big{(}\sum_{\ell=1}^{3}a_{\ell}(\underline{x})n_{\ell}(\underline{x})\partial_{x_{\ell}}+b(\underline{x})I\Big{)}.\end{split}$

This kind of approach has several advantages.

  * •

It generates the fractional Fourier law from the Fourier law

$q(\underline{x},t)=T^{\alpha}(\underline{x})v(x,t)$ (27)

using the boundary conditions of the problem and without modifying the
conservation of energy law.

  * •

We can define the fractional heat equation for nonhomogeneous materials.

  * •

The fractional differential equation remains in the divergence form, so the
definition of a weak solution in obtained in a simple way.

  * •

It turns out that the approach through the quaternionic functional calculus
for defining the fractional heat equations is consistent with the classical
one. Indeed, we have that for any $\alpha\in(0,1)$

$2\operatorname{div}(\nabla^{\alpha})=(-\Delta)^{\frac{1}{2}+\frac{\alpha}{2}}.$

Now we present how the quaternions can be used to describe the vector
operators. Let $e_{\ell}$, for $\ell=1,2,3$, be an orthogonal basis for the
quaternions $\mathbb{H}$. We identify the vector operator $T$, described in
(20), with the quaternionic gradient operator with non constant coefficients

$\left(\begin{split}a_{1}(\underline{x})\partial_{x_{1}}\\\
a_{2}(\underline{x})\partial_{x_{2}}\\\
a_{3}(\underline{x})\partial_{x_{3}}\end{split}\right)\equiv\sum_{\ell=1}^{3}e_{\ell}T_{\ell},$
(28)

where the components $T_{\ell}$, $\ell=1,2,3$, are defined by
$T_{\ell}:=a_{\ell}(\underline{x})\partial_{x_{\ell}}$,
$\underline{x}\in\overline{\Omega}$. From the physical point of view the
operator $T$, defined in (28), can represent the Fourier law for
nonhomogeneous materials, but it can represent also different physical laws.
Our goal is to generate the fractional powers of $T$, that we denote with
$P_{\alpha}(T)$ for $\alpha\in(0,1)$, when the operators $T_{\ell}$, for
$\ell=1,2,3$ do not commute among themselves.

######  Remark 5.2.

The notation $P_{\alpha}(T)$, for the fractional powers of $T$ ,is more
precise with respect to the formal notation $T^{\alpha}$ is a sense that will
be clear just in the following with the precise definition. The formal
notation $T^{\alpha}$ is used in (23) and (24) (see (27)).

######  Definition 5.3.

The vector part of the fractional powers $P_{\alpha}(T)$ is called the
fractional Fourier law associated with $T$.

Now we present the general theory of the $S$-spectrum to construct the
fractional power of a quaternionic right linear operator. Let $V$ be a two-
sided quaternionic Banach space and $\mathcal{K}(V)$ the set of closed
quaternionic right linear operators on $V$. The Banach space of all bounded
right linear operators on $V$ is indicated by the symbol $\mathcal{B}(V)$ and
is endowed with the natural operator norm. For $T\in\mathcal{K}(V)$, we define
the operator associated with the $S$-spectrum as:

$\mathcal{Q}_{s}(T):=T^{2}-2\mathrm{Re}(s)T+|s|^{2}\mathcal{I},\qquad\text{for
$s\in\mathbb{H}$}$ (29)

where $\mathcal{Q}_{s}(T):dom(T^{2})\to V$, where $dom(T^{2})$ is the domain
of $T^{2}$. We define the $S$-resolvent set of $T$ as

$\rho_{S}(T):=\\{s\in\mathbb{H}:\mathcal{Q}_{s}(T)\ {\rm is\ invertible\ and\
}\mathcal{Q}_{s}(T)^{-1}\in\mathcal{B}(V)\\}$

and the $S$-spectrum of $T$ as

$\sigma_{S}(T):=\mathbb{H}\setminus\rho_{S}(T).$

The operator $\mathcal{Q}_{s}(T)^{-1}$ is called the pseudo $S$-resolvent
operator. For $s\in\rho_{S}(T)$, the left $S$-resolvent operator is defined as

$S_{L}^{-1}(s,T):=\mathcal{Q}_{s}(T)^{-1}\overline{s}-T\mathcal{Q}_{s}(T)^{-1}$
(30)

and the right $S$-resolvent operator is given by

$S_{R}^{-1}(s,T):=-(T-\mathcal{I}\overline{s})\mathcal{Q}_{s}(T)^{-1}.$ (31)

The fractional powers of $T$, denoted by $P_{\alpha}(T)$, are defined as
follows: for any $I\in\mathbb{S}$, for $\alpha\in(0,1)$ and
$v\in\operatorname{dom}(T)$ we set

$P_{\alpha}(T)v:=\frac{1}{2\pi}\int_{-I\mathbb{R}}S_{L}^{-1}(s,T)\,ds_{I}\,s^{\alpha-1}Tv,$
(32)

or

$P_{\alpha}(T)v:=\frac{1}{2\pi}\int_{-I\mathbb{R}}s^{\alpha-1}\,ds_{I}\,S_{R}^{-1}(s,T)Tv,$
(33)

where $ds_{j}=ds/I$. These formulas are a consequence of the quaternionic
version of the $H^{\infty}$-functional calculus based on the $S$-spectrum, see
the book [23] for more details. For the generation of the fractional powers
$P_{\alpha}(T)$ a crucial assumption on the $S$-resolvent operators is that,
for $s\in\mathbb{H}\setminus\\{0\\}$ with ${\rm Re}(s)=0$, the estimates

$\left\|S_{L}^{-1}(s,T)\right\|_{\mathcal{B}(V)}\leq\frac{\Theta}{|s|}\quad\text{and}\quad\left\|S_{R}^{-1}(s,T)\right\|_{\mathcal{B}(V)}\leq\frac{\Theta}{|s|},$
(34)

hold with a constant $\Theta>0$ that does not depend on the quaternion $s$. It
is important to observe that the conditions (34) assure that the integrals
(32) and (33) are convergent and so the fractional powers are well defined.

For the definition of the fractional powers of the operator $T$ we can use
equivalently the integral representation in (32) or the one in (33). Moreover,
they correspond to a modified version of Balakrishnan’s formula that takes
only spectral points with positive real part into account.

We want to apply the previous theory to the case $V:=L^{2}(\Omega,\mathbb{H})$
and $T\in\mathcal{K}(L^{2}(\Omega,\mathbb{H}))$ defined as in (28)
($\operatorname{dom}(T)\subset L^{2}(\Omega,\mathbb{H})$ is a densely subset).
A crucial problem is to determine the conditions on the coefficients $a_{1}$,
$a_{2}$, $a_{3}:\overline{\Omega}\subset\mathbb{R}^{3}\to\mathbb{R}$ such that
(32) and (33) are convergent. This problem is splitted into two problems

  * •

the first is to find appropriate conditions for the coefficients $a_{i}$’s
such that the purely imaginary quaternions are in the $S$-resolvent set
$\rho_{S}(T)$ (i.e. $\mathcal{Q}_{s}(T):\operatorname{dom}(T^{2})\to
L^{2}(\Omega,\mathbb{H})$ is invertible and bounded). This is a necessary
condition, see formulas (32) and (33). Then, since in the quaternionic case
the map $s\mapsto s^{\alpha}$, for $\alpha\in(0,1)$ is not defined for
$s\in(-\infty,0)$ and, unlike in the complex setting, it is not possible to
choose different branches of $s^{\alpha}$ in order to avoid this problem. For
this reason it is of great importance to assume the condition
$\mathrm{Re}(s)\geq 0$ that avoids the half real line $(-\infty,0]$.

  * •

The second crucial fact is to determine the conditions on the coefficients
$a_{i}$’s such that the estimate (34) for the $\mathcal{S}$-resolvent operator
of $T$ holds true.

Both these problems are solved by considering the following approach.
According to the initial condition of the boundary-value problems we invert
the operator $\mathcal{Q}_{s}(T)$ on the space $H^{1}_{0}(\Omega,\mathbb{H})$,
when we consider the Dirichlet boundary condition, and on the space

$\mathcal{H}:=\\{u\in
H^{1}(\Omega)|\,\int_{\Omega}u(\underline{x})\,dV(\underline{x})=0\,\\}$

when we consider the Robin boundary condition. The invertibility of
$\mathcal{Q}_{s}(T)$ is thus reduced to solve in a weak sense the following
two partial differential equations: given $F\in L^{2}(\Omega,\mathbb{H})$ and
$s\in\mathbb{H}\setminus\\{0\\}$ such that $\operatorname{Re}(s)=0$

$\begin{cases}&Q_{s}(T)(u)=\big{(}T^{2}-2s_{0}T+|s|^{2}\mathcal{I}\big{)}u(\underline{x})=F(\underline{x}),\
\ \ \underline{x}\in\Omega,\\\ &u\in H^{1}_{0}(\Omega,\mathbb{H}),\end{cases}$
(35)

and

$\begin{cases}&Q_{s}(T)(u)=\big{(}T^{2}-2s_{0}T+|s|^{2}\mathcal{I}\big{)}u(\underline{x})=F(\underline{x}),\
\ \ \underline{x}\in\Omega,\\\ &u\in\mathcal{H}(\Omega,\mathbb{H}),\\\
&b(\underline{x})v(\underline{x})+\sum_{\ell=1}^{3}a^{2}_{\ell}(\underline{x})n_{\ell}(\underline{x})\partial_{x_{\ell}}v(\underline{x})=0,\
\ \underline{x}\in\partial\Omega.\end{cases}$ (36)

To solve in the weak sense (35) (resp. (36)) means that for any $F\in
L^{2}(\Omega,\mathbb{H})$ we have to find $u_{F}\in
H^{1}_{0}(\Omega,\mathbb{H})$ (resp. $u_{F}\in\mathcal{H}(\Omega,\mathbb{H})$)
such that: for any $v\in H^{1}_{0}(\Omega,\mathbb{H})$ (resp.
$v\in\mathcal{H}(\Omega,\mathbb{H})$) we have

$\langle
Q_{s}(T)(u_{F}),v\rangle=(F,v)_{L^{2}}=\int_{\Omega}\overline{F}v\,dV(\underline{x}),$
(37)

where the angle-brackets means that $\mathcal{Q}_{s}(T)$ is applied to $u_{F}$
in the sense of distribution. If we solve (37), we can define

$Q_{s}(T)^{-1}(F):=u_{F}$

In order to solve (37) we apply the Lax-Milgram Lemma to the sesquilinear
form: $\langle Q_{s}(T)(u_{F}),v\rangle$. Thus it is crucial to prove an
explicit formula for the left hand side of (37). This formula can be deduced
from an arguments of integration by parts and using the Dirichlet boundary
condition for (35):

$\begin{split}&\langle
Q_{s}(T)(u),v\rangle=\sum_{\ell=1}^{3}\int_{\Omega}\overline{a_{\ell}(\underline{x})\partial_{x_{\ell}}(u(\underline{x}))}\,a_{\ell}(\underline{x})\partial_{x_{\ell}}(v(\underline{x}))\,dV(\underline{x})+\\\
&\frac{1}{2}\sum_{\ell=1}^{3}\int_{\Omega}\overline{\partial_{x_{\ell}}(u(\underline{x}))}\partial_{x_{\ell}}\left(a^{2}_{\ell}(\underline{x})\right)v(\underline{x})\,dV(\underline{x})+({\rm
Vect}(Q_{s}(T))u,v)_{L^{2}}\\\
&+|s|^{2}(u,v)_{L^{2}}(:=b_{s,1}(u,v))\end{split}$ (38)

or the Robin boundary condition for (36)

$\begin{split}&\langle
Q_{s}(T)(u),v\rangle=\sum_{\ell=1}^{3}\int_{\Omega}\overline{a_{\ell}(\underline{x})\partial_{x_{\ell}}(u(\underline{x}))}\,a_{\ell}(\underline{x})\partial_{x_{\ell}}(v(\underline{x}))\,dV(\underline{x})\\\
&+\frac{1}{2}\sum_{\ell=1}^{3}\int_{\Omega}\overline{\partial_{x_{\ell}}(u(\underline{x}))}\partial_{x_{\ell}}\left(a^{2}_{\ell}(\underline{x})\right)v(\underline{x})\,dV(\underline{x})+({\rm
Vect}(Q_{s}(T))u,v)_{L^{2}}\\\
&+\int_{\partial\Omega}a(\underline{x})\overline{u(\underline{x})}v(\underline{x})\,dS(\underline{x})+|s|^{2}(u,v)_{L^{2}}(=:b_{s,2}(u,v)).\end{split}$
(39)

In [19], [28], [29], [17] and [18] we found suitable conditions for the
coefficients $a_{i}$’s such that the two sesquilinear forms $b_{s,1}$ and
$b_{s,2}$ are coercive and continuous when $\Omega$ is bounded or unbounded.
In conclusion by the Lax-Milgram lemma we obtain the solvability of the
equation (37) (i.e. the invertibility of $\mathcal{Q}_{s}(t)$) and the
estimate (34) for the $\mathcal{S}$-resolvent operator. Thus the problem of
the convergence of (32) and (33) is solved. In the next three paragraphs we
summarize the conditions we found on the coefficients of $T$ to obtain the
convergence of (32) and (33) .

The fractional Fourier’s law in the problem (23) with $\Omega$ bounded. In the
paper [19] it was considered the commutative Fourier’s law $T_{com}$, that is
an operator of the form

$T_{com}=a_{1}(x_{1})\partial_{x_{1}}e_{1}+a_{2}(x_{2})\partial_{x_{2}}e_{2}+a_{3}(x_{3})\partial_{x_{3}}e_{3}$
(40)

where the real operators $a_{1}(x)\partial_{x_{1}}$,
$a_{2}(x_{2})\partial_{x_{2}}$ and $a_{3}(x_{3})\partial_{x_{3}}$ commute
among themselves. It has been shown that if the coefficients
$a_{\ell}:\overline{\Omega}\to\mathbb{R}$, for $\ell=1,2,3$ belong to
$\mathcal{C}^{1}(\overline{\Omega},\mathbb{R})$ and if $a_{\ell}$, for
$\ell=1,2,3$ are suitably large and the their derivative are suitably small
then the integrals in (32) and (33) are convergent.

In [28] we replace the commutative Fourier’s law $T_{com}$ by the more general
Fourier’s law

$T(x)=a_{1}(\underline{x})\partial_{x_{1}}e_{1}+a_{2}(\underline{x})\partial_{x_{2}}e_{2}+a_{3}(\underline{x})\partial_{x_{3}}e_{3}$
(41)

where now the real operators $a_{1}(\underline{x})\partial_{x_{1}}$,
$a_{2}(\underline{x})\partial_{x_{2}}e_{2}$ and
$a_{3}(\underline{x})\partial_{x_{3}}$ do not commute among themselves. In
this case the conditions for the existence of the fractional powers are more
complicated.

The main result is summarized in the following theorem (see for more details
Theorems $4.1$, $4.4$ and $4.5$ in [28]).

######  Theorem 5.4.

Let $\Omega$ be a bounded $\mathcal{C}^{1}$-domain in $\mathbb{R}^{3}$, let
$T=\sum_{i=1}^{3}a_{i}(\underline{x})\partial_{x_{i}}e_{i}$ with
$a_{i}\in\mathcal{C}^{1}(\overline{\Omega})$ for any $i=1,\,2,\,3$ and set

$F_{(a_{1},a_{2},a_{3})}:=\sum_{i=1}^{3}e_{i}\partial_{x_{i}}(a_{i}).$

Let $a_{1},a_{2},a_{3}\geq m>0$, and assume that

$\min\\{\inf_{\underline{x}\in\Omega}a_{1}^{2},\inf_{\underline{x}\in\Omega}a_{2}^{2},\inf_{\underline{x}\in\Omega}a_{3}^{2}\\}-(2\max\\{\sup_{\underline{x}\in\Omega}a_{1}^{2},\sup_{\underline{x}\in\Omega}a_{2}^{2},\sup_{\underline{x}\in\Omega}a_{3}^{2}\\})^{1/2}C_{\Omega}\|F_{(a_{1},a_{2},a_{3})}\|_{L^{\infty}}>0$
(42)

and

$1-2\|F_{(a_{1},a_{2},a_{3})}\|_{L^{\infty}}\Big{(}1+4C^{2}_{\Omega}\max\\{\sup_{\underline{x}\in\Omega}(1/a_{1}^{2}),\sup_{\underline{x}\in\Omega}(1/a_{2}^{2}),\sup_{\underline{x}\in\Omega}(1/a_{3}^{2})\\}\Big{)}>0,$
(43)

where $C_{\Omega}$ in the Poincaré constant of $\Omega$ and

$\|F_{(a_{1},a_{2},a_{3})}\|_{L^{\infty}}:=\sup_{\underline{x}\in\Omega}(|\partial_{x_{1}}(a_{1})|+|\partial_{x_{2}}(a_{2})|+|\partial_{x_{3}}(a_{3})|).$

Then for any $\alpha\in(0,1)$ and for any $v\in\operatorname{dom}(T)$, the
integrals (32) and (33) converge absolutely.

The sesquilinear form $b_{s,1}(u,v)$, defined in (38), associated with the
invertibility of the operator
$\mathcal{Q}_{s}(T):=T^{2}-2s_{0}T+|s|^{2}\mathcal{I}$, with homogeneous
Dirichlet boundary conditions, has to be considered with care. We summarized
in Remark 5.5 some considerations associated with $b_{s,1}(u,v)$.

######  Remark 5.5.

We point out some fact that appear in the application of the Lax–Milgram lemma
according to the dimension $n$ of $\Omega$.

(I) The quadratic form $b_{s,1}(u,v)$ associated to the operator
$\mathcal{Q}_{s}(T)$ is in general degenerate on
$H^{1}_{0}(\Omega,\mathbb{H})$.

(II) In dimension $n=3$ , when $\Omega$ is a $\mathcal{C}^{1}$ bounded set in
$\mathbb{R}^{3}$ and $a_{1}\not=0$, $a_{2}\not=0$, $a_{3}\not=0$, it turns out
that $b_{s,1}(u,v)$ is continuous and coercive under suitable conditions on
the coefficients of $\mathcal{Y}\times\mathcal{Y}$, where

$\mathcal{Y}:=\\{v\in H^{1}_{0}(\Omega,\mathbb{H})\ :\
v_{0}=v_{1}=v_{2}=v_{3}\\}$

is a closed subspace of $H^{1}_{0}(\Omega,\mathbb{H})$ and the $S$-resolvent
operators satisfy suitable growth conditions which ensure the existence of the
fractional powers.

(III) In dimension $n=2$, when $\Omega$ is a $\mathcal{C}^{1}$ bounded set in
$\mathbb{R}^{2}$ and $a_{1}\not=0$, $a_{2}\not=0$, it turns out that
$b_{s,1}(u,v)$ is continuous and coercive under suitable conditions on the
coefficients of $\mathcal{X}\times\mathcal{X}$, where

$\mathcal{X}:=\\{v\in H^{1}_{0}(\Omega,\mathbb{H})\ :\ \ v_{0}=v_{2}\ \ {\rm
and}\ \ v_{1}=v_{3}\\}$

is a closed subspace of $H^{1}_{0}(\Omega,\mathbb{H})$ and the $S$-resolvent
operators satisfy suitable growth conditions which ensure the existence of the
fractional powers.

(IV) If we consider the quadratic form in dimension $n=3$, that is when
$\Omega$ is a $\mathcal{C}^{1}$ bounded set in $\mathbb{R}^{3}$ and
$a_{1}\not=0$, $a_{2}\not=0$, $a_{3}=0$, then the quadratic form is not
coercive because of $a_{3}=0$. It seems that this case cannot be treated using
Lax–Milgram Lemma, but a suitable method for degenerate equations has to be
used.

(V) In [28] we proved, under more restrictive hypothesis on the coefficients
$a_{i}$’s, that the sesquilinear form $b_{s,1}(u,v)$ is continuous and
coercive in $H^{1}_{0}(\Omega,\mathbb{H})$.

(VI) From the physical point of view the case for $a_{1}\not=0$,
$a_{2}\not=0$, $a_{3}=0$, in dimension $n=3$ is the case in which the
conductivity is the direction $z$ goes to zero.

(VII) The proofs for the continuity and coercivity are similar in any
dimension, the estimate for the $S$-resolvent operators have some differences
according to the fact that we work in $\mathcal{Y}$ or in $\mathcal{X}$.

The fractional Fourier’s law in the problem (24) with $\Omega$ bounded.
Regarding the initial boundary value problem of Robin-type, the conditions on
the coefficients $a_{i}$’s depend on two positive constants which appear in
the following two inequalities:

  * •

for all $u\in H^{1}(\Omega,\mathbb{R})$ the following inequality holds:

$\|u\|_{H^{1/2}(\partial\Omega,\mathbb{R})}\leq
C_{\partial\Omega}\|u\|_{H^{1}(\Omega,\mathbb{R})}.$ (44)

where $C_{\partial\Omega}$ does not depend on $u$

  * •

for all $u\in H^{1}(\Omega,\mathbb{R})$ the following inequality holds:

$\left\|u-|\Omega|^{-1}\int_{\Omega}u(x)dx\right\|_{L^{2}(\Omega,\mathbb{R})}\leq
C_{P}\|\nabla u\|_{L^{2}(\Omega,\mathbb{R})},$ (45)

where $C_{P}$ does not depend on $u$.

We obtained in [18] the following result (see Theorems $4.4$, $5.1$ and $5.2$
in [18]).

######  Theorem 5.6.

Let $\Omega$ be a bounded domain in $\mathbb{R}^{3}$ with boundary
$\partial\Omega$ of class $\mathcal{C}^{1}$. Assume that
$a\in\mathcal{C}^{0}(\partial\Omega,\mathbb{R})$ and let $T$ be the operator
defined in (20) with coefficients $a_{1}$, $a_{2}$,
$a_{3}\in\mathcal{C}^{1}(\overline{\Omega},\mathbb{R})$. Define the following
constants:

$C_{T}:=\min_{\ell=1,2,3}\inf_{x\in\Omega}(a^{2}_{\ell}(x)),\quad
C_{T}^{\prime}:=\sum_{i,\ell=1}^{3}\|a_{\ell}\partial_{x_{\ell}}a_{i}\|_{\infty},\quad
K_{a,\,\Omega}:=C^{2}_{\partial\Omega}\|a\|_{\infty},$

where $\|\cdot\|_{\infty}$ denotes the sup norm and $C_{\partial\Omega}$ is
the constant in (44). Moreover, assume that

$C_{T}-C_{T}^{\prime}C_{P}-K_{a,\Omega}\Big{(}1+C_{P}^{2}\Big{)}>0\ \ \ and\ \
\ \ \ C_{T}>0,$ (46)

where $C_{P}$ is the constant in (45). Then for any $\alpha\in(0,1)$ and for
any $v\in\operatorname{dom}(T)$, the integrals (32) and (33) converge
absolutely.

######  Remark 5.7.

In [17] we treated the case of the operator $T$ with commutative coefficients
and of $\Omega$ bounded with Robin-type boundary condition.

The fractional Fourier’s law in the problem (23) with $\Omega$ unbounded.
Regarding the initial boundary value problem of Dirichlet-type for the
unbounded domains, we obtained in [18] the following result (see Theorems
$4.8$, $5.1$ and $5.2$ in [18]).

######  Theorem 5.8.

Let $\Omega$ be an unbounded domain in $\mathbb{R}^{3}$ with boundary
$\partial\Omega$ of class $\mathcal{C}^{1}$. Let $T$ be the operator defined
in (20) with coefficients $a_{1}$, $a_{2}$,
$a_{3}\in\mathcal{C}^{1}(\overline{\Omega},\mathbb{R})$. Suppose that

$M:=\sum_{i,j=1}^{3}\|a_{i}\partial_{x_{i}}(a_{j})\|_{L^{3}(\Omega)}<+\infty$
(47)

and

$C_{T}:=\min_{\ell=1,2,3}\inf_{\underline{x}\in\Omega}(a^{2}_{\ell}(\underline{x}))>0,\
\ \ \ C_{T}-4M>0.$ (48)

Then for any $\alpha\in(0,1)$ and for any $v\in\operatorname{dom}(T)$, the
integrals (32) and (33) converge absolutely.

## References

  * [1] D. Alpay, F. Colombo, J. Gantner, S. Sabadini, A new resolvent equation for the S-functional calculus, J. Geom. Anal., 25 (2015), no. 3, 1939–1968.
  * [2] D. Alpay, F. Colombo, D.P. Kimsey, The spectral theorem for quaternionic unbounded normal operators based on the $S$-spectrum, J. Math. Phys., 57 (2016), no. 2, 023503, 27 pp.
  * [3] D. Alpay, F. Colombo, D.P. Kimsey, I. Sabadini, The spectral theorem for unitary operators based on the $S$-spectrum, Milan J. Math., 84 (2016), no. 1, 41–61.
  * [4] D. Alpay, F. Colombo, I. Sabadini, Slice Hyperholomorphic Schur Analysis, Operator Theory: Advances and Applications, 256. Birkhäuser/Springer, Cham, 2016. xii+362 pp.
  * [5] D. Alpay, F. Colombo, I. Sabadini, Quaternionic de Branges spaces and characteristic operator function, SpringerBriefs in Mathematics, Springer, Cham, 2020/21.
  * [6] D. Alpay, F. Colombo, I. Sabadini, Hilbert spaces of slice hyperholomorphic functions, Preprint 2020.
  * [7] D. Alpay, F. Colombo, T. Qian, I. Sabadini, The $H^{\infty}$ functional calculus based on the S-spectrum for quaternionic operators and for n-tuples of noncommuting operators, J. Funct. Anal., 271 (2016), no. 6, 1544–1584.
  * [8] D. Alpay, M. Shapiro, Reproducing kernel quaternionic Pontryagin spaces, Integral Equations Operator Theory, 50 (2004), no. 4, 431–476.
  * [9] D. Alpay, M. Shapiro, D. Volok, Rational hyperholomorphic functions in $R^{4}$, J. Funct. Anal., 221 (2005), no. 1, 122-149.
  * [10] D. Alpay, M. Shapiro, D. Volok, Reproducing kernel spaces of series of Fueter polynomials. Operator theory in Krein spaces and nonlinear eigenvalue problems, 19–45, Oper. Theory Adv. Appl., 162, Birkhäuser, Basel, 2006.
  * [11] D. Baohua, K. I. Kou, T. Qian, I. Sabadini, On the inversion of Fueter’s theorem, J. Geom. Phys., 108 (2016), 102–116.
  * [12] D. Baohua, K. I. Kou, T. Qian, I. Sabadini, The inverse Fueter mapping theorem for axially monogenic functions of degree k, J. Math. Anal. Appl. 476 (2019), 819–835.
  * [13] G. Birkhoff, J. von Neumann, The logic of quantum mechanics, Ann. of Math., 37 (1936), 823–843.
  * [14] N. Bourbaki, Éléments de mathématique. Fasc. XXXII. Théories spectrales. Chapitre I: Algèbres normées. Chapitre II: Groupes localement compacts commutatifs. (French) Actualités Scientifiques et Industrielles, No. 1332 Hermann, Paris 1967 iv+166 pp.
  * [15] F. Brackx, R. Delanghe, F. Sommen, Clifford analysis, Research Notes in Mathematics, 76. Pitman (Advanced Publishing Program), Boston, MA, 1982. x+308 pp.
  * [16] P. Cerejeiras, F. Colombo, U. Kähler, I. Sabadini, Perturbation of normal quaternionic operators, Trans. Amer. Math. Soc., 372 (2019), no. 5, 3257–3281.
  * [17] F. Colombo, D. Deniz-Gonzales, S. Pinton, Fractional powers of vector operators with first order boundary conditions, J. Geom. Phys., 151 (2020), 103618, 18 pp.
  * [18] F. Colombo, D. Deniz-Gonzales, S. Pinton, Non commutative fractional Fourier law in bounded and unbounded domains, Preprint 2020.
  * [19] F. Colombo and J. Gantner, Fractional powers of vector operators and fractional Fourier’s law in a Hilbert space, J. Phys. A, 51 (2018), 305201 (25pp).
  * [20] F. Colombo, J. Gantner, An application of the S-functional calculus to fractional diffusion processes, Milan J. Math., 86 (2018), no. 2, 225–303.
  * [21] F. Colombo, J. Gantner, Formulations of the F-functional calculus and some consequences, Proc. Roy. Soc. Edinburgh Sect. A, 146 (2016), no. 3, 509–545.
  * [22] F. Colombo, J. Gantner, Fractional powers of quaternionic operators and Kato’s formula using slice hyperholomorphicity, Trans. Amer. Math. Soc., 370 (2018), no. 2, 1045–1100.
  * [23] F. Colombo, J. Gantner, Quaternionic closed operators, fractional powers and fractional diffusion processes, Operator Theory: Advances and Applications, 274. Birkhäuser/Springer, Cham, 2019. viii+322 pp.
  * [24] F. Colombo, J. Gantner, D. P. Kimsey, Spectral theory on the S-spectrum for quaternionic operators, Operator Theory: Advances and Applications, 270. Birkhäuser/Springer, Cham, 2018. ix+356 pp.
  * [25] F. Colombo, G. Gentili, I. Sabadini, D.C. Struppa, Extension results for slice regular functions of a quaternionic variable, Adv. Math., 222 (2009), no. 5, 1793–1808.
  * [26] F. Colombo, O.J. Gonzalez-Cervantes, I. Sabadini, A nonconstant coefficients differential operator associated to slice monogenic functions, Trans. Amer. Math. Soc., 365 (2013), no. 1, 303–318.
  * [27] F. Colombo, R. Lavicka, I. Sabadini, V. Soucek, The Radon transform between monogenic and generalized slice monogenic functions, Math. Ann., 363 (2015), no. 3-4, 733–752.
  * [28] F. Colombo, S. Mongodi, M. Peloso, S. Pinton, Fractional powers of the non commutative Fourier’s laws by the S-spectrum approach, Math. Methods Appl. Sci., 42 (2019), no. 5, 1662–1686.
  * [29] F. Colombo, M. Peloso, S. Pinton, The structure of the fractional powers of the noncommutative Fourier law, Math. Methods Appl. Sci., 42 (2019), 6259–6276
  * [30] F. Colombo, I. Sabadini, D.C. Struppa, Michele Sce’s Works in Hypercomplex Analysis 1955-1973. A translation with commentaries, Birkhäuser, 2020, Hardcover ISBN 978-3-030-50215-7.
  * [31] F. Colombo, D. Pena Pena, I. Sabadini, F. Sommen, A new integral formula for the inverse Fueter mapping theorem, J. Math. Anal. Appl., 417 (2014), no. 1, 112–122.
  * [32] F. Colombo, I. Sabadini, A structure formula for slice monogenic functions and some of its consequences, Hypercomplex analysis, 101–114, Trends Math., Birkhäuser Verlag, Basel, 2009.
  * [33] F. Colombo, I. Sabadini, The Cauchy formula with $s$-monogenic kernel and a functional calculus for noncommuting operators, J. Math. Anal. Appl., 373 (2011), 655–679.
  * [34] F. Colombo, I. Sabadini, The F-functional calculus for unbounded operators, J. Geom. Phys., 86 (2014), 392–407.
  * [35] F. Colombo, I. Sabadini, The F-spectrum and the SC-functional calculus, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), no. 3, 479–500.
  * [36] F. Colombo, I. Sabadini, On some properties of the quaternionic functional calculus, J. Geom. Anal., 19(3)(2009), pp. 601–627.
  * [37] F. Colombo, I. Sabadini, On the formulations of the quaternionic functional calculus, J. Geom. Phys., 60(10)(2010), pp. 1490–1508.
  * [38] F. Colombo, I. Sabadini, F. Sommen, The Fueter mapping theorem in integral form and the F-functional calculus, Math. Methods Appl. Sci., 33 (2010), 2050–2066.
  * [39] F. Colombo, I. Sabadini, F. Sommen, The inverse Fueter mapping theorem, Commun. Pure Appl. Anal., 10 (2011), 1165–1181.
  * [40] F. Colombo, I. Sabadini, F. Sommen, The inverse Fueter mapping theorem using spherical monogenics, Israel J. Math., 194 (2013), 485–505.
  * [41] F. Colombo, I. Sabadini, F. Sommen, The Fueter primitive of biaxially monogenic functions, Commun. Pure Appl. Anal., 13 (2014), 657–672.
  * [42] F. Colombo, I. Sabadini, F. Sommen, D.C. Struppa, Analysis of Dirac systems and computational algebra, Progress in Mathematical Physics, 39. Birkhäuser Boston, Inc., Boston, MA, 2004. xiv+332 pp.
  * [43] F. Colombo, I. Sabadini, D.C. Struppa, A new functional calculus for noncommuting operators, J. Funct. Anal., 254 (2008), no. 8, 2255–2274.
  * [44] F. Colombo, I. Sabadini, D. C. Struppa, Entire slice regular functions, SpringerBriefs in Mathematics. Springer, Cham, 2016. v+118 pp.
  * [45] F. Colombo, I. Sabadini, D.C. Struppa, Slice monogenic functions, Israel J. Math., 171 (2009), 385–403.
  * [46] F. Colombo, I. Sabadini, D.C. Struppa, Noncommutative functional calculus. Theory and applications of slice hyperholomorphic functions, Progress in Mathematics, 289. Birkhäuser/Springer Basel AG, Basel, 2011. vi+221 pp.
  * [47] F. Colombo, I. Sabadini, D.C. Struppa, An extension theorem for slice monogenic functions and some of its consequences, Israel J. Math., 177 (2010), 369–389.
  * [48] F. Colombo, I. Sabadini, D.C. Struppa, Duality theorems for slice hyperholomorphic functions, J. Reine Angew. Math., 645 (2010), 85–105.
  * [49] C. G. Cullen, An integral theorem for analytic intrinsic functions on quaternions, Duke Math. J., 32 (1965), 139–148.
  * [50] R. Delanghe, F. Sommen, V. Soucek, Clifford algebra and spinor-valued functions. A function theory for the Dirac operator, Related REDUCE software by F. Brackx and D. Constales. With 1 IBM-PC floppy disk (3.5 inch). Mathematics and its Applications, 53. Kluwer Academic Publishers Group, Dordrecht, 1992. xviii+485 pp.
  * [51] D. R. Farenick, B. A. F. Pidkowich, The spectral theorem in quaternions, Linear Algebra Appl., 371 (2003), 75–102.
  * [52] T. Friedrich, Dirac operators in Riemannian geometry. Translated from the 1997 German original by Andreas Nestke. Graduate Studies in Mathematics, 25. American Mathematical Society, Providence, RI, 2000. xvi+195 pp.
  * [53] R. Fueter, Die Funktionentheorie der Differentialgleichungen $\Delta u=0$ und $\Delta\Delta u=0$ mit vier reellen Variablen, Comment. Math. Helv., 7 (1934-35), 307–330.
  * [54] S. Gal, I. Sabadini, Quaternionic approximation. With application to slice regular functions, Frontiers in Mathematics. Birkhäuser/Springer, Cham, 2019. x+221 pp.
  * [55] J. Gantner, On the equivalence of complex and quaternionic quantum mechanics, Quantum Stud. Math. Found.,5 (2018), no. 2, 357–390.
  * [56] J. Gantner, Operator Theory on One-Sided Quaternionic Linear Spaces: Intrinsic S-Functional Calculus and Spectral Operators, to appear in Mem. Amer. Math. Soc. 2020, arXiv:1803.10524.
  * [57] G. Gentili, D.C. Struppa, A new theory of regular functions of a quaternionic variable, Adv. Math., 216 (2007), 279–301.
  * [58] G. Gentili, C. Stoppato, D.C. Struppa, Regular functions of a quaternionic variable, Springer Monographs in Mathematics. Springer, Heidelberg, 2013. x+185 pp.
  * [59] R. Ghiloni, A. Perotti, Slice regular functions on real alternative algebras, Adv. Math., 226 (2011), no. 2, 1662–1691.
  * [60] K. Gürlebeck, W. Sprössig, Quaternionic Analysis and Elliptic Boundary Value Problems, International Series of Numerical Mathematics, 89. Birkhäuser Verlag, Basel, 1990, 253pp.
  * [61] J. E. Gilbert, M. A. M. Murray, Clifford algebras and Dirac operators in harmonic analysis, Cambridge Studies in Advanced Mathematics, 26. Cambridge University Press, Cambridge, 1991. viii+334 pp.
  * [62] K. Gürlebeck, K. Habetha, W. Sprößig, Application of holomorphic functions in two and higher dimensions, Birkhääuser/Springer, [Cham], 2016. xv+390 pp.
  * [63] G. Laville, I. Ramadanoff, Holomorphic Cliffordian functions, Adv. Appl. Clifford Algebras 8 (1998), no. 2, 323–340.
  * [64] C. Li, A. McIntosh, T. Qian, Clifford algebras, Fourier transforms and singular convolution operators on Lipschitz surfaces, Rev. Mat. Iberoamericana, 10 (1994), 665–721.
  * [65] B. Jefferies, A. McIntosh, The Weyl calculus and Clifford analysis, Bull. Austral. Math. Soc., 57 (1998), 329–341.
  * [66] B. Jefferies, A. McIntosh, J. Picton-Warlow, The monogenic functional calculus, Studia Math., 136 (1999), 99–119.
  * [67] A. McIntosh, Operators which have an $H^{\infty}$ functional calculus. Proc. Centre Math. Anal. Austral. Nat. Univ., 14, Austral. Nat. Univ., Canberra, (1986).
  * [68] A. McIntosh, A. Pryde, A functional calculus for several commuting operators, Indiana U. Math. J., 36 (1987), 421–439.
  * [69] B. Jefferies, Spectral properties of noncommuting operators, Lecture Notes in Mathematics, 1843, Springer-Verlag, Berlin, 2004.
  * [70] V. Kisil, Möbius transformations and monogenic functional calculus. Electron. Res. Announc. Amer. Math. Soc., 2 (1996), no. 1, 26–33.
  * [71] D. Pena Pena, I. Sabadini, F. Sommen, Fueter’s theorem for monogenic functions in biaxial symmetric domains, Results Math., 72 (2017), no. 4, 1747–1758.
  * [72] D. Pena Pena, F. Sommen, Biaxial monogenic functions from Funk-Hecke’s formula combined with Fueter’s theorem, Math. Nachr., 288 (2015), no. 14–15, 1718–1726.
  * [73] D. Pena Pena, F. Sommen, A generalization of Fueter’s theorem, Results Math., 49 (2006), no. 3–4, 301–311.
  * [74] T. Qian, Generalization of Fueter’s result to $R^{n+1}$, Rend. Mat. Acc. Lincei 9 (1997), 111–117.
  * [75] T. Qian, Singular integrals on star-shaped Lipschitz surfaces in the quaternionic space, Math. Ann., 310 (1998), 601–630.
  * [76] T. Qian, Fueter Mapping Theorem in Hypercomplex Analysis, in Operator Theory, D. Alpay ed., (2015), 1491–1507.
  * [77] T. Qian, P. Li, Singular integrals and Fourier theory on Lipschitz boundaries, Science Press Beijing, Beijing; Springer, Singapore, 2019. xv+306 pp.
  * [78] R. Rocha-Chavez, M. Shapiro, F. Sommen, Integral theorems for functions and differential forms, in Cm. Chapman & Hall/CRC Research Notes in Mathematics, 428. Chapman & Hall/CRC, Boca Raton, FL, 2002. x+204 pp.
  * [79] M. Sce, Osservazioni sulle serie di potenze nei moduli quadratici, Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Nat., (8) 23 (1957), 220–225.
  * [80] K. Schmüdgen, _Unbounded self-adjoint operators on Hilbert space_ , Graduate Texts in Mathematics, vol. 265, Springer, Dordrecht, 2012.
  * [81] O. Teichmüller, Operatoren im Wachsschen Raum (German), J. Reine Angew. Math. 174 (1936), 73–124.
  * [82] K. Viswanath, Normal operations on quaternionic Hilbert spaces, Trans. Amer. Math. Soc. 162 (1971), 337–350.