# The harmonic $H^{\infty}$-functional calculus
based on the $S$-spectrum

Antonino De Martino  (AM) Politecnico di Milano, Dipartimento di Matematica,
Via E. Bonardi 9, 20133 Milano, Italy<EMAIL_ADDRESS>,  Stefano
Pinton  (SP) Politecnico di Milano, Dipartimento di Matematica, Via E. Bonardi
9, 20133 Milano, Italy<EMAIL_ADDRESS>and  Peter Schlosser  (PS)
Politecnico di Milano, Dipartimento di Matematica, Via E. Bonardi 9, 20133
Milano, Italy<EMAIL_ADDRESS>
###### Abstract.

The aim of this paper is to introduce the $H^{\infty}$-functional calculus for
harmonic functions over the quaternions. More precisely, we give meaning to
$Df(T)$ for unbounded sectorial operators $T$ and polynomially growing
functions of the form $Df$, where $f$ is a slice hyperholomorphic function and
$D=\partial_{q_{0}}+e_{1}\partial_{q_{1}}+e_{2}\partial_{q_{2}}+e_{3}\partial_{q_{3}}$
is the Cauchy-Fueter operator. The harmonic functional calculus can be viewed
as a modification of the well known $S$-functional calculus $f(T)$, with a
different resolvent operator. The harmonic $H^{\infty}$-functional calculus is
defined in two steps: First, for functions with a certain decay property, one
can make sense of the bounded operator $Df(T)$ directly via a Cauchy-type
formula. In a second step, a regularization procedure is used to extend the
functional calculus to polynomially growing functions and consequently
unbounded operators $Df(T)$. The harmonic functional calculus is an important
functional calculus of the quaternionic fine structures on the $S$-spectrum,
which arise also in the Clifford setting and they encompass a variety of
function spaces and the corresponding functional calculi. These function
spaces emerge through all possible factorizations of the second map of the
Fueter-Sce extension theorem. This field represents an emerging and expanding
research area that serves as a bridge connecting operator theory, harmonic
analysis, and hypercomplex analysis.

AMS Classification: 47A10, 47A60

Keywords: H-infinity functional calculus, harmonic functional calculus,
Cauchy-Fueter operator, $S$-spectrum, fine structures.

Acknowledgements: The research of P. Schlosser was funded by the Austrian
Science Fund (FWF) under Grant No. J 4685-N and by the European Union –
NextGenerationEU.

##  1\. Introduction

The complex Riesz-Dunford functional calculus [37] is based on the Cauchy
integral formula and gives meaning to holomorphic functions of operators via
the integral

$f(A):=\frac{1}{2\pi i}\int_{\partial\Omega}(z\mathcal{I}-A)^{-1}f(z)dz,$ (1)

where $\Omega\subset\mathbb{C}$ is a suitable open set containing the spectrum
of the complex bounded linear operator $A$. Extending the holomorphic
functional calculus to quaternionic operators presented a formidable challenge
because of the various potential concepts of quaternionic holomorphicity.
Indeed, in hypercomplex analysis, unlike in complex analysis, it is important
to recognize the existence of multiple notions of analyticity, including slice
hyperholomorphic functions and Cauchy-Fueter regular functions. Moreover, in
the slice hyperholomorphic setting, due to the lack of commutativity, there
exist two different hyperholomorphic Cauchy kernels

$S_{L}^{-1}(s,q):=(s-\overline{q})(s^{2}-2q_{0}s+|q|^{2})^{-1}\quad\text{and}\quad
S_{R}^{-1}(s,q):=(s^{2}-2q_{0}s+|q|^{2})^{-1}(s-\overline{q}).$ (2)

For $f$ in the space $\mathcal{SH}_{L}(U)$ of left slice hyperholomorphic
functions, the left Cauchy kernel $S_{L}^{-1}(s,q)$ gives raise to the Cauchy
integral formula

$f(q)=\frac{1}{2\pi}\int_{\partial(U\cap\mathbb{C}_{J})}S_{L}^{-1}(s,q)ds_{J}f(s),$
(3)

while for right slice hyperholomorphic functions a similar formula holds true
using the right Cauchy kernel $S_{R}^{-1}(s,q)$, see for example [21, Theorem
2.1.32]. For the interpretation of the integral and the notion of slice
hyperholomorphicity we refer to Definition 2.3 and Definition 2.2. Motivated
by the integral (3), the quaternionic extension of (1), called $S$-functional
calculus for a quaternionic linear operator $T$, is defined by

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

where the left $S$-resolvent operator $S_{L}^{-1}(s,T)$ is associated with (2)
and properly defined in (12). In this paper we establish a functional calculus
similar to (4), for functions in the space

$\mathcal{AH}_{L}(U)=\Set{Df}{f\in\mathcal{SH}_{L}(U)},$

where $D$ is the Cauchy-Fueter operator

$D=\frac{\partial}{\partial q_{0}}+e_{1}\frac{\partial}{\partial
q_{1}}+e_{2}\frac{\partial}{\partial q_{2}}+e_{3}\frac{\partial}{\partial
q_{3}}.$ (5)

Due to the Fueter mapping theorem [38], the space $\mathcal{AH}_{L}(U)$ is
exactly the one of axially harmonic functions. Applying the operator $D$ to
the formula (3), carrying the derivatives inside the integral and using the
explicit formula

$DS_{L}^{-1}(s,q)=-2(s^{2}-2q_{0}s+|q|^{2})^{-1}=:-2Q_{c,s}^{-1}(q)$

of the Cauchy-kernel, we obtain the Cauchy-type formula for axially harmonic
functions

$Df(q)=\frac{-1}{\pi}\int_{\partial(U\cap\mathbb{C}_{J})}Q_{c,s}^{-1}(q)ds_{J}f(s).$

Replacing the quaternion $q$ by a suitable quaternionic operator $T$ the above
formula motivates the definition of the harmonic functional calculus

$Df(T)=\frac{-1}{\pi}\int_{\partial(U\cap\mathbb{C}_{J})}Q_{c,s}^{-1}(T)ds_{J}f(s).$
(6)

For a rigorous definition of the operator $Q_{c,s}^{-1}(T)$ see (10) and (11).
For bounded operators $T$, this functional calculus is already well
established in [11, 35]. In this paper we consider the class of unbounded
sectorial operators (also called operators of type $\omega$). The set $U$
mentioned in the boundary integration within equation (6) must encompass the
$S$-spectrum of $T$. So for this class of operators the $S$-spectrum is
unbounded and so we have to require additional decay properties on the
function $f$ to ensure the convergence of the integral.

Nonetheless, by employing a regularization procedure known as the
$H^{\infty}$-functional calculus, originally introduced by A. McIntosh in the
complex context [44], we can subsequently extend the harmonic functional
calculus to include functions $f$ that exhibit polynomial growth. To explore
the $H^{\infty}$-functional calculus in the complex context, one can refer to
the following sources: [40, 41, 42, 48]. Additionally, for an extension of the
$H^{\infty}$-functional calculus in the monogenic Clifford setting and its
applications, one can consider the books [43, 46].

We now intend to offer a comprehensive perspective on our theory by
elucidating its initial motivations, detailing recent developments, and
introducing a new branch known as the fine structures on the $S$-spectrum.
Within this new branch, we encounter the development of the harmonic
functional calculus, whose $H^{\infty}$-version has been elaborated in this
work.

The study of quaternionic operators is motivated by their relevance to the
formulation of quantum mechanics. The foundational work by Birkhoff and von
Neumann [8] showed that there are essentially two ways to formulate quantum
mechanics, one using complex numbers and the other using quaternions. This
result is significant since it establishes the importance of quaternionic
operators as a fundamental part of quantum mechanics. The notion of
$S$-spectrum was discovered in 2006, only using methods in hypercomplex
analysis even though its existence was suggested by quaternionic quantum
mechanics. For more details see the introduction of the book [21]. The notion
of $S$-spectrum extends also to Clifford operators, see [30], and recently it
has been shown that the quaternionic and the Clifford settings are just
particular cases of a more general framework in which the spectral theory on
the $S$-spectrum can be developed, see [22, 24] and the references therein.
Using the notion of $S$-spectrum it was also possible to prove the
quaternionic version of the spectral theorem. Precisely, unitary operators,
using Herglotz’s functions, were considered in [2], while quaternionic normal
operators are studied in [1]. More recently, the spectral theorem based on the
$S$-spectrum was extended also to Clifford operators, see [23].

The development of the spectral theory on the $S$-spectrum has opened up
several research directions in hypercomplex analysis and operator theory,
without claiming completeness, we mention the quaternionic perturbation theory
and invariant subspaces [9] and new classes of fractional diffusion problems
[15, 16, 19, 20, 25] that are based on the $H^{\infty}$-version of the
$S$-functional calculus [3, 19]. Finally, we mention that the spectral theory
on the $S$-spectrum is systematically organized in the books [20, 21, 30].

In recent times a new branch of the spectral theory on the $S$-spectrum has
been developed, that is called fine structures on the $S$-spectrum. It turns
out that the space $\mathcal{AH}_{L}(U)$ of axially harmonic functions is only
one of the spaces arising from the factorization of the operators in the
Fueter-Sce theorem [38, 45, 47], which connects the class of slice
hyperholomorphic functions with the class of axially monogenic functions
$\mathcal{AM}_{L}(U)$ via the powers $\Delta^{\frac{n-1}{2}}$ of the Laplace
operator in dimension $n+1$, where $n$ is the number of imaginary units in the
Clifford algebra $\mathbb{R}_{n}$. Their corresponding integral
representations in turn give raise to various functional calculi, see [11, 14,
34, 35]. Note, that for odd $n$ the operator $\Delta^{\frac{n-1}{2}}$ is a
pointwise differential operator, see [29, 47], while for even values of $n$ we
are dealing with fractional powers of the Laplace operator, see [45]. For more
information on the work of M. Sce see the translation of his work with in
[29], and for a different description of the Fueter-Sce theorem [32, 33].

In the special case of the quaternions $\mathbb{H}$ (which are classically
identified with $\mathbb{R}_{2}$), the Fueter-Sce theorem, called Fueter
mapping theorem in this case, holds true with $n=3$. This means, the
connection between slice hyperholomorphic and axially monogenic functions is
via the four dimensional Laplace operator

$\Delta=\frac{\partial^{2}}{\partial q_{0}^{2}}+\frac{\partial^{2}}{\partial
q_{1}^{2}}+\frac{\partial^{2}}{\partial
q_{2}^{2}}+\frac{\partial^{2}}{\partial q_{3}^{2}}.$

This operator can now be factorized using the Cauchy-Fueter operator $D$ from
(5) and its conjugate $\overline{D}$ in the two different ways

$\Delta=D\overline{D}=\overline{D}D.$ (7)

The space $\mathcal{AH}_{L}(U)$, for which we motivated the harmonic
functional calculus in (6), now arises from the first of the above
factorizations, namely

$\begin{CD}\mathcal{SH}_{L}(U)@>{D}>{}>\mathcal{AH}_{L}(U)@>{\overline{D}}>{}>\mathcal{AM}_{L}(U).\end{CD}$

At this point we also want to mention that the polyanalytic functional
calculus of the space

$\mathcal{AP}_{2,L}(U)=\Set{\overline{D}f}{f\in\mathcal{SH}_{L}(U)},$

arising from the second factorization in (7), is constructed with similar
strategy in [10, 34] and the $\mathcal{F}$-functional calculus for the space
$\mathcal{AM}_{L}(U)$ was established long ago in [28], while more recent
investigations can be found in [12, 13, 17, 26, 27].

Plan of the paper. The aim of Section 2 is to collect the basic facts about
quaternionic function theory and in particular to fix the notations used in
the paper. In Section 3 we investigate the integral (6) for operators $T$ of
type $\omega$ and slice hyperholomorphic functions $f$ which have some order
of decay at $0$ and at $\infty$. Beside the welldefinedness of this functional
calculus stated in Theorem 3.4, we also prove its basic properties as the
commutation with other operators in Proposition 3.7 and Corollary 3.8, the
product rule in Theorem 3.10 and the equivalence to the rational functional
calculus in Proposition 3.12. In Section 4 we extend the theory of Section 3
to functions which are allowed to grow polynomially at $0$ and at $\infty$,
using certain regularizer functions. Again, the commutation relations in
Proposition 4.7 and Corollary 4.8, the product rule in Theorem 4.9 and the
equivalence to the rational functional calculus in Proposition 4.10 are also
proven for this extended theory.

##  2\. Preliminary results on quaternionic function theory

The setting in which we will work in this paper is the one of the quaternions

$\mathbb{H}:=\Set{s_{0}+s_{1}e_{1}+s_{2}e_{2}+s_{3}e_{3}}{s_{0},s_{1},s_{2},s_{3}\in\mathbb{R}}$

with the three imaginary units $e_{1},e_{2},e_{3}$ satisfying the relations

$e_{1}^{2}=e_{2}^{2}=e_{3}^{2}=-1\qquad\text{and}\qquad\begin{array}[]{l}e_{1}e_{2}=-e_{2}e_{1}=e_{3},\\\
e_{2}e_{3}=-e_{3}e_{2}=e_{1},\\\ e_{3}e_{1}=-e_{1}e_{3}=e_{2}.\end{array}$

For every quaternion $s\in\mathbb{H}$ we set

$\displaystyle\operatorname{Re}(s)$ $\displaystyle:=s_{0},$
$\displaystyle(\textit{real part})$ $\displaystyle\operatorname{Im}(s)$
$\displaystyle:=s_{1}e_{1}+s_{2}e_{2}+s_{3}e_{3},$
$\displaystyle(\textit{imaginary part})$ $\displaystyle\overline{s}$
$\displaystyle:=s_{0}-s_{1}e_{1}-s_{2}e_{2}-s_{3}e_{3},$
$\displaystyle(\textit{conjugate})$ $\displaystyle|s|$
$\displaystyle:=\sqrt{s_{0}^{2}+s_{1}^{2}+s_{2}^{2}+s_{3}^{2}}.$
$\displaystyle(\textit{modulus})$

The unit sphere of purely imaginary quaternions is defined as

$\mathbb{S}:=\Set{s\in\mathbb{H}}{s_{0}=0\text{ and }|s|=1},$

and for every $J\in\mathbb{S}$ consider the complex plane

$\mathbb{C}_{J}:=\Set{u+Jv}{u,v\in\mathbb{R}},$

which is an isomorphic copy of the complex numbers, since every
$J\in\mathbb{S}$ satisfies $J^{2}=-1$. Moreover, for every quaternion
$s\in\mathbb{H}$ we consider the corresponding $2$-sphere

$[s]:=\Set{\operatorname{Re}(s)+J}{\operatorname{Im}(s)||J\in\mathbb{S}}.$

Next, we introduce the so called slice hyperholomorphic functions, which are a
quaternionic analog to the complex holomorphic functions. The sets on which
those functions are defined and suitable for operator theory are the following
axially symmetric sets.

######  Definition 2.1 (Axially symmetric sets).

A subset $U\subseteq\mathbb{H}$ is called axially symmetric, if $[s]\subseteq
U$ for every $s\in U$.

######  Definition 2.2 (Slice hyperholomorphic functions).

Let $U\subseteq\mathbb{H}$ be axially symmetric and open. A function
$f:U\rightarrow\mathbb{H}$ is called left (resp. right) slice
hyperholomorphic, if there exists two continuously differentiable functions
$\alpha,\beta:\mathcal{U}:=\Set{(u,v)\in\mathbb{R}^{2}}{u+\mathbb{S}v\subseteq
U}\rightarrow\mathbb{H}$, with

$f(u+Jv)=\alpha(u,v)+J\beta(u,v),\quad\Big{(}\text{resp.}\;f(u+Jv)=\alpha(u,v)+\beta(u,v)J\Big{)},$
(8)

for every $(u,v)\in\mathcal{U}$, $J\in\mathbb{S}$, and if the functions
$\alpha,\beta$ satisfy the symmetry conditions

$\alpha(u,-v)=\alpha(u,v)\quad\text{and}\quad\beta(u,-v)=-\beta(u,v),$

as well as the Cauchy-Riemann equations

$\frac{\partial}{\partial u}\alpha(u,v)=\frac{\partial}{\partial
v}\beta(u,v),\quad\text{and}\quad\frac{\partial}{\partial
v}\alpha(u,v)=-\frac{\partial}{\partial u}\beta(u,v).$ (9)

The class of left (resp. right) slice hyperholomorphic functions on $U$ is
denoted by $\mathcal{SH}_{L}(U)$ (resp. $\mathcal{SH}_{R}(U)$). In the special
case that the functions $\alpha$ and $\beta$ are real valued, we call the
function $f$ intrinsic and denote the space of intrinsic functions by
$\mathcal{N}(U)$.

Next we introduce path integrals of slice hyperholomorphic functions. Since it
is sufficient to consider paths embedded in only one complex plane
$\mathbb{C}_{J}$, the idea is to reduce it to a classical complex path
integral.

######  Definition 2.3.

Let $U\subseteq\mathbb{H}$ be axially symmetric, $J\in\mathbb{S}$. Then for
any $f\in\mathcal{SH}_{R}(U^{\prime})$, $g\in\mathcal{SH}_{L}(U^{\prime})$ for
some open set $U^{\prime}\supseteq\overline{U}$, we define the integral

$\int_{\partial(U\cap\mathbb{C}_{J})}f(s)ds_{J}g(s):=\int_{a}^{b}f(\gamma(t))\frac{\gamma^{\prime}(t)}{J}g(\gamma(t))dt,$

where $\gamma:[a,b]\rightarrow\mathbb{C}_{J}$ is a parametrization of the
boundary $\partial(U\cap\mathbb{C}_{J})$.

From now on $V$ always denotes a two-sided Banach space over the quaternions
$\mathbb{H}$. In the following we will specify the class of operators with
commuting components, which will be of interest in this paper.

######  Definition 2.4 (Operators with commuting components).

A right-linear closed operator $T:V\rightarrow V$ with a two-sided linear
domain $\operatorname{dom}(T)$ is said to be an operator with commuting
components, if there exist two-sided linear operators $T_{i}:V\rightarrow V$,
$i\in\\{0,\dots,3\\}$, with $\operatorname{dom}(T_{i})=\operatorname{dom}(T)$,
such that

$T=T_{0}+e_{1}T_{1}+e_{2}T_{2}+e_{3}T_{3},$

and the property that for all $i,j\in\\{0,\dots,3\\}$

$\operatorname{dom}(T^{2})\subseteq\operatorname{dom}(T_{i}T_{j})\quad\text{and}\quad
T_{i}T_{j}=T_{j}T_{i}\text{ on }\operatorname{dom}(T^{2}).$

We will denote the class of closed operators with commuting components by
$\mathcal{KC}(V)$.

For any operator $T\in\mathcal{KC}(V)$ we additionally define the conjugate
operator

$\overline{T}:=T_{0}-e_{1}T_{1}-e_{2}T_{2}-e_{3}T_{3},$

with $\operatorname{dom}(\overline{T}):=\operatorname{dom}(T)$. Then for every
$T\in\mathcal{KC}(V)$ there obviously holds the properties

$\operatorname{dom}(\overline{T}T)=\operatorname{dom}(T^{2})\subseteq\operatorname{dom}(T\overline{T}),$

as well as

$|T|^{2}:=\overline{T}T=T\overline{T}=T_{0}^{2}+T_{1}^{2}+T_{2}^{2}+T_{3}^{2},\quad\text{on
}\operatorname{dom}(T^{2}).$

For any operator $T\in\mathcal{KC}(V)$ we now use the operator

$Q_{c,s}(T):=s^{2}\mathcal{I}-2T_{0}s+|T|^{2},\qquad\text{with
}\operatorname{dom}Q_{c,s}(T)=\operatorname{dom}(T^{2}),$ (10)

to define the $S$-resolvent set

$\rho_{S}(T):=\Set{s\in\mathbb{H}}{Q_{c,s}(T)\text{ is bijective}}.$ (11)

The complement of this set will be called the $S$-spectrum

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

It is proven in [20, Theorem 3.1.6] that $\rho_{S}(T)$ is an axially symmetric
open subset of $\mathbb{H}$ and that the mapping $s\mapsto Q_{c,s}^{-1}(T)$ is
intrinsic on $\rho_{S}(T)$. Moreover, motivated by the Cauchy kernels (2) we
define for every $s\in\rho_{S}(T)$ the left and the right $S$-resolvent

$S_{L}^{-1}(s,T):=(s\mathcal{I}-\overline{T})Q_{c,s}^{-1}(T)\quad\text{and}\quad
S_{R}^{-1}(s,T):=sQ_{c,s}^{-1}(T)-\sum\limits_{i=0}^{3}T_{i}Q_{c,s}^{-1}(T)\overline{e_{i}}.$
(12)

Note, that on $\operatorname{dom}(T)$ we are allowed to interchange
$T_{i}Q_{c,s}^{-1}(T)=Q_{c,s}^{-1}(T)T_{i}$, which gives the more elegant form
of the right $S$-resolvent

$S_{R}^{-1}(s,T)=Q_{c,s}^{-1}(T)(s\mathcal{I}-\overline{T}),\quad\text{on
}\operatorname{dom}(T).$ (13)

######  Remark 2.5.

Note, that the original definition of the $S$-resolvent set is

$\rho_{S}(T):=\Set{s\in\mathbb{H}}{Q_{s}(T)\text{ is bijective}},$ (14)

using the operator

$Q_{s}(T):=T^{2}-2s_{0}T+|s|^{2}.$

This definition has the advantage that it is well defined for any closed
operator $T:V\rightarrow V$, not only for those with commuting components.
However, it is proven in [20, Theorem 3.3.4] that for operators
$T\in\mathcal{KC}(V)$ the two definitions (11) and (14) coincide. Moreover,
also the left (resp. right) $S$-resolvent operators (12) are originally
defined as

$S_{L}^{-1}(s,T):=Q_{s}^{-1}(T)\overline{s}-TQ_{s}^{-1}(T)\quad\text{and}\quad
S_{R}^{-1}(s,T):=(\overline{s}\mathcal{I}-T)Q_{s}^{-1}(T),$ (15)

but turn out to be equivalent to (12) in the case of operators
$T\in\mathcal{KC}(V)$.

In order to finally define the class of operators, for which we will introduce
the harmonic $H^{\infty}$-functional calculus in this paper, we denote for
every angle $\omega\in(0,\pi)$ the open sector

$S_{\omega}:=\Set{s\in\mathbb{H}\setminus\\{0\\}}{|\operatorname{Arg}(s)|<\omega},$

where $\operatorname{Arg}(s)=0$ if $\operatorname{Im}(s)=0$, and in the case
$\operatorname{Im}(s)\neq 0$, understood as the argument of a complex number
in the complex plane $s\in\mathbb{C}_{J}$, with
$J=\frac{\operatorname{Im}(s)}{|\operatorname{Im}(s)|}$.

######  Definition 2.6 (Operators of type $\omega$).

Any $T\in\mathcal{KC}(V)$ is called operator of type $\omega\in(0,\pi)$, if
$\sigma_{S}(T)\subseteq\overline{S_{\omega}}$ and if for every
$\theta\in(\omega,\pi)$ there exists some $C_{\theta}\geq 0$, such that

$\|S_{L}^{-1}(s,T)\|\leq\frac{C_{\theta}}{|s|}\quad\text{and}\quad\|S_{R}^{-1}(s,T)\|\leq\frac{C_{\theta}}{|s|},\quad\text{for
every }s\in S_{\theta}^{c}\setminus\\{0\\}.$ (16)

A similar estimate as in [18, Lemma 3.7] now shows, that the estimates (16) on
the $S$-resolvent operators imply a similar estimate on the commutative
$Q$-resolvent $Q_{c,s}^{-1}(T)$, which will be crucial for the existence of
the integral (19a) of the harmonic functional calculus.

######  Lemma 2.7.

Let $T\in\mathcal{KC}(V)$ such that $T,\overline{T}$ are of type $\omega$.
Then for every $\theta\in(\omega,\pi)$ there exists some $C_{\theta}\geq 0$,
such that

$\|Q_{c,s}^{-1}(T)\|\leq\frac{C_{\theta}}{|s|^{2}},\quad\text{for every }s\in
S_{\theta}^{c}\setminus\\{0\\}.$ (17)

###### Proof.

Using the representations (12) for $S_{L}^{-1}(s,T)$ and (13) for
$S_{R}^{-1}(s,T)$, gives

$\displaystyle\big{(}S_{R}^{-1}(s,T)+S_{R}^{-1}(s,\overline{T})\big{)}$
$\displaystyle\big{(}S_{L}^{-1}(s,T)+S_{L}^{-1}(s,\overline{T})\big{)}$
$\displaystyle=Q_{c,s}^{-1}(T)\big{(}(s\mathcal{I}-\overline{T})+(s\mathcal{I}-T)\big{)}\big{(}(s\mathcal{I}-\overline{T})+(s\mathcal{I}-T)\big{)}Q_{c,s}^{-1}(T)$
$\displaystyle=4Q_{c,s}^{-1}(T)(s\mathcal{I}-T_{0})^{2}Q_{c,s}^{-1}(T),$

as well as

$\displaystyle\big{(}S_{R}^{-1}(s,T)-S_{R}^{-1}(s,\overline{T})\big{)}$
$\displaystyle\big{(}S_{L}^{-1}(s,T)-S_{L}^{-1}(s,\overline{T})\big{)}$
$\displaystyle=Q_{c,s}^{-1}(T)\big{(}(s\mathcal{I}-\overline{T})-(s\mathcal{I}-T)\big{)}\big{(}(s\mathcal{I}-\overline{T})-(s\mathcal{I}-T)\big{)}Q_{c,s}^{-1}(T)$
$\displaystyle=4Q_{c,s}^{-1}(T)(T_{0}^{2}-|T|^{2})Q_{c,s}^{-1}(T).$

Note, that it is justified by
$\operatorname{ran}(S_{L}^{-1}(s,T))\subseteq\operatorname{dom}(T)$ to use the
representation (13) of $S_{R}^{-1}(T)$. By subtracting these two equations we
then conclude the identity

$\displaystyle\big{(}S_{R}^{-1}(s,T)+$ $\displaystyle
S_{R}^{-1}(s,\overline{T})\big{)}\big{(}S_{L}^{-1}(s,T)+S_{L}^{-1}(s,\overline{T})\big{)}$
$\displaystyle-\big{(}S_{R}^{-1}(s,T)-S_{R}^{-1}(s,\overline{T})\big{)}\big{(}S_{L}^{-1}(s,T)-S_{L}^{-1}(s,\overline{T})\big{)}$
$\displaystyle\hskip
56.9055pt=4Q_{c,s}^{-1}(T)(s^{2}-2sT_{0}+|T|^{2})Q_{c,s}^{-1}(T)=4Q_{c,s}^{-1}(T).$

Since $T$ as well as $\overline{T}$ are assumed to be of type $\omega$, the
estimates (16) of the $S$-resolvents hold for $T$ as well as for
$\overline{T}$ (let’s say for the same constant $C_{\theta}$) and we get the
final estimate

$\|Q_{c,s}^{-1}(T)\|\leq\frac{1}{4}\bigg{(}\bigg{(}\frac{C_{\theta}}{|s|}+\frac{C_{\theta}}{|s|}\bigg{)}\bigg{(}\frac{C_{\theta}}{|s|}+\frac{C_{\theta}}{|s|}\bigg{)}+\bigg{(}\frac{C_{\theta}}{|s|}+\frac{C_{\theta}}{|s|}\bigg{)}\bigg{(}\frac{C_{\theta}}{|s|}+\frac{C_{\theta}}{|s|}\bigg{)}\bigg{)}=\frac{2C_{\theta}^{2}}{|s|^{2}}.\qed$

##  3\. Harmonic functional calculus for decaying functions

In this section we take the first step in establishing the harmonic functional
calculus for unbounded operators of type $\omega$, by giving a direct meaning
to the integral (6) for the following classes of slice hyperholomorphic
functions

  1. i)

$\Psi_{L}^{Q}(S_{\theta}):=\Set{f\in\mathcal{SH}_{L}(S_{\theta})}{\exists\alpha>0,\,C_{\alpha}\geq
0:|f(s)|\leq\frac{C_{\alpha}|s|^{1+\alpha}}{1+|s|^{1+2\alpha}}\text{ for every
}s\in S_{\theta}}$,

  2. ii)

$\Psi_{R}^{Q}(S_{\theta}):=\Set{f\in\mathcal{SH}_{R}(S_{\theta})}{\exists\alpha>0,\,C_{\alpha}\geq
0:|f(s)|\leq\frac{C_{\alpha}|s|^{1+\alpha}}{1+|s|^{1+2\alpha}}\text{ for every
}s\in S_{\theta}}$,

  3. iii)

$\Psi^{Q}(S_{\theta}):=\Set{f\in\mathcal{N}(S_{\theta})}{\exists\alpha>0,\,C_{\alpha}\geq
0:|f(s)|\leq\frac{C_{\alpha}|s|^{1+\alpha}}{1+|s|^{1+2\alpha}}\text{ for every
}s\in S_{\theta}}$.

The decay at $0$ and at $\infty$ is necessary in order to make the integrals
(18a) and (19a) converge.

######  Remark 3.1.

Since the $S$-resolvents $S_{L}^{-1}(s,T)$ and $S_{R}^{-1}(s,T)$ admit at
$s=0$ only a $\frac{1}{|s|}$-singularity due to (16), instead of the
$\frac{1}{|s|^{2}}$-singularity of $Q_{c,s}^{-1}(s,T)$ in (17), the
$\mathcal{O}(|s|^{1+\alpha})$ decay in the space $\Psi^{Q}(S_{\theta})$ can be
reduced to an $\mathcal{O}(|s|^{\alpha})$ decay for the $S$-functional
calculus (18a). This means classically, the $S$-functional calculus is defined
for the larger class of functions

  1. i)

$\Psi_{L}(S_{\theta}):=\Set{f\in\mathcal{SH}_{L}(S_{\theta})}{\exists\alpha>0,\,C_{\alpha}\geq
0:|f(s)|\leq\frac{C_{\alpha}|s|^{\alpha}}{1+|s|^{1+2\alpha}}\text{ for every
}s\in S_{\theta}}$,

  2. ii)

$\Psi_{R}(S_{\theta}):=\Set{f\in\mathcal{SH}_{R}(S_{\theta})}{\exists\alpha>0,\,C_{\alpha}\geq
0:|f(s)|\leq\frac{C_{\alpha}|s|^{\alpha}}{1+|s|^{1+2\alpha}}\text{ for every
}s\in S_{\theta}}$,

  3. iii)

$\Psi(S_{\theta}):=\Set{f\in\mathcal{N}(S_{\theta})}{\exists\alpha>0,\,C_{\alpha}\geq
0:|f(s)|\leq\frac{C_{\alpha}|s|^{\alpha}}{1+|s|^{1+2\alpha}}\text{ for every
}s\in S_{\theta}}$.

However, since we will use the $S$-functional typically in combination with
the harmonic functional calculus (19a), we will consider functions in the
smaller spaces $\Psi_{L}^{Q}(S_{\theta})$, $\Psi_{R}^{Q}(S_{\theta})$,
$\Psi^{Q}(S_{\theta})$ most of the time.

The following well known $S$-functional calculus is motivated by the Cauchy
formula (3) and for example worked out in [3].

######  Definition 3.2 ($S$-functional calculus for decaying functions).

Let $T\in\mathcal{KC}(V)$ be an operator of type $\omega$. Then for any
$f\in\Psi_{L}(S_{\theta})$ (resp. $f\in\Psi_{R}(S_{\theta})$),
$\theta\in(\omega,\pi)$, the $S$-functional calculus is defined as the
bounded, everywhere defined operator

$\displaystyle
f(T):=\frac{1}{2\pi}\int_{\partial(S_{\varphi}\cap\mathbb{C}_{J})}S_{L}^{-1}(s,T)ds_{J}f(s),$
(18a)
$\displaystyle\bigg{(}\text{resp.}\;f(T):=\frac{1}{2\pi}\int_{\partial(S_{\varphi}\cap\mathbb{C}_{J})}f(s)ds_{J}S_{R}^{-1}(s,T)\bigg{)},$
(18b)

where $\varphi\in(\omega,\theta)$ and $J\in\mathbb{S}$ are arbitrary and the
integral is independent of those parameters.

The novelty of this paper is now the following harmonic functional calculus,
which replaces the left and right $S$-resolvent operators in (18a) by the
commutative pseudo resolvent $Q_{c,s}^{-1}(T)$. This leads to a functional
calculus for the axially harmonic functions $\mathcal{AH}(U)$, i.e. for
$Df(T)$ for any slice hyperholomorphic function $f\in\mathcal{SH}(U)$. Note,
that simply plugging $Df$ into the $S$-functional calculus (18a) is not
allowed since in general $Df$ is no longer a slice hyperholomorphic function.

######  Definition 3.3 (Harmonic functional calculus for decaying functions).

Let $T\in\mathcal{KC}(V)$ with $T,\overline{T}$ being operators of type
$\omega$. Then for every $f\in\Psi_{L}^{Q}(S_{\theta})$ (resp.
$f\in\Psi_{R}^{Q}(S_{\theta})$), $\theta\in(\omega,\pi)$, the harmonic
functional calculus is defined as

$\sigma_{S}(T)$$\omega$$\varphi$$\theta$$\mathbb{C}_{J}$$\operatorname{dom}(f)$

$\displaystyle
Df(T):=\frac{-1}{\pi}\int_{\partial(S_{\varphi}\cap\mathbb{C}_{J})}Q_{c,s}^{-1}(T)ds_{J}f(s),$
(19a)
$\displaystyle\bigg{(}\text{resp.}\;(fD)(T):=\frac{-1}{\pi}\int_{\partial(S_{\varphi}\cap\mathbb{C}_{J})}f(s)ds_{J}Q_{c,s}^{-1}(T)\bigg{)},$
(19b)

where $\varphi\in(\omega,\theta)$ and $J\in\mathbb{S}$ are arbitrary.

The following theorem shows that the Definition 3.3 is well posed.

######  Theorem 3.4.

Let $T\in\mathcal{KC}(V)$ such that $T,\overline{T}$ are of type $\omega$.
Then for every $f\in\Psi_{L}^{Q}(S_{\theta})$ (resp.
$f\in\Psi_{R}^{Q}(S_{\theta})$), $\theta\in(\omega,\pi)$, the integrals (19a)
are absolute convergent and do neither depend on the angle
$\varphi\in(\omega,\theta)$ nor on the choice of the imaginary unit
$J\in\mathbb{S}$. Moreover, if two functions $f,g\in\Psi_{L}^{Q}(S_{\theta})$
(resp. $f,g\in\Psi_{R}^{Q}(S_{\theta})$) satisfy $Df=Dg$ (resp. $fD=gD$), then
also the functional calculi $Df(T)=Dg(T)$ (resp. $(fD)(T)=(gD)(T)$) coincide.

###### Proof.

We will only consider $f\in\Psi_{L}^{Q}(S_{\theta})$, i.e. the first integral
in (19a), the calculations for the second integral are the same.

For the absolute convergence of integral (19a), we use the path
$\gamma(t):=\begin{cases}-te^{J\varphi},&t<0,\\\
te^{-J\varphi},&t>0,\end{cases}$ along the boundary of
$S_{\varphi}\cap\mathbb{C}_{J}$. Then the estimate (17) of the operator
$Q_{c,s}^{-1}(T)$ and the fact that $f\in\Psi_{L}^{Q}(S_{\theta})$, give the
absolute convergence of integral

$\int_{\mathbb{R}\setminus\\{0\\}}\big{\|}Q_{c,\gamma(t)}^{-1}(T)\big{\|}\,|\gamma^{\prime}(t)|\,|f(\gamma(t))|dt\leq
C_{\varphi}C_{\alpha}\int_{\mathbb{R}\setminus\\{0\\}}\frac{|t|^{\alpha-1}}{1+|t|^{1+2\alpha}}dt<\infty.$
(20)

For the independence of the angle $\varphi$, we consider two angles
$\varphi_{1}<\varphi_{2}\in(\omega,\theta)$ and for every $0<\varepsilon<R$ we
consider the curves

$\mathbb{C}_{J}$$\varphi_{1}$$\varphi_{2}$$\sigma_{\varepsilon}$$\sigma_{\varepsilon}$$\gamma_{1,\varepsilon,R}$$\gamma_{1,\varepsilon,R}$$\gamma_{2,\varepsilon,R}$$\gamma_{2,\varepsilon,R}$$\sigma_{R}$$\sigma_{R}$

$\displaystyle\sigma_{\varepsilon}(\varphi)$ $\displaystyle:=\varepsilon
e^{J\varphi},\hskip
29.30629pt\varphi\in(-\varphi_{2},-\varphi_{1})\cup(\varphi_{1},\varphi_{2}),$
$\displaystyle\sigma_{R}(\varphi)$ $\displaystyle:=Re^{J\varphi},\hskip
25.6073pt\varphi\in(-\varphi_{2},-\varphi_{1})\cup(\varphi_{1},\varphi_{2}),$
$\displaystyle\gamma_{1,\varepsilon,R}(t)$
$\displaystyle:=\begin{cases}-te^{J\varphi_{1}},&t\in(-R,-\varepsilon),\\\
te^{-J\varphi_{1}},&t\in(\varepsilon,R),\end{cases}$
$\displaystyle\gamma_{2,\varepsilon,R}(t)$
$\displaystyle:=\begin{cases}-te^{J\varphi_{2}},&t\in(-R,-\varepsilon),\\\
te^{-J\varphi_{2}},&t\in(\varepsilon,R).\end{cases}$

Then the Cauchy integral theorem gives

$\int_{\gamma_{1,\varepsilon,R}}Q_{c,s}^{-1}(T)ds_{J}f(s)=\int_{\sigma_{R}\oplus\gamma_{2,\varepsilon,R}\ominus\sigma_{\varepsilon}}Q_{c,s}^{-1}(T)ds_{J}f(s).$
(21)

In the limit $\varepsilon\rightarrow 0^{+}$, the integral along
$\sigma_{\varepsilon}$ vanishes, because of

$\displaystyle\bigg{|}\int_{\sigma_{\varepsilon}}Q_{c,s}^{-1}(T)ds_{J}f(s)\bigg{|}$
$\displaystyle=\bigg{|}\int_{\varphi_{1}<|\varphi|<\varphi_{2}}Q_{c,\varepsilon
e^{J\varphi}}^{-1}(T)\varepsilon e^{J\varphi}f(\varepsilon
e^{J\varphi})d\varphi\bigg{|}$ $\displaystyle\leq
C_{\varphi_{1}}C_{\alpha}\int_{\varphi_{1}<|\varphi|<\varphi_{2}}\frac{1}{\varepsilon^{2}}\,\varepsilon\,\frac{\varepsilon^{1+\alpha}}{1+\varepsilon^{1+2\alpha}}d\varphi$
$\displaystyle=2C_{\varphi_{1}}C_{\alpha}(\varphi_{2}-\varphi_{1})\frac{\varepsilon^{\alpha}}{1+\varepsilon^{1+2\alpha}}\overset{\varepsilon\rightarrow
0^{+}}{\longrightarrow}0.$ (22)

In the same way also the integral along $\sigma_{R}$ vanishes in the limit
$R\rightarrow\infty$,

$\bigg{|}\int_{\sigma_{R}}Q_{c,s}^{-1}(T)ds_{J}f(s)\bigg{|}\leq
2C_{\varphi_{1}}C_{\alpha}(\varphi_{2}-\varphi_{1})\frac{R^{\alpha}}{1+R^{1+2\alpha}}\overset{R\rightarrow\infty}{\longrightarrow}0.$
(23)

Performing now the limits $\varepsilon\rightarrow 0^{+}$ and
$R\rightarrow\infty$ in (21) and using that the integrals (22) and (23)
vanish, we obtain the independency of the angle

$\int_{\partial(S_{\varphi_{1}}\cap\mathbb{C}_{J})}Q_{c,s}^{-1}(T)ds_{J}f(s)=\int_{\partial(S_{\varphi_{2}}\cap\mathbb{C}_{J})}Q_{s,c}^{-1}(T)ds_{J}f(s).$

For the independence on the imaginary unit $J\in\mathbb{S}$, we consider two
imaginary units $J,I\in\mathbb{S}$. For any angles
$\varphi_{1}<\varphi_{2}<\varphi_{3}\in(\omega,\theta)$ we fix now
$\varepsilon>0$ and define the paths

$\mathbb{C}_{I}$$\sigma_{\varepsilon}$$\sigma_{\varepsilon}$$\gamma_{1,\varepsilon}$$\gamma_{1,\varepsilon}$$\gamma_{3,\varepsilon}$$\gamma_{3,\varepsilon}$$\gamma_{2,\varepsilon}$$[s]$$[s]$

$\displaystyle\gamma_{1,\varepsilon}(t)$
$\displaystyle:=\begin{cases}-te^{I\varphi_{1}},&t<-\varepsilon,\\\
te^{-I\varphi_{1}},&t>\varepsilon,\end{cases}$
$\displaystyle\gamma_{2,\varepsilon}(t)$
$\displaystyle:=\begin{cases}-te^{J\varphi_{2}},&t<-\varepsilon,\\\
te^{-J\varphi_{2}},&t>\varepsilon,\end{cases}$ (24)
$\displaystyle\gamma_{3,\varepsilon}(t)$
$\displaystyle:=\begin{cases}-te^{I\varphi_{3}},&t<-\varepsilon,\\\
te^{-I\varphi_{3}},&t>\varepsilon,\end{cases}$
$\displaystyle\sigma_{\varepsilon}(\varphi)$ $\displaystyle:=\varepsilon
e^{I\varphi},\quad\varphi\in(-\varphi_{3},-\varphi_{1})\cup(\varphi_{1},\varphi_{3}).$

Note, that $\gamma_{1,\varepsilon}$, $\gamma_{3,\varepsilon}$,
$\sigma_{\varepsilon}$ are curves in $\mathbb{C}_{I}$, while
$\gamma_{2,\varepsilon}$ is in $\mathbb{C}_{J}$.

The Cauchy formula (3) then gives the representation

$f(s)=\frac{1}{2\pi}\int_{\gamma_{3,\varepsilon}\ominus\sigma_{\varepsilon}\ominus\gamma_{1,\varepsilon}}S_{L}^{-1}(p,s)dp_{I}f(p),\qquad
s\in\operatorname{ran}(\gamma_{2,\varepsilon}),$ (25)

where the integral along $\infty$, which closes the path on the right,
vanishes due to the asymptotics

$|S_{L}^{-1}(p,s)f(p)|=\frac{|p-\overline{s}|}{|p^{2}-2s_{0}p+|s|^{2}|}\frac{C_{\alpha}|p|^{1+\alpha}}{1+|p|^{1+2\alpha}}=\mathcal{O}(|p|^{-1-\alpha}),\quad\text{as
}p\rightarrow\infty.$ (26)

Next we consider the curves

$\mathbb{C}_{J}$$[p]$$[p]$$\gamma_{3,\varepsilon}$$\gamma_{1,\varepsilon}$$\gamma_{2,\varepsilon}$$\gamma_{2,\varepsilon}$$\kappa_{\varepsilon}$$\kappa_{\varepsilon}$$\tau_{\frac{\varepsilon}{2}}$

$\displaystyle\tau_{\frac{\varepsilon}{2}}(\varphi)$
$\displaystyle:=\frac{\varepsilon}{2}e^{J\varphi},\qquad\varphi\in(\varphi_{2},2\pi-\varphi_{2}),$
$\displaystyle\kappa_{\varepsilon}(t)$
$\displaystyle:=\begin{cases}-te^{J\varphi_{2}},&t\in(-\varepsilon,-\frac{\varepsilon}{2}),\\\
te^{-J\varphi_{2}},&t\in(\frac{\varepsilon}{2},\varepsilon).\end{cases}$ (27)

In this setting the Cauchy formula (3) gives

$\displaystyle Q_{c,p}^{-1}(T)$
$\displaystyle=\frac{-1}{2\pi}\int_{\gamma_{2,\varepsilon}\oplus\kappa_{\varepsilon}\oplus\tau_{\frac{\varepsilon}{2}}}Q_{c,s}^{-1}(T)ds_{J}S_{R}^{-1}(s,p)$
$\displaystyle=\frac{1}{2\pi}\int_{\gamma_{2,\varepsilon}\oplus\kappa_{\varepsilon}\oplus\tau_{\frac{\varepsilon}{2}}}Q_{c,s}^{-1}(T)ds_{J}S_{L}^{-1}(p,s),\quad
p\in\operatorname{ran}(\gamma_{3,\varepsilon}),$ (28)

where the negative sign in the first line comes from the fact that the curve
$\gamma_{2,\varepsilon}\oplus\kappa_{\varepsilon}\oplus\tau_{\frac{\varepsilon}{2}}$
(closed at $\infty$ on the left) surrounds the points $[p]\cap\mathbb{C}_{J}$
in the negative sense, and in the second line we used the connection
$S_{R}^{-1}(s,p)=-S_{L}^{-1}(p,s)$ between the left and the right Cauchy
kernel. The integral along $\infty$, which closes the path on the left,
vanishes due to the asymptotics

$\|Q_{c,s}^{-1}(T)\|\,|S_{R}^{-1}(s,p)|\leq\frac{C_{\varphi_{2}}}{|s|^{2}}\frac{|s-\overline{p}|}{|s^{2}-2p_{0}s+|p|^{2}|}=\mathcal{O}(|s|^{-2}),\quad\text{as
}s\rightarrow\infty.$

Analogously we obtain

$0=\frac{1}{2\pi}\int_{\gamma_{2,\varepsilon}\oplus\kappa_{\varepsilon}\oplus\tau_{\frac{\varepsilon}{2}}}Q_{c,s}^{-1}(T)ds_{J}S_{L}^{-1}(p,s),\qquad
p\in\operatorname{ran}(\gamma_{1,\varepsilon}),$ (29)

since in this case the points $[p]\cap\mathbb{C}_{J}$ lie outside the
integration path and hence the Cauchy integral vanishes. The combination of
(25), (28) and (29), leads to the formula

$\displaystyle\int_{\gamma_{2,\varepsilon}}Q_{c,s}^{-1}(T)ds_{J}f(s)$
$\displaystyle=\frac{1}{2\pi}\int_{\gamma_{2,\varepsilon}}Q_{c,s}^{-1}(T)ds_{J}\bigg{(}\int_{\gamma_{3,\varepsilon}\ominus\sigma_{\varepsilon}\ominus\gamma_{1,\varepsilon}}S_{L}^{-1}(p,s)dp_{I}f(p)\bigg{)}$
$\displaystyle=\int_{\gamma_{3,\varepsilon}}\bigg{(}Q_{c,p}^{-1}(T)-\frac{1}{2\pi}\int_{\kappa_{\varepsilon}\oplus\tau_{\frac{\varepsilon}{2}}}Q_{c,s}^{-1}(T)ds_{J}S_{L}^{-1}(p,s)\bigg{)}dp_{I}f(p)$
$\displaystyle\quad-\frac{1}{2\pi}\int_{\sigma_{\varepsilon}}\bigg{(}\int_{\gamma_{2,\varepsilon}}Q_{c,s}^{-1}(T)ds_{J}S_{L}^{-1}(p,s)\bigg{)}dp_{I}f(p)$
$\displaystyle\quad+\frac{1}{2\pi}\int_{\gamma_{1,\varepsilon}}\bigg{(}\int_{\kappa_{\varepsilon}\oplus\tau_{\frac{\varepsilon}{2}}}Q_{c,s}^{-1}(T)ds_{J}S_{L}^{-1}(p,s)\bigg{)}dp_{I}f(p)$
$\displaystyle=\int_{\gamma_{3,\varepsilon}}Q_{c,p}^{-1}(T)dp_{I}f(p)-\frac{1}{2\pi}\int_{\sigma_{\varepsilon}}\int_{\gamma_{2,\varepsilon}\oplus\kappa_{\varepsilon}\oplus\tau_{\frac{\varepsilon}{2}}}Q_{c,s}^{-1}(T)ds_{J}S_{L}^{-1}(p,s)dp_{I}f(p),$
(30)

where in the last equation we combined the integration path
$\gamma_{1,\varepsilon}\ominus\gamma_{3,\varepsilon}$ and replaced it by
$\sigma_{\varepsilon}$, see also the graphic in (24). This is allowed by the
Cauchy theorem since the integral along $\infty$, which closes the path on the
right, vanishes due the asymptotics (26), and since the singularities of
$S_{L}^{-1}(p,s)$ lie on
$\kappa_{\varepsilon}\oplus\tau_{\frac{\varepsilon}{2}}$, which is outside the
integration path, see the graphic in (27). Finally, we now perform the limit
$\varepsilon\rightarrow 0^{+}$ to this equation and show that the double
integral in (30) vanishes. Therefore, we estimate the integrand by

$\big{|}Q_{c,s}^{-1}(T)S_{L}^{-1}(p,s)f(p)\big{|}\leq\frac{C_{\varphi_{1}}}{|s|^{2}}\frac{|p|+|s|}{|p^{2}-2s_{0}p+|s|^{2}|}\frac{C_{\alpha}|p|^{1+\alpha}}{1+|p|^{1+2\alpha}}\leq\frac{C_{\varphi_{1}}C_{\alpha}(|p|+|s|)|p|^{1+\alpha}}{|s|^{2}|p-s_{j}||p-\overline{s_{j}}|},$

where $\\{s_{I},\overline{s_{I}}\\}=[s]\cap\mathbb{C}_{I}$ are the
intersections of $[s]$ with the complex plane $\mathbb{C}_{I}$. The first part
of the integral then vanishes because of

$\displaystyle\bigg{|}\int_{\sigma_{\varepsilon}}\int_{\gamma_{2,\varepsilon}\oplus\kappa_{\varepsilon}}$
$\displaystyle Q_{c,s}^{-1}(T)ds_{J}S_{L}^{-1}(p,s)dp_{I}f(p)\bigg{|}$
$\displaystyle\leq
2C_{\varphi_{1}}C_{\alpha}\int_{\varphi_{1}<|\varphi|<\varphi_{3}}\int_{\frac{\varepsilon}{2}}^{\infty}\frac{(\varepsilon+r)\varepsilon^{1+\alpha}}{r^{2}|\varepsilon
e^{I\varphi}-re^{I\varphi_{2}}||\varepsilon
e^{I\varphi}-re^{-I\varphi_{2}}|}\varepsilon drd\varphi$
$\displaystyle=2C_{\varphi_{1}}C_{\alpha}\varepsilon^{\alpha}\int_{\varphi_{1}<|\varphi|<\varphi_{3}}\int_{\frac{1}{2}}^{\infty}\frac{1+\rho}{\rho^{2}|e^{I\varphi}-\rho
e^{I\varphi_{2}}||e^{I\varphi}-\rho e^{-I\varphi_{2}}|}d\rho
d\varphi\overset{\varepsilon\rightarrow 0^{+}}{\longrightarrow}0,$

where the double integral in the last line exists since the singularities
$(\rho,\varphi)=(1,\pm\varphi_{2})$ of the denominator is of order $1$, which
is integrable in the two-dimensional integral. Also the second part of the
double integral in (30) vanishes, since

$\displaystyle\bigg{|}\int_{\sigma_{\varepsilon}}\int_{\tau_{\frac{\varepsilon}{2}}}$
$\displaystyle Q_{c,s}^{-1}(T)ds_{J}S_{L}^{-1}(p,s)dp_{I}f(p)\bigg{|}$
$\displaystyle\leq
C_{\varphi_{1}}C_{\alpha}\int_{\varphi_{1}<|\varphi|\varphi_{3}}\int_{\varphi_{2}}^{2\pi-\varphi_{2}}\frac{(\varepsilon+\frac{\varepsilon}{2})\varepsilon^{1+\alpha}}{\frac{\varepsilon^{2}}{4}|\varepsilon
e^{I\varphi}-\frac{\varepsilon}{2}e^{I\phi}||\varepsilon
e^{I\varphi}-\frac{\varepsilon}{2}e^{-I\phi}|}\varepsilon\frac{\varepsilon}{2}d\phi
d\varphi$
$\displaystyle=3C_{\varphi_{1}}C_{\alpha}\varepsilon^{\alpha}\int_{\varphi_{1}<|\varphi|<\varphi_{3}}\int_{\varphi_{2}}^{2\pi-\varphi_{2}}\frac{1}{|e^{I\varphi}-\frac{1}{2}e^{I\phi}||e^{I\varphi}-\frac{1}{2}e^{-I\phi}|}d\phi
d\varphi\overset{\varepsilon\rightarrow 0^{+}}{\longrightarrow}0.$

Hence, in the limit $\varepsilon\rightarrow 0^{+}$ the equation (30) turns
into the desired independence of the imaginary unit

$\int_{\partial(S_{\varphi_{2}}\cap\mathbb{C}_{J})}Q_{c,s}^{-1}(T)ds_{J}f(s)=\int_{\partial(S_{\varphi_{3}}\cap\mathbb{C}_{I})}Q_{c,p}^{-1}(T)dp_{I}f(p).$

For the independence of the kernel of $D$, as in [10], we consider two
functions $f,g\in\mathcal{SH}_{L}(S_{\theta})$ with $Df(s)=Dg(s)$. Due to the
decomposition (8) we can write

$f(u+Jv)-g(u+Jv)=\alpha(u,v)+J\beta(u,v).$

Moreover, we can write the Cauchy-Fueter operator in spherical coordinates
with respect to this decomposition $x=u+Jv$, with $u\in\mathbb{R}$ and $v>0$,
as

$D=\frac{\partial}{\partial x_{0}}+e_{1}\frac{\partial}{\partial
x_{1}}+e_{2}\frac{\partial}{\partial x_{2}}+e_{3}\frac{\partial}{\partial
x_{3}}=\frac{\partial}{\partial u}+J\frac{\partial}{\partial
v}+\frac{J}{v}\Gamma_{J},$

where $\Gamma_{J}$ is a symbol for the angular derivatives. Using now the
identity $J\Gamma_{J}J=\Gamma_{J}-2$, see [31, Paragraph 1.12.1], it follows
from $D(f-g)(s)=0$ that the functions $\alpha$ and $\beta$ satisfy the Vekua-
type differential equations

$\displaystyle\frac{\partial}{\partial u}\alpha(u,v)-\frac{\partial}{\partial
v}\beta(u,v)$ $\displaystyle=\frac{2}{v}\beta(u,v),$
$\displaystyle\frac{\partial}{\partial v}\alpha(u,v)+\frac{\partial}{\partial
u}\beta(u,v)$ $\displaystyle=0.$

However, $\alpha$ and $\beta$ also satisfy the Cauchy-Riemann equations (9)
since $f-g$ is slice hyperholomorphic, and hence we conclude $\beta(u,v)=0$
and consequently also $\frac{\partial}{\partial
u}\alpha(u,v)=\frac{\partial}{\partial v}\alpha(u,v)=0$ have to vanish. Since
the domain $S_{\theta}$ on which the function $f-g$ is holomorphic is
connected, this implies, that the function

$f(s)-g(s)=c\quad\text{is constant for every }s\in S_{\theta}\text{ with
}\operatorname{Im}(s)\neq 0.$ (31)

By continuity, the difference $f(s)-g(s)=c$ then has to be constant for every
$s\in S_{\theta}$. However, this is only possible for $c=0$, since otherwise
$f-g\notin\Psi_{L}^{Q}(S_{\theta})$. This means that $f=g$ and then
$Df(T)=Dg(T)$ follows from the integral (19a). ∎

In the next lemma we state same basic properties of the functional calculus
(19a).

######  Lemma 3.5.

Let $T\in\mathcal{KC}(V)$ with $T,\overline{T}$ being operators of type
$\omega$, $f,g\in\Psi^{Q}_{L}(S_{\theta})$, for some $\theta\in(\omega,\pi)$.
Then

  1. i)

$Df(T)$ is a bounded operator with commuting components.

  2. ii)

$D(f+g)(T)=Df(T)+Dg(T)$,

  3. iii)

$Df(\overline{T})=Df(T)$,

  4. iv)

If $f$ is intrinsic, then $\overline{Df(T)}=Df(T)$.

The same results hold true for $f,g\in\Psi_{R}^{Q}(S_{\theta})$ and the
corresponding functional calculus (19b).

###### Proof.

The boundedness of $Df(T)$ in the statement i) follows from the estimate (20).
The linearity in ii) follows from the linearity of the integral (19a). In
order to show that $Df(T)$ has commuting components, we fix $J\in\mathbb{S}$
and $\varphi\in(\omega,\pi)$ and write the integral (19a) as

$\displaystyle Df(T)$
$\displaystyle=\frac{-1}{\pi}\int_{\partial(S_{\varphi}\cap\mathbb{C}_{J})}Q_{c,s}^{-1}(T)ds_{J}f(s)$
$\displaystyle=\frac{-1}{\pi}\bigg{(}\int_{-\infty}^{0}Q_{c,-te^{J\varphi}}^{-1}(T)Je^{J\varphi}f(-te^{J\varphi})dt-\int_{0}^{\infty}Q_{c,te^{-J\varphi}}^{-1}(T)Je^{-J\varphi}f(te^{-J\varphi})dt\bigg{)}$
$\displaystyle=\frac{-1}{\pi}\int_{0}^{\infty}\Big{(}Q_{c,te^{J\varphi}}^{-1}(T)Je^{J\varphi}f(te^{J\varphi})-Q_{c,te^{-J\varphi}}^{-1}(T)Je^{-J\varphi}f(te^{-J\varphi})\Big{)}dt$
$\displaystyle=\frac{-1}{\pi}\int_{0}^{\infty}\Big{(}Q_{c,te^{-J\varphi}}(T)Je^{J\varphi}f(te^{J\varphi})-Q_{c,te^{J\varphi}}(T)Je^{-J\varphi}f(te^{-J\varphi})\Big{)}|Q_{c,te^{J\varphi}}(T)|^{-2}dt,$
(32)

where in the last line we rewrote the commutative pseudo resolvent as

$Q_{c,s}^{-1}(T)=Q_{c,\overline{s}}(T)Q_{c,\overline{s}}^{-1}(T)Q_{c,s}^{-1}(T)=Q_{c,\overline{s}}(T)|Q_{c,s}(T)|^{-2},$

with

$|Q_{c,s}(T)|^{2}=|s|^{4}\mathcal{I}-4s_{0}|s|^{2}T_{0}+4s_{0}^{2}|T|^{2}-2|s|^{2}|T|^{2}-4s_{0}T_{0}|T|^{2}+4|s|^{2}T_{0}^{2}+|T|^{4}.$
(33)

Using also the explicit value

$Q_{c,te^{\pm J\varphi}}(T)=t^{2}e^{\pm 2J\varphi}-2te^{\pm
J\varphi}T_{0}+|T|^{2}$ (34)

from (10) as well as the decomposition

$f(te^{J\varphi})=\alpha(t\cos\varphi,t\sin\varphi)+J\beta(t\cos\varphi,t\sin\varphi),$
(35)

from (8), makes this integral

$\displaystyle Df(T)$
$\displaystyle=\frac{-1}{\pi}\int_{0}^{\infty}\Big{(}\big{(}Q_{c,te^{-J\varphi}}(T)e^{J\varphi}-Q_{c,te^{J\varphi}}(T)e^{-J\varphi}\big{)}J\alpha(t\cos\varphi,t\sin\varphi)$
$\displaystyle\hskip
65.44142pt-\big{(}Q_{c,te^{-J\varphi}}(T)e^{J\varphi}+Q_{c,te^{J\varphi}}(T)e^{-J\varphi}\big{)}\beta(t\cos\varphi,t\sin\varphi)\Big{)}|Q_{c,te^{J\varphi}}(T)|^{-2}dt$
$\displaystyle=\frac{2}{\pi}\int_{0}^{\infty}\Big{(}(|T|^{2}-t^{2})\sin\varphi\alpha(t\cos\varphi,t\sin\varphi)$
$\displaystyle\hskip
71.13188pt+\big{(}(|T|^{2}+t^{2})\cos\varphi-2tT_{0}\big{)}\beta(t\cos\varphi,t\sin\varphi)\Big{)}|Q_{c,te^{J\varphi}}(T)|^{-2}dt.$
(36)

If we now decompose

$\alpha(u,v)=\alpha_{0}(u,v)+\sum\limits_{i=1}^{3}e_{i}\alpha_{i}(u,v)\qquad\text{and}\qquad\beta(u,v)=\beta_{0}(u,v)+\sum\limits_{i=1}^{3}e_{i}\beta_{i}(u,v),$

into their components, the components $Df(T)_{i}$, $i\in\\{0,\dots,3\\}$ of
the harmonic functional calculus are explicitly given by

$\displaystyle Df(T)_{i}$
$\displaystyle=\frac{2}{\pi}\int_{0}^{\infty}\Big{(}(|T|^{2}-t^{2})\sin\varphi\alpha_{i}(t\cos\varphi,t\sin\varphi)$
$\displaystyle\hskip
65.44142pt+\big{(}(|T|^{2}+t^{2})\cos\varphi-2tT_{0}\big{)}\beta_{i}(t\cos\varphi,t\sin\varphi)\Big{)}|Q_{c,te^{J\varphi}}(T)|^{-2}dt,$
(37)

which are obviously commuting, since the components of $T$ are commuting and
there are no imaginary units involved.

The statement ii) follows from the fact, that the resolvent
$Q_{c,s}^{-1}(T)=Q_{c,s}^{-1}(\overline{T})$ is the same for $T$ and for
$\overline{T}$ by definition (10). Hence also $Dg(T)=Dg(\overline{T})$ follows
immediately.

For the proof of iii) we note that by Definition 2.2, the functions $\alpha$
and $\beta$ in the decomposition (35) are real valued since $f$ is intrinsic.
Then it follows from the explicit representation (37) that
$Df(T)_{1}=Df(T)_{2}=Df(T)_{3}=0$. Hence only the term $Df(T)=Df(T)_{0}$
remains and the conjugate operator is $\overline{Df(T)}=Df(T)$. ∎

To the best of our knowledge, a similar statement as Lemma 3.5 is not known
for the $S$-functional calculus (18a). However, we will need those results in
Section 4, and hence prove the following lemma regarding properties of the
commuting components of the $S$-functional calculus.

######  Lemma 3.6.

Let $T\in\mathcal{KC}(V)$ with $T,\overline{T}$ being operators of type
$\omega$, and $f,g\in\Psi_{L}(S_{\theta})$ or $f,g\in\Psi_{R}(S_{\theta})$,
for some $\theta\in(\omega,\pi)$. Then

  1. i)

$f(T)$ is a bounded operator with commuting components.

  2. ii)

$(f+g)(T)=f(T)+g(T)$

  3. iii)

If $f$ is intrinsic, then $\overline{f(T)}=f(\overline{T})$.

###### Proof.

The boundedness i) of the $S$-functional calculus (18a) follows from the
estimate

$\int_{\mathbb{R}\setminus\\{0\\}}\big{\|}S_{L}^{-1}(\gamma(t),T)\big{\|}\,|\gamma^{\prime}(t)|\,|f(\gamma(t))|dt\leq
C_{\varphi}C_{\alpha}\int_{\mathbb{R}\setminus\\{0\\}}\frac{|t|^{\alpha-1}}{1+|t|^{1+2\alpha}}dt<\infty,$

which is similar to (20), using the bound (16) of the $S$-resolvent and the
definition of the space $\Psi_{L}(S_{\theta})$ in Remark 3.1. The linearity
ii) is also clear by the linearity of the integral (18a). For the proof of
iii) we fix $J\in\mathbb{S}$, $\varphi\in(\omega,\pi)$ and derive in a similar
way as in (32) with $Q_{c,s}^{-1}(T)$ replaced by
$S_{L}^{-1}(s,T)=(s\mathcal{I}-\overline{T})Q_{c,s}^{-1}(T)$ the formula

$\displaystyle f(T)$
$\displaystyle=\frac{1}{2\pi}\int_{0}^{\infty}\Big{(}(te^{J\varphi}-\overline{T})Q_{c,te^{-J\varphi}}(T)Je^{J\varphi}f(te^{J\varphi})$
$\displaystyle\hskip
65.44142pt-(te^{-J\varphi}-\overline{T})Q_{c,te^{J\varphi}}(T)Je^{-J\varphi}f(te^{-J\varphi})\Big{)}|Q_{c,te^{J\varphi}}(T)|^{-2}dt.$
(38)

Using the explicit value (34) of $Q_{c,te^{\pm J\varphi}}(T)$ and the
decomposition (35) of $f$, we further rewrite the integral (38) in the same as
we derived (33), such that we get

$\displaystyle f(T)$
$\displaystyle=\frac{1}{\pi}\int_{0}^{\infty}\Big{(}\big{(}2tT_{0}\sin\varphi-t|T|^{2}\sin(2\varphi)\big{)}\alpha(t\cos\varphi,t\sin\varphi)$
$\displaystyle\hskip
65.44142pt-\big{(}t^{2}\sin\varphi-|T|^{2}\sin\varphi\big{)}\overline{T}\,\alpha(t\cos\varphi,t\sin\varphi)$
$\displaystyle\hskip
65.44142pt-\big{(}t^{3}-2t^{2}T_{0}\cos\varphi+t|T|^{2}\cos(2\varphi)\big{)}\beta(t\cos\varphi,t\sin\varphi)$
$\displaystyle\hskip
65.44142pt+\big{(}t^{2}\cos\varphi-2tT_{0}+|T|^{2}\cos\varphi\big{)}\overline{T}\,\beta(t\cos\varphi,t\sin\varphi)\Big{)}|Q_{c,te^{J\varphi}}(T)|^{-2}dt.$
(39)

If we finally write

$\alpha=\alpha_{0}+\sum\limits_{i=1}^{3}e_{i}\alpha_{i},\quad\beta=\beta_{0}+\sum\limits_{i=1}^{3}e_{i}\beta_{i}\quad\text{and}\quad\overline{T}=T_{0}-\sum\limits_{i=1}^{3}e_{i}T_{i},$

with real valued functions
$\alpha_{i},\beta_{i}:\mathcal{U}\rightarrow\mathbb{R}$, $i\in\\{0,1,2,3\\}$,
and sort (39) with respect to the imaginary units, we end up with components
of $f(T)$ which are real linear combinations of powers of $T_{i}$,
$i\in\\{0,1,2,3\\}$. Due to the commutativity of the components $T_{i}$, also
the components of $f(T)$ are then commuting. Hence i) is proven. For the proof
of iii), we note that $\alpha$ and $\beta$ are real valued, since $f$ is
assumed to be intrinsic. Hence we immediately obtain the identity
$\overline{f(T)}=f(\overline{T})$ from (39). ∎

######  Proposition 3.7.

Let $T\in\mathcal{KC}(V)$ with $T,\overline{T}$ being operators of type
$\omega$, $B:V\rightarrow V$ an everywhere defined bounded operator and
$f\in\Psi^{Q}(S_{\theta})$, for some $\theta\in(\omega,\pi)$. Then there holds

$\begin{array}[]{c}\forall j\in\\{0,\dots,3\\}:BT_{j}=T_{j}B\text{ on
}\operatorname{dom}(T)\\\ \Downarrow\\\ BDf(T)=Df(T)B.\end{array}$

###### Proof.

From the assumption it follows that $B$ in particular commutes with the term
$|Q_{c,s}(T)|^{2}$ from (33) on $\operatorname{dom}(T^{2})$, and hence also
with its inverse

$|Q_{c,s}(T)|^{-2}B=B|Q_{c,s}(T)|^{-2},\quad\text{on }V.$

Using this fact in the representation (36) of $Df(T)$, immediately gives the
commutation

$Df(T)B=BDf(T)$

since there are no imaginary units in the integrand of (36), not even in the
functions $\alpha$ and $\beta$, which are real valued since $f$ is assumed to
be intrinsic. ∎

######  Corollary 3.8.

Let $T\in\mathcal{KC}(V)$ with $T,\overline{T}$ being operators of type
$\omega$, $f\in\Psi^{Q}(S_{\theta})$ and $g\in\Psi_{L}^{Q}(S_{\theta})$ or
$g\in\Psi_{R}^{Q}(S_{\theta})$, for some $\theta\in(\omega,\pi)$. Then there
holds

$g(T)Df(T)=Df(T)g(T).$

###### Proof.

We only consider $g\in\Psi_{L}^{Q}(S_{\theta})$, the case
$g\in\Psi_{R}^{Q}(S_{\theta})$ follows analogously. It is clear by definition
(12), that $T_{j}S_{L}^{-1}(s,T)=S_{L}^{-1}(s,T)T_{j}$ commutes on
$\operatorname{dom}(T)$ for every $j\in\\{0,\dots,3\\}$ and hence also

$g(T)T_{j}=\frac{1}{2\pi}\int_{\partial(S_{\varphi}\cap\mathbb{C}_{J})}S_{L}^{-1}(s,T)ds_{J}g(s)T_{j}=\frac{1}{2\pi}\int_{\partial(S_{\varphi}\cap\mathbb{C}_{J})}T_{j}S_{L}^{-1}(s,T)ds_{J}g(s)=T_{j}g(T),$

on $\operatorname{dom}(T)$. From Proposition 3.7 we then conclude
$g(T)Df(T)=Df(T)g(T)$. ∎

Next we want to derive the very important product rule of the harmonic
functional calculus. The basic ingredient will be the following resolvent
equation.

######  Lemma 3.9.

Let $T\in\mathcal{KC}(V)$, $s,p\in\rho_{S}(T)$ with $s\notin[p]$. Then

$\displaystyle Q_{c,s}^{-1}(T)S_{L}^{-1}(p,s)+S_{R}^{-1}(s,p)Q_{c,p}^{-1}(T)$
$\displaystyle=Q_{c,s}^{-1}(T)S_{L}^{-1}(p,T)+S_{R}^{-1}(s,\overline{T})Q_{c,p}^{-1}(T)$
(40a)
$\displaystyle=Q_{c,s}^{-1}(T)S_{L}^{-1}(p,\overline{T})+S_{R}^{-1}(s,T)Q_{c,p}^{-1}(T).$
(40b)

###### Proof.

Using the left $S$-resolvent in (12) and the representation (13) of the right
resolvent, we obtain

$\displaystyle Q_{c,s}^{-1}(T)S_{L}^{-1}$
$\displaystyle(p,T)+S_{R}^{-1}(s,\overline{T})Q_{c,p}^{-1}(T)$
$\displaystyle=Q_{c,s}^{-1}(T)\big{(}S_{L}^{-1}(p,T)Q_{c,p}(T)+Q_{c,s}(T)S_{R}^{-1}(s,\overline{T})\big{)}Q_{c,p}^{-1}(T)$
$\displaystyle=Q_{c,s}^{-1}(T)\big{(}(p-\overline{T})Q_{c,p}^{-1}(T)Q_{c,p}(T)+Q_{c,s}(T)Q_{c,s}^{-1}(T)(s-T)\big{)}Q_{c,p}^{-1}(T)$
$\displaystyle=Q_{c,s}^{-1}(T)(p-2T_{0}+s)Q_{c,p}^{-1}(T).$ (41)

Since the right hand side of this equation does not change when we replace
$T\rightarrow\overline{T}$, the equation (40b) is proven. Moreover, using (12)
for $S_{L}^{-1}(p,s)$ and (15) for $S_{R}^{-1}(s,p)$, analogously gives

$\displaystyle Q_{c,s}^{-1}(T)S_{L}^{-1}$
$\displaystyle(p,s)+S_{R}^{-1}(s,p)Q_{c,p}^{-1}(T)$
$\displaystyle=Q_{c,s}^{-1}(T)\big{(}S_{L}^{-1}(p,s)Q_{c,p}(T)+Q_{c,s}(T)S_{R}^{-1}(s,p)\big{)}Q_{c,p}^{-1}(T)$
$\displaystyle=Q_{c,s}^{-1}(T)\big{(}(p-\overline{s})Q_{c,p}(T)+Q_{c,s}(T)(\overline{s}-p)\big{)}(p^{2}-2s_{0}p+|s|^{2})^{-1}Q_{c,p}^{-1}(T)$
$\displaystyle=Q_{c,s}^{-1}(T)(p-2T_{0}+s)Q_{c,p}^{-1}(T).$ (42)

Comparison of (41) and (42) gives the resolvent identity (40a). ∎

With the resolvent equation (40), we are now ready to prove the product rule
of the harmonic functional calculus.

######  Theorem 3.10.

Let $T\in\mathcal{KC}(V)$ with $T,\overline{T}$ being operators of type
$\omega$ and $\theta\in(\omega,\pi)$. Then for every
$f\in\Psi^{Q}(S_{\theta})$, $g\in\Psi_{L}^{Q}(S_{\theta})$ with
$\theta\in(\omega,\pi)$ we obtain the product rule

$\displaystyle D(fg)(T)$ $\displaystyle=Df(T)g(T)+f(\overline{T})Dg(T)$ (43a)
$\displaystyle=Df(T)g(\overline{T})+f(T)Dg(T).$ (43b)

The same product rule also holds for $f\in\Psi^{Q}_{R}(S_{\theta})$ and
$g\in\Psi^{Q}(S_{\theta})$ using the respective functional calculi (18b) and
(19b).

###### Proof.

We only prove the product rule (43a), since (43b) follows from (43a) when we
replace $T$ by $\overline{T}$ and using Lemma 3.5 iii).

Since $f\in\Psi^{Q}(S_{\theta})$ is intrinsic and
$g\in\Psi_{L}^{Q}(S_{\theta})$ is left slice hyperholomorphic, the product
$fg$ is also left slice hyperholomorphic and satisfies the needed estimates in
order to be in the space $fg\in\Psi_{L}(S_{\theta})$. Consider now two angles
$\varphi_{2}<\varphi_{1}\in(\omega,\theta)$ and imaginary units
$J,I\in\mathbb{S}$. Then, using the functional calculi (18a) and (19a), we
have

$\displaystyle Df$ $\displaystyle(T)g(T)+f(\overline{T})Dg(T)$
$\displaystyle=\frac{-1}{2\pi^{2}}\bigg{(}\int_{\partial(S_{\varphi_{1}}\cap\mathbb{C}_{J})}f(s)ds_{J}Q_{c,s}^{-1}(T)\int_{\partial(S_{\varphi_{2}}\cap\mathbb{C}_{I})}S_{L}^{-1}(p,T)dp_{I}g(p)$
$\displaystyle\hskip
56.9055pt+\int_{\partial(S_{\varphi_{1}}\cap\mathbb{C}_{J})}f(s)ds_{J}S_{R}^{-1}(s,\overline{T})\int_{\partial(S_{\varphi_{2}}\cap\mathbb{C}_{I})}Q_{c,p}^{-1}(T)dp_{I}g(p)\bigg{)}$
$\displaystyle=\frac{-1}{2\pi^{2}}\int_{\partial(S_{\varphi_{1}}\cap\mathbb{C}_{J})}f(s)ds_{J}\int_{\partial(S_{\varphi_{2}}\cap\mathbb{C}_{I})}\Big{(}Q_{c,s}^{-1}(T)S_{L}^{-1}(p,T)+S_{R}^{-1}(s,\overline{T})Q_{c,p}^{-1}(T)\Big{)}dp_{I}g(p)$
$\displaystyle=\frac{-1}{2\pi^{2}}\int_{\partial(S_{\varphi_{1}}\cap\mathbb{C}_{J})}f(s)ds_{J}\int_{\partial(S_{\varphi_{2}}\cap\mathbb{C}_{I})}\Big{(}Q_{c,s}^{-1}(T)S_{L}^{-1}(p,s)+S_{R}^{-1}(s,p)Q_{c,p}^{-1}(T)\Big{)}dp_{I}g(p),$
(44)

where in the last line we used the resolvent equation (40a). Since we chose
$\varphi_{2}<\varphi_{1}$, every point
$s\in\partial(S_{\varphi_{1}}\cap\mathbb{C}_{J})$ lies outside
$\overline{S_{\varphi_{2}}}$ and hence we get

$\int_{\partial(S_{\varphi_{2}}\cap\mathbb{C}_{I})}S_{L}^{-1}(p,s)dp_{I}g(p)=0,\qquad
s\in\partial(S_{\varphi_{1}}\cap\mathbb{C}_{J}),$ (45)

where the integral along $\infty$, which closes the set on the right, vanishes
due to the asymptotics

$\big{|}S_{L}^{-1}(p,s)g(p)\big{|}\leq\frac{C_{\beta}|p-\overline{s}||p|^{1+\beta}}{|p^{2}-2s_{0}p+|s|^{2}|(1+|p|^{1+2\beta})}=\mathcal{O}\Big{(}\frac{1}{|p|^{1+\beta}}\Big{)},\qquad\text{as
}p\rightarrow\infty.$

Analogously, every point $p\in\partial(S_{\varphi_{2}}\cap\mathbb{C}_{I})$ is
contained in $S_{\varphi_{1}}$, which gives

$\frac{1}{2\pi}\int_{\partial(S_{\varphi_{1}}\cap\mathbb{C}_{J})}f(s)ds_{J}S_{R}^{-1}(s,p)=f(p),\qquad
p\in\partial(S_{\varphi_{2}}\cap\mathbb{C}_{I}).$ (46)

Plugging now (45) and (46) into (44) reduces the double integral to

$\displaystyle Df(T)g(T)+f(\overline{T})Dg(T)$
$\displaystyle=\frac{-1}{\pi}\int_{\partial(S_{\varphi_{2}}\cap\mathbb{C}_{I})}f(p)Q_{c,p}^{-1}(T)dp_{I}g(p),$
$\displaystyle=\frac{-1}{\pi}\int_{\partial(S_{\varphi_{2}}\cap\mathbb{C}_{I})}Q_{c,p}^{-1}(T)dp_{I}f(p)g(p)=D(fg)(T),$

where in the second line we used that $f$ is intrinsic and interchanges with
$Q_{c,p}^{-1}(T)dp_{J}$. ∎

In the last part of this section we investigate how the harmonic functional
calculus acts on intrinsic rational functions, i.e. on functions of the form
$f(s)=\frac{p(s)}{q(s)}$, consisting of polynomials $p,q$ with real valued
coefficients. To do so, we consider for every polynomial
$q(s)=\sum_{j=0}^{m}b_{j}s^{j}$ and every right-linear closed operator $T$ the
polynomial functional calculus

$q[T]:=\sum\limits_{j=0}^{m}q_{j}T^{j},$ (47)

with $\operatorname{dom}(q[T]):=\operatorname{dom}(T^{m})$ and start with a
lemma, ensuring its invertibility.

######  Lemma 3.11.

Let $T:V\rightarrow V$ be a right-linear closed operator with a two-sided
linear domain and $q\not\equiv 0$ an intrinsic polynomial. If $q$ does not
admit any zeros in $\sigma_{S}(T)$, then $q[T]$ is bijective.

###### Proof.

For the constant polynomial $q(s)=q_{0}$, we have $q_{0}\neq 0$ by the
assumption $q\not\equiv 0$. Hence the operator $q[T]=q_{0}$ is bijective. Let
us now consider $\deg(q)=m\geq 1$ and choose some arbitrary bounded slice
Cauchy domain $U\subseteq\rho_{S}(T)$, containing all the zeros of $q$. Then
define for some fixed $J\in\mathbb{S}$ the operator

$B:=\frac{-1}{2\pi}\int_{\partial(U\cap\mathbb{C}_{J})}S_{L}^{-1}(s,T)ds_{J}\frac{1}{q(s)}.$
(48)

Then we will prove by induction that
$\operatorname{ran}(B)\subseteq\operatorname{dom}(T^{j})$ and

$T^{j}B=\frac{-1}{2\pi}\int_{\partial(U\cap\mathbb{C}_{J})}S_{L}^{-1}(s,T)ds_{J}\frac{s^{j}}{q(s)},\qquad
j\in\\{0,\dots,m-1\\}.$ (49)

The induction start $j=0$ is clear by definition (48). For the induction step
$j-1\rightarrow j$ we use the identity

$S_{L}^{-1}(s,T)s=TS_{L}^{-1}(s,T)+\mathcal{I},$ (50)

see [39, Theorem 2.33], to get

$\int_{\partial(U\cap\mathbb{C}_{J})}S_{L}^{-1}(s,T)ds_{J}\frac{s^{j}}{q(s)}=\int_{\partial(U\cap\mathbb{C}_{J})}TS_{L}^{-1}(s,T)ds_{J}\frac{s^{j-1}}{q(s)}+\int_{\partial(U\cap\mathbb{C}_{J})}ds_{J}\frac{s^{j-1}}{q(s)}.$
(51)

Since the last integral does no longer contain the $S$-resolvent operator
$S_{L}^{-1}(s,T)$, it is holomorphic everywhere except the zeros of the
denominator $q(s)$, which all lie inside $U$. Hence it is allowed by the
Cauchy theorem to replace $U$ by a ball $B_{0,R}$ centered at the origin and
with large enough radius $R$ such that it contains all zeros of $q$. This then
leads to

$\int_{\partial(U\cap\mathbb{C}_{J})}ds_{J}\frac{s^{j-1}}{q(s)}=\int_{\partial(B_{0,R}\cap\mathbb{C}_{J})}ds_{J}\frac{s^{j-1}}{q(s)}\overset{R\rightarrow\infty}{\longrightarrow}0,$
(52)

where the right hand side vanishes in the limit $R\rightarrow\infty$ due to
the asymptotics $\frac{s^{j-1}}{q(s)}=\mathcal{O}(|s|^{j-1-m})$ and $j\leq
m-1$. Hence the left hand side, which does not depend on the radius $R$, has
to vanish identically. This reduces equation (51) to

$\displaystyle\int_{\partial(U\cap\mathbb{C}_{J})}S_{L}^{-1}(s,T)ds_{J}\frac{s^{j}}{q(s)}$
$\displaystyle=\int_{\partial(U\cap\mathbb{C}_{J})}TS_{L}^{-1}(s,T)ds_{J}\frac{s^{j-1}}{q(s)}$
$\displaystyle=T\int_{\partial(U\cap\mathbb{C}_{J})}S_{L}^{-1}(s,T)ds_{J}\frac{s^{j-1}}{q(s)}=-2\pi
T^{j}B,$

where in the second equation we used Hille’s theorem [36, Theorem II.2.6] and
in the third equation the induction assumption (49) for $j-1$. Hence (49) is
proven.

Next, we once more use the resolvent identity (50) to write

$\displaystyle S_{L}^{-1}(s,T)q(s)$
$\displaystyle=b_{0}S_{L}^{-1}(s,T)+\sum\limits_{j=1}^{m}b_{j}(TS_{L}^{-1}(s,T)+\mathcal{I})s^{j-1}$
$\displaystyle=b_{0}S_{L}^{-1}(s,T)+T\sum\limits_{j=1}^{m}b_{j}S_{L}^{-1}(s,T)s^{j-1}+\frac{q(s)-b_{0}}{s}.$

By the Cauchy theorem and the already proven identity (49), we then have

$\displaystyle 0$
$\displaystyle=\frac{-1}{2\pi}\int_{\partial(U\cap\mathbb{C}_{J})}S_{L}^{-1}(s,T)ds_{J}1=\frac{-1}{2\pi}\int_{\partial(U\cap\mathbb{C}_{J})}S_{L}^{-1}(s,T)ds_{J}\frac{q(s)}{q(s)}$
$\displaystyle=b_{0}B+T\sum\limits_{j=1}^{m}b_{j}T^{j-1}B-\frac{1}{2\pi}\int_{\partial(U\cap\mathbb{C}_{J})}ds_{J}\frac{q(s)-b_{0}}{sq(s)}$
$\displaystyle=q[T]B-\frac{1}{2\pi}\int_{\partial(U\cap\mathbb{C}_{J})}ds_{J}\frac{q(s)-b_{0}}{sq(s)}.$
(53)

In the same way as in (52), we are allowed to exchange $U$ by a ball $B_{0,R}$
centered at the origin and with a large enough radius to contain all zeros of
$q$. Here it is important that $s=0$ is not an additional singularity of the
integrand since it is also a zero of the nominator. This then leads to the
integral

$\displaystyle\frac{1}{2\pi}\int_{\partial(U\cap\mathbb{C}_{J})}ds_{J}\frac{q(s)-b_{0}}{sq(s)}$
$\displaystyle=\frac{1}{2\pi}\int_{\partial(B_{0,R}\cap\mathbb{C}_{J})}ds_{J}\frac{q(s)-b_{0}}{sq(s)}$
$\displaystyle=1-\frac{b_{0}}{2\pi}\int_{\partial(B_{0,R}\cap\mathbb{C}_{J})}ds_{J}\frac{1}{sq(s)}\overset{R\rightarrow\infty}{\longrightarrow}1,$

where we used that the last integral vanishes in the limit
$R\rightarrow\infty$ since $\frac{1}{sq(s)}=\mathcal{O}(s^{-1-m})$ and $m\geq
1$. Plugging this into (53) gives $q[T]B=1$. Moreover, the operator $B$
commutes with $T$ on $\operatorname{dom}(T)$, because

$\displaystyle TB$
$\displaystyle=\frac{-1}{2\pi}\int_{\partial(U\cap\mathbb{C}_{J})}TS_{L}^{-1}(s,T)ds_{J}\frac{1}{q(s)}=\frac{-1}{2\pi}\int_{\partial(U\cap\mathbb{C}_{J})}(S_{L}^{-1}(s,T)s-\mathcal{I})ds_{J}\frac{1}{q(s)}$
$\displaystyle=\frac{-1}{2\pi}\int_{\partial(U\cap\mathbb{C}_{J})}\frac{1}{q(s)}ds_{J}(sS_{R}^{-1}(s,T)-\mathcal{I})=\frac{-1}{2\pi}\int_{\partial(U\cap\mathbb{C}_{J})}\frac{1}{q(s)}ds_{J}S_{R}^{-1}(s,T)T$
$\displaystyle=\frac{-1}{2\pi}\int_{\partial(U\cap\mathbb{C}_{J})}S_{L}^{-1}(s,T)ds_{J}\frac{1}{q(s)}T=BT,$

where we used the resolvent equation
$sS_{R}^{-1}(s,T)-\mathcal{I}=S_{R}^{-1}(s,T)T$ from [39, Theorem 2.33], as
well as the fact that for any intrinsic function $g$ the two integrals

$\int_{\partial(U\cap\mathbb{C}_{J})}S_{L}^{-1}(s,T)ds_{J}g(s)=\int_{\partial(U\cap\mathbb{C}_{J})}g(s)ds_{J}S_{R}^{-1}(s,T)$

coincide. With this commutativity we can rewrite the already proven $q[T]B=1$
also as $Bq[T]=1$ on $\operatorname{dom}(T^{m})$, which proves that $q[T]$ is
bijective with inverse

$q[T]^{-1}=B=\frac{-1}{2\pi}\int_{\partial(U\cap\mathbb{C}_{J})}S_{L}^{-1}(s,T)ds_{J}\frac{1}{q(s)}.\qed$
(54)

######  Proposition 3.12.

Let $T\in\mathcal{KC}(V)$ with $T,\overline{T}$ being operators of type
$\omega$ and consider two intrinsic polynomials $p,q$ with the properties

  1. i)

$\deg(q)\geq\deg(p)+1$;

  2. ii)

$p$ admits a zero of at least order $2$ at the origin;

  3. iii)

$q$ does not admit any zeros in $\overline{S_{\omega}}$.

Then $q[T]$, $q[\overline{T}]$ are bijective,
$\frac{p}{q}\in\Psi^{Q}(S_{\theta})$ for some $\theta\in(\omega,\pi)$ and

$\displaystyle D\Big{(}\frac{p}{q}\Big{)}(T)$
$\displaystyle=\big{(}Dp[T]q[T]-p[T]Dq[T]\big{)}q[T]^{-1}q[\overline{T}]^{-1}$
(55a)
$\displaystyle=\big{(}Dp[T]q[\overline{T}]-p[\overline{T}]Dq[T]\big{)}q[T]^{-1}q[\overline{T}]^{-1}.$
(55b)

######  Remark 3.13.

In (55) the left hand side is understood as the harmonic functional calculus
(19a) (indicated by the round brackets), while the terms on the two right hand
sides are understood as the polynomial functional calculus (47) and as the
harmonic polynomial functional calculus

$Dp[T]:=-2\sum\limits_{i=0}^{n}p_{i}\sum\limits_{k=0}^{i-1}T^{k}\overline{T}^{i-1-k},$
(56)

with $\operatorname{dom}(Dp[T]):=\operatorname{dom}(T^{n-1})$, (indicated by
square brackets). Note, that the operator (56) is motivated by the action
$Ds^{i}=-2\sum_{k=0}^{i-1}s^{k}\overline{s}^{i-1-k}$ of the Cauchy Fueter
operator (5) on monomials, see [7, Lemma 1].

###### Proof of Proposition 3.12.

Note, that the second equation (55b) follows from the first one (55a) when we
replace $T$ by $\overline{T}$ and use that all terms involving $D$ stay the
same due to Lemma 3.5 iii) and the definitions (47) and (56) of the polynomial
functional calculi. Hence it is left to prove the equality (55a).

In the first step we will change the integration path
$\partial(S_{\varphi}\cap\mathbb{C}_{J})$ in the integrals (18a) and (19a) of
$(\frac{p}{q})(T)$ and $D(\frac{p}{q})(T)$, to some finite path in
$\overline{S_{\omega}}^{c}$ surrounding all the zeros of $q$. Since the closed
set $\overline{S_{\omega}}$ does not contain any zeros of the polynomial $q$
by assumption iii), we can choose $\theta\in(\omega,\pi)$ small enough such
that $\overline{S_{\theta}}$ does not contain any zeros of $q$ either. This
means that $\frac{p}{q}$ is intrinsic on $S_{\theta}$. Moreover, $p$ admits a
zeros of at least order $2$ at the origin by assumption ii), $q$ does not
admit a zero at the origin by assumption iii) and $\deg(q)\geq\deg(p)+1$ by
assumption i). These three properties ensure the existence of a constant
$C_{1}\geq 0$ such that

$\Big{|}\frac{p(s)}{q(s)}\Big{|}\leq C_{1}\frac{|s|^{2}}{1+|s|^{3}},\quad s\in
S_{\theta},$ (57)

and hence $\frac{p}{q}\in\Psi^{Q}(S_{\theta})$. By choosing $\varepsilon$
small and $R$ large enough, one can ensure that all the zeros of $q$ lie
inside the set

zeros of
$q$$\omega$$\varphi$$\theta$$\sigma_{S}(T)$$U_{\varepsilon,R}$$\mathbb{C}_{J}$$\gamma_{\varepsilon,R}$$\gamma_{\varepsilon,R}$$\tau_{\varepsilon}$$\tau_{R}$

$U_{\varepsilon,R}:=\Set{s\in\overline{S_{\varphi}}^{c}}{\varepsilon<|s|<R}.$

If we decompose
$\partial(U_{\varepsilon,R}\cap\mathbb{C}_{J})=\tau_{R}\ominus\gamma_{\varepsilon,R}\ominus\tau_{\varepsilon}$,
using the curves

$\displaystyle\gamma_{\varepsilon,R}(t)$
$\displaystyle:=\begin{cases}-te^{J\varphi},&t\in(-R,-\varepsilon),\\\
te^{-J\varphi},&t\in(\varepsilon,R),\end{cases}$
$\displaystyle\tau_{\varepsilon}(\phi)$ $\displaystyle:=\varepsilon
e^{J\phi},\hskip 22.76228pt\phi\in(\varphi,2\pi-\varphi),$
$\displaystyle\tau_{R}(\phi)$ $\displaystyle:=Re^{J\phi},\hskip
19.91684pt\phi\in(\varphi,2\pi-\varphi),$

we note that the two integrals

$\lim\limits_{\varepsilon\rightarrow
0^{+}}\int_{\tau_{\varepsilon}}Q_{c,s}^{-1}(T)ds_{J}\frac{p(s)}{q(s)}=\lim\limits_{R\rightarrow\infty}\int_{\tau_{R}}Q_{c,s}^{-1}(T)ds_{J}\frac{p(s)}{q(s)}=0,$

vanish due to the estimates (17) and (57). Hence we obtain

$\displaystyle D\Big{(}\frac{p}{q}\Big{)}(T)$
$\displaystyle=\frac{-1}{\pi}\int_{\partial(S_{\varphi}\cap\mathbb{C}_{J})}Q_{c,s}^{-1}(T)ds_{J}\frac{p(s)}{q(s)}$
$\displaystyle=\frac{-1}{\pi}\lim\limits_{\varepsilon\rightarrow
0}\lim\limits_{R\rightarrow\infty}\int_{\gamma_{\varepsilon,R}}Q_{c,s}^{-1}(T)ds_{J}\frac{p(s)}{q(s)}$
$\displaystyle=\frac{1}{\pi}\lim\limits_{\varepsilon\rightarrow
0}\lim\limits_{R\rightarrow\infty}\int_{\partial(U_{\varepsilon,R}\cap\mathbb{C}_{J})}Q_{c,s}^{-1}(T)ds_{J}\frac{p(s)}{q(s)}$
$\displaystyle=\frac{1}{\pi}\int_{\partial(U_{\varepsilon,R}\cap\mathbb{C}_{J})}Q_{c,s}^{-1}(T)ds_{J}\frac{p(s)}{q(s)},$
(58)

where in the last line we used that by the Cauchy theorem the integral does
not depend on the choice of $\varepsilon$ and $R$ as long all zeros of $q(s)$
are inside $U_{\varepsilon,R}$.

In the second step we consider the operator

$B:=\frac{1}{\pi}\int_{\partial(U_{\varepsilon,R}\cap\mathbb{C}_{J})}Q_{c,s}^{-1}(T)ds_{J}\frac{1}{q(s)},$
(59)

for which we will prove that
$\operatorname{ran}(B)\subseteq\operatorname{dom}(\overline{T}^{j})$ and

$\overline{T}^{j}B=\frac{1}{\pi}\int_{\partial(U_{\varepsilon,R}\cap\mathbb{C}_{J})}Q_{c,s}^{-1}(T)ds_{J}\frac{s^{j}}{q(s)}+2\sum\limits_{k=0}^{j-1}T^{k}\overline{T}^{j-1-k}q[T]^{-1},\qquad
j\in\\{0,\dots,m\\},$ (60)

where for $j=0$ the sum $\sum_{k=0}^{-1}$ is understood to be $0$. First of
all, we note that $q[T]$ is bijective due to Lemma 3.11. The induction start
$j=0$ is clear by the definition (59) of the operator $B$. For the induction
step $j-1\rightarrow j$ we use the identity

$Q_{c,s}^{-1}(T)s=\overline{T}Q_{c,s}^{-1}(T)+S_{L}^{-1}(s,T),$

which follows immediately from the definition (12) of the left $S$-resolvent,
to get

$\displaystyle\frac{1}{\pi}\int_{\partial(U_{\varepsilon,R}\cap\mathbb{C}_{J})}Q_{c,s}^{-1}(T)ds_{J}\frac{s^{j}}{q(s)}$
$\displaystyle=\frac{1}{\pi}\int_{\partial(U_{\varepsilon,R}\cap\mathbb{C}_{J})}\big{(}\overline{T}Q_{c,s}^{-1}(T)+S_{L}^{-1}(s,T)\big{)}ds_{J}\frac{s^{j-1}}{q(s)}$
$\displaystyle=\overline{T}\Big{(}\overline{T}^{j-1}B-2\sum\limits_{k=0}^{j-2}T^{k}\overline{T}^{j-2-k}q[T]^{-1}\Big{)}-2T^{j-1}q[T]^{-1}$
$\displaystyle=\overline{T}^{j}B-2\sum\limits_{k=0}^{j-1}T^{k}\overline{T}^{j-1-k}q[T]^{-1},$

where in the second line we used the induction assumption (60) for $j-1$ as
well as the identities (49) and (54). Bringing the very right term to the left
hand side finishes the induction step.

In the third step we show that

$B=-Dq[T]q[T]^{-1}q[\overline{T}]^{-1}.$ (61)

Since $\overline{U_{\varepsilon,R}}\subseteq\rho_{S}(T)$, where $s\mapsto
Q_{c,s}(T)$ is slice hyperholomorphic, we get

$\displaystyle 0$
$\displaystyle=\frac{1}{\pi}\int_{\partial(U_{\varepsilon,R}\cap\mathbb{C}_{J})}Q_{c,s}^{-1}(T)ds_{J}1=\frac{1}{\pi}\int_{\partial(U_{\varepsilon,R}\cap\mathbb{C}_{J})}Q_{c,s}^{-1}(T)ds_{J}\frac{q(s)}{q(s)}$
$\displaystyle=\sum\limits_{j=0}^{m}\frac{b_{j}}{\pi}\int_{\partial(U_{\varepsilon,R}\cap\mathbb{C}_{J})}Q_{c,s}^{-1}(T)ds_{J}\frac{s^{j}}{q(s)}$
$\displaystyle=\sum\limits_{j=0}^{m}b_{j}\Big{(}\overline{T}^{j}B-2\sum\limits_{k=0}^{j-1}T^{k}\overline{T}^{j-1-k}q[T]^{-1}\Big{)}=q[\overline{T}]B+Dq[T]q[T]^{-1},$

where in the last line we used (60) and the definition (56) of $Dq[T]$. Using
the bijectivity of $q[\overline{T}]$ from Lemma 3.11, this is exactly equation
in (61).

In the fourth step we combine the results (58), (60) and (61) to get the
stated equation

$\displaystyle D\Big{(}\frac{p}{q}\Big{)}(T)$
$\displaystyle=\frac{1}{\pi}\int_{\partial(U_{\varepsilon,R}\cap\mathbb{C}_{J})}Q_{c,s}^{-1}(T)ds_{J}\frac{p(s)}{q(s)}=\sum\limits_{i=0}^{n}\frac{a_{i}}{\pi}\int_{\partial(U_{\varepsilon,R}\cap\mathbb{C}_{J})}Q_{c,s}^{-1}(T)ds_{J}\frac{s^{i}}{q(s)}$
$\displaystyle=\sum\limits_{i=0}^{n}a_{i}\Big{(}\overline{T}^{i}B-2\sum\limits_{k=0}^{i-1}T^{k}\overline{T}^{i-1-k}q[T]^{-1}\Big{)}=p[\overline{T}]B+Dp[T]q[T]^{-1}$
$\displaystyle=\Big{(}Dp[T]q[T]-p[\overline{T}]Dq[T]\Big{)}q[T]^{-1}q[\overline{T}]^{-1}.\qed$

##  4\. The harmonic $H^{\infty}$-functional calculus

In this section we want to enlarge the harmonic functional calculus of
Definition 3.3 to slice hyperholomorphic functions on a sector, which are
polynomially growing at $0$ and at $\infty$. More precisely we consider the
following function spaces.

######  Definition 4.1.

For every $\theta\in(0,\pi)$ we define the function spaces

  1. i)

$\mathcal{F}_{L}(S_{\theta}):=\Set{f\in\mathcal{SH}_{L}(S_{\theta})}{\exists\alpha>0,\,C_{\alpha}\geq
0:|f(s)|\leq
C_{\alpha}\big{(}|s|^{\alpha}+\frac{1}{|s|^{\alpha}}\big{)},\,s\in
S_{\theta}}$

  2. ii)

$\mathcal{F}_{R}(S_{\theta}):=\Set{f\in\mathcal{SH}_{R}(S_{\theta})}{\exists\alpha>0,\,C_{\alpha}\geq
0:|f(s)|\leq
C_{\alpha}\big{(}|s|^{\alpha}+\frac{1}{|s|^{\alpha}}\big{)},\,s\in
S_{\theta}}$

  3. iii)

$\mathcal{F}(S_{\theta}):=\Set{f\in\mathcal{N}(S_{\theta})}{\exists\alpha>0,\,C_{\alpha}\geq
0:|f(s)|\leq
C_{\alpha}\big{(}|s|^{\alpha}+\frac{1}{|s|^{\alpha}}\big{)},\,s\in
S_{\theta}}$

First, we generalize the $S$-functional calculus of Definition 3.2 to these
polynomially growing functions by some regularizing procedure. This is called
the $H^{\infty}$-functional calculus and was for example already considered in
[21, Section 6.3].

######  Definition 4.2 (The $H^{\infty}$-functional calculus).

Let $T\in\mathcal{KC}(V)$ be an injective operator of type $\omega$. Then for
every $f\in\mathcal{F}_{L}(S_{\theta})$, $\theta\in(\omega,\pi)$, we define
the $H^{\infty}$-functional calculus

$f(T):=e(T)^{-1}(ef)(T),$ (62)

where $e\in\Psi(S_{\theta})$ is such that $e(T)$ is injective and
$ef\in\Psi_{L}(S_{\theta})$. The operators $e(T)$ and $(ef)(T)$ are understood
by the $S$-functional calculus (18a) for decaying functions.

It is then proven in [21, Theorem 6.3.4] that such a regularizer function $d$
exists and that (62) is independent of $e$.

Motivated by the product rule (43), we will now also generalize the harmonic
functional calculus from Definition 3.3 to the functions from Definition 4.1.

######  Definition 4.3 (The harmonic $H^{\infty}$-functional calculus).

Let $T\in\mathcal{KC}(V)$ be such that $T,\overline{T}$ are injective
operators of type $\omega$. Then for every $f\in\mathcal{F}_{L}(S_{\theta})$,
$\theta\in(\omega,\pi)$ we define the harmonic $H^{\infty}$-functional
calculus

$\displaystyle Df(T):=$ $\displaystyle
e(\overline{T})^{-1}\big{(}D(ef)(T)-f(T)De(T)\big{)}$ (63a) $\displaystyle=$
$\displaystyle e(T)^{-1}\big{(}D(ef)(T)-f(\overline{T})De(T)\big{)},$ (63b)

where $e\in\Psi^{Q}(S_{\theta})$, such that $e(T),e(\overline{T})$ are
injective and $ef\in\Psi_{L}^{Q}(S_{\theta})$. Here $D(ef)(T),De(T)$ are
understood by the harmonic functional calculus (19a) for decaying functions,
$e(T),e(\overline{T})$ by the $S$-functional calculus (18a) for decaying
functions and $f(T),f(\overline{T})$ by the $H^{\infty}$-functional calculus
(62).

######  Remark 4.4.

Note, that in the Definitions 4.2 & 4.3 only left slice hyperholomorphic
functions $f\in\mathcal{F}_{L}(S_{\theta})$ are considered. The reason why the
natural analogues

$f(T):=(fe)(T)e(T)^{-1}\quad\text{and}\quad(fD)(T):=\big{(}((fe)D)(T)-f(\overline{T})De(T)\big{)}e(T)^{-1},$

of (62) and (63b) for right slice hyperholomorphic functions
$f\in\mathcal{F}_{R}(S_{\theta})$ are missing, is that these terms would be
defined only on $\operatorname{dom}(e(T)^{-1})$ and hence not independent of
the choice of the regularizer; see also [20, Remark 7.2.2].

######  Theorem 4.5.

Let $T\in\mathcal{KC}(V)$ be such that $T,\overline{T}$ are injective
operators of type $\omega$. Then for every $f\in\mathcal{F}_{L}(S_{\theta})$,
$\theta\in(\omega,\pi)$ a regularizer function $e$ in the sense of Definition
4.3 exists, (63a) and (63b) equal and do not depend on the choice of the
regularizer $e$. Moreover, for every $f,g\in\Psi_{L}(S_{\theta})$ with
$Df(s)=Dg(s)$, also the operators $Df(T)=Dg(T)$ coincide.

###### Proof.

For the independence of the regularizer $e$ let
$e_{1},e_{2}\in\Psi^{Q}(S_{\theta})$ be two regularizers such that
$e_{1}(T),e_{1}(\overline{T}),e_{2}(T),e_{2}(\overline{T})$ are injective and
$e_{1}f,e_{2}f\in\Psi_{L}^{Q}(S_{\theta})$. From the two versions of the
product rule (43a) and (43b) applied to $D(e_{1}e_{2}f)(T)$, we get

$De_{1}(T)(e_{2}f)(T)+e_{1}(\overline{T})D(e_{2}f)(T)=De_{2}(T)(e_{1}f)(\overline{T})+e_{2}(T)D(e_{1}f)(T).$
(64)

Rearranging this equation and interchanging $De_{1}(T)$ and $(e_{2}f)(T)$ as
well as $De_{2}(T)$ and $(e_{1}f)(\overline{T})$, which is allowed by
Corollary 3.8 and Lemma 3.5 iii), gives

$e_{2}(T)D(e_{1}f)(T)-(e_{2}f)(T)De_{1}(T)=e_{1}(\overline{T})D(e_{2}f)(T)-(e_{1}f)(\overline{T})De_{2}(T).$
(65)

Next, we note that $e_{1}(\overline{T})e_{2}(T)=e_{2}(T)e_{1}(\overline{T})$
commute due to [19, Lemma 3.9] and so do their inverses
$e_{1}^{-1}(\overline{T})e_{2}^{-1}(T)=e_{2}^{-1}(T)e_{1}^{-1}(\overline{T})$.
Multiplying these two inverses from the left to (65) and using the definition
of the $H^{\infty}$-functional calculus (62) gives

$e_{1}(\overline{T})^{-1}\big{(}D(e_{1}f)(T)-f(T)De_{1}(T)\big{)}=e_{2}(T)^{-1}\big{(}D(e_{2}f)(T)-f(\overline{T})De_{2}(T)\big{)}.$
(66)

This equation with the particular choice $e_{1}=e_{2}$ gives the identity

$e_{2}(\overline{T})^{-1}\big{(}D(e_{2}f)(T)-f(T)De_{2}(T)\big{)}=e_{2}(T)^{-1}\big{(}D(e_{2}f)(T)-f(\overline{T})De_{2}(T)\big{)}.$

This equation on the one hand shows the equivalence of the definitions (63a)
and (63b), and on the other hand plugging it into (66), gives the independence
of the regularizer

$e_{1}(\overline{T})^{-1}\big{(}D(e_{1}f)(T)-f(T)De_{1}(T)\big{)}=e_{2}(\overline{T})^{-1}\big{(}D(e_{2}f)(T)-f(T)De_{2}(T)\big{)}.$

It is left to show that a regularizer function $e\in\Psi^{Q}(S_{\theta})$ with
$e(\overline{T})$ injective and $ef\in\Psi_{L}^{Q}(S_{\theta})$ exists. Since
$f\in\mathcal{F}_{L}(S_{\theta})$, it admits the estimate

$|f(s)|\leq C_{\alpha}\Big{(}|s|^{\alpha}+\frac{1}{|s|^{\alpha}}\Big{)},\qquad
s\in S_{\theta},$

for some $\alpha>0$, $C_{\alpha}\geq 0$. For arbitrary $n\in\mathbb{N}$ with
$n>1+\alpha$ we choose the regularizer

$e(s):=\frac{s^{n}}{(1+s)^{2n-1}}.$ (67)

Then obviously $e\in\Psi^{Q}(S_{\theta})$ and due to the asymptotics

$e(s)f(s)=\begin{cases}\mathcal{O}(|s|^{-n+1+\alpha}),&\text{as
}s\rightarrow\infty,\\\ \mathcal{O}(|s|^{n-\alpha}),&\text{as }s\rightarrow
0,\end{cases}$

we also have $ef\in\Psi_{L}^{Q}(S_{\theta})$. In order to show that
$e(T)=T^{n}(1+T)^{-2n+1}$ is injective, we note that since $T$ is injective,
the $n$-th power $T^{n}$ is injective as well. Since $(1+T)^{-2n+1}$ is
bijective due to $-1\in\rho_{S}(T)$, we conclude that $e(T)$ is injective. The
injectivity of $e(\overline{T})$ follows the same argument.

In order to prove the independence of the kernel of $D$, let
$f,g\in\Psi_{L}(S_{\theta})$ with $Df(s)=Dg(s)$. Assume furthermore that $e$
is a regularizer for $f$ as well as for $g$. In the same way as in (31) one
shows that $f=g+c$ only differ by a constant. Plugging this into (63) gives

$\displaystyle Df(T)$
$\displaystyle=e(\overline{T})^{-1}\big{(}D(ef)(T)-f(T)De(T)\big{)}$
$\displaystyle=e(\overline{T})^{-1}\big{(}D(eg+ce)(T)-(g+c)(T)De(T)\big{)}$
$\displaystyle=e(\overline{T})^{-1}\big{(}D(eg(T)+cDe(T)-g(T)De(T)-cDe(T)\big{)}$
$\displaystyle=e(\overline{T})^{-1}\big{(}D(eg(T)-g(T)De(T)\big{)}=Dg(T),$

where in the third equation we used the linearity of the harmonic functional
calculus in Lemma 3.5 ii) and the linearity $(g+c)(T)=g(T)+c$, which holds
with equality if one of the operators is bounded. ∎

######  Lemma 4.6.

Let $T\in\mathcal{KC}(V)$ with $T,\overline{T}$ being injective operators of
type $\omega$. Then for every $f,g\in\mathcal{F}_{L}(S_{\theta})$,
$\theta\in(\omega,\pi)$, there holds

  1. i)

$Df(T)$ is a closed operator;

  2. ii)

$Df(T)+Dg(T)\subseteq D(f+g)(T)$;

  3. iii)

$Df(T)=Df(\overline{T})$.

  4. iv)

If $f$ is intrinsic, then $Df(T)\in\mathcal{KC}(V)$ with
$\overline{Df(T)}=Df(T)$.

###### Proof.

Let $e$ be a regularizer of $f$ according to Definition 4.3. For the proof of
i), we first use the definition (62) of the $H^{\infty}$-functional calculus
to rewrite (63a) as

$Df(T)=\big{(}e(T)e(\overline{T})\big{)}^{-1}\big{(}e(T)D(ef)(T)-(ef)(T)De(T)\big{)}.$
(68)

Since this is the combination of an unbounded closed with a bounded operator,
also its decomposition $Df(T)$ turns out to be a closed operator again. Next,
we note that due to Lemma 3.6 iii), we know that $e(T)$ and $e(\overline{T})$
are bounded operators with commuting components and that

$e(T)e(\overline{T})=e(T)\overline{e(T)}=|e(T)|^{2}$

only admits the component $(e(T)e(\overline{T}))_{0}$, while
$(e(T)e(\overline{T}))_{i}=0$, $i\in\\{1,2,3\\}$, vanish. Hence also the
inverse $(e(T)e(\overline{T}))^{-1}=(e(T)e(\overline{T}))_{0}^{-1}$ is an
unbounded operator only consisting of the inverse of that one real component.
Moreover, $e(T),D(ef)(T),(ef)(T),De(T)$ are operators with commuting
components as well, see Lemma 3.5 and Lemma 3.6, and the respective components
are combinations of powers of the components $T_{0},T_{1},T_{2},T_{3}$, see
the explicit representations (37) and (39). Hence also the decomposition (68)
is an operator with commuting component, which prove that
$Df(T)\in\mathcal{KC}(V)$.

For the proof of ii) we assume that $e$ is a regularizer of $f$ and of $g$.
Then $e$ is a regularizer of the sum $f+g$ as well. Combining now the facts

  1. $\circ$

$AB+AC\subseteq A(B+C)$, for unbounded operators $A,B,C$ in $V$,

  2. $\circ$

$D(ef)(T)+D(eg)(T)=D(ef+eg)(T)$, by Lemma 3.5,

  3. $\circ$

$f(T)+g(T)\subseteq(f+g)(T)$, by [3, Theorem 5.6],

leads us to the inclusion

$\displaystyle Df(T)+Dg(T)$
$\displaystyle=e(\overline{T})^{-1}\big{(}D(ef)(T)-f(T)De(T)\big{)}+e(\overline{T})^{-1}\big{(}D(eg)(T)-g(T)De(T)\big{)}$
$\displaystyle\subseteq
e(\overline{T})^{-1}\Big{(}D(ef)(T)-f(T)De(T)+D(eg)(T)-g(T)De(T)\Big{)}$
$\displaystyle\subseteq
e(\overline{T})^{-1}\Big{(}D(ef+eg)(T)-(f+g)(T)De(T)\Big{)}=D(f+g)(T).$

The proof of iii) follows from the two equivalent definitions (63a) and (63b)
and Lemma 3.5 iii).

Finally, for the statement iv) it first follows from Lemma 3.5 iii) and Lemma
3.6 iii), that

$\displaystyle\overline{e(T)D(ef)(T)-(ef)(T)De(T)}$
$\displaystyle=e(\overline{T})D(ef)(T)-(ef)(\overline{T})De(T)$
$\displaystyle=e(T)D(ef)(T)-(ef)(T)De(T),$

where the second equation follows from the two versions of the product rule
(43a) and (43b). Hence the operator $e(T)D(ef)(T)-(ef)(T)De(T)$, and
consequently also $Df(T)$ in the representation (68), is two-sided linear.
This means that $Df(T)$ only consists of one component and automatically
implies $Df(T)\in\mathcal{KC}(V)$ as well as $\overline{Df(T)}=Df(T)$. ∎

######  Proposition 4.7.

Let $T\in\mathcal{KC}(V)$ with $T,\overline{T}$ being injective operators of
type $\omega$, $B:V\rightarrow V$ an everywhere defined bounded operator and
$f\in\mathcal{F}(S_{\theta})$, for some $\theta\in(\omega,\pi)$. Then there
holds

$\begin{array}[]{c}\forall j\in\\{0,\dots,3\\}:BT_{j}=T_{j}B\quad\text{on
}\operatorname{dom}(T)\\\ \Downarrow\\\ BDf(T)\subseteq Df(T)B.\end{array}$

###### Proof.

First of all, we note that since $BT_{j}=T_{j}B$ commutes with every component
$T_{j}$ on $\operatorname{dom}(T)$, the operator $B$ commutes also with every
component $e(T)_{i}$ of $e(T)$, see the representation (39). Hence $B$ also
commutes with

$e(T)e(\overline{T})=e(T)\overline{e(T)}=e_{1}(T)_{0}^{2}+e(T)_{1}^{2}+e(T)_{2}^{2}+e(T)_{3}^{2}.$

Consequently for the inverse we get the inclusion

$\displaystyle B\big{(}e(T)e(\overline{T})\big{)}^{-1}$
$\displaystyle=\big{(}e(T)e(\overline{T})\big{)}^{-1}e(T)e(\overline{T})B\big{(}e(T)e(\overline{T})\big{)}^{-1}$
$\displaystyle=\big{(}e(T)e(\overline{T})\big{)}^{-1}Be(T)e(\overline{T})\big{(}e(T)e(\overline{T})\big{)}^{-1}\subseteq\big{(}e(T)e(\overline{T})\big{)}^{-1}B.$

Since for the same reason, $B$ also commutes with the components of $e(T)$,
$D(ef)(T)$, $(ef)(T)$ and $De(T)$, it also commutes with the components of the
bracket term in (68). However, since $Df(T)=\overline{Df(T)}$ by Lemma 4.6
iv), we know that $Df(T)$ and hence also $e(T)D(ef)(T)-(ef)(T)De(T)$ only
consists of the real term, while all the components which are denoted on the
imaginary units $e_{1},e_{2},e_{3}$ vanish identically. Hence

$B\big{(}e(T)D(ef)(T)-(ef)(T)De(T)\big{)}=\big{(}e(T)D(ef)(T)-(ef)(T)De(T)\big{)}B,\quad\text{on
}V,$

commutes with the whole bracket term in (68) and hence also with

$\displaystyle BDf(T)$
$\displaystyle=B\big{(}e(T)e(\overline{T})\big{)}^{-1}\big{(}e(T)D(ef)(T)-(ef)(T)De(T)\big{)}$
$\displaystyle\subseteq\big{(}e(T)e(\overline{T})\big{)}^{-1}B\big{(}e(T)D(ef)(T)-(ef)(T)De(T)\big{)}$
$\displaystyle=\big{(}e(T)e(\overline{T})\big{)}^{-1}\big{(}e(T)D(ef)(T)-(ef)(T)De(T)\big{)}B=Df(T)B.\qed$

######  Corollary 4.8.

Let $T\in\mathcal{KC}(V)$ with $T,\overline{T}$ being injective operators of
type $\omega$. Then

  1. i)

$Df(T)g(T)\subseteq g(T)Df(T)$ for
$f\in\Psi^{Q}(S_{\theta}),\,g\in\mathcal{F}_{L}(S_{\theta})$

  2. ii)

$g(T)Df(T)\subseteq Df(T)g(T)$ for
$f\in\mathcal{F}^{Q}(S_{\theta}),\,g\in\Psi_{L}^{Q}(S_{\theta})$ or
$g\in\Psi_{R}^{Q}(S_{\theta})$

###### Proof.

For the proof of i) let $e$ be a regularizer of $g$ according to Definition
4.2. Then

$Df(T)g(T)=Df(T)e(T)^{-1}(eg)(T)\subseteq e(T)^{-1}(eg)(T)Df(T)=g(T)Df(T),$
(69)

where we commuted

$Df(T)e(T)^{-1}=e(T)^{-1}e(T)Df(T)e(T)^{-1}=e(T)^{-1}Df(T)e(T)e(T)^{-1}\subseteq
e(T)^{-1}Df(T),$ (70)

and $Df(T)(eg)(T)=(eg)(T)Df(T)$, which is allowed by Corollary 3.8.

The inclusion in ii) follows from Proposition 4.7, since $g(T)T_{j}=T_{j}g(T)$
obviously commute on $\operatorname{dom}(T)$ by the integral definition of the
$S$-functional calculus (18a). ∎

######  Theorem 4.9.

Let $T\in\mathcal{KC}(V)$ with $T,\overline{T}$ being injective operators of
type $\omega$. Then for every $f\in\mathcal{F}(S_{\theta})$,
$g\in\mathcal{F}_{L}(S_{\theta})$, $\theta\in(\omega,\pi)$, there holds the
product rules

$\displaystyle D(fg)(T)$ $\displaystyle\supseteq
Df(T)g(T)+f(\overline{T})Dg(T),\quad\text{and}$ (71a) $\displaystyle D(fg)(T)$
$\displaystyle\supseteq Df(T)g(\overline{T})+f(T)Dg(T).$ (71b)

###### Proof.

First of all, the second rule (71b) follows immediately when we replace $T$ by
$\overline{T}$ in (71a) and use Lemma 4.6 iii). For the prove of (71a) let
$e_{1}$ be a regularizer of $f$ and $e_{2}$ a regularizer of $g$ according to
Definition 4.3. Using the product rule $(fg)(T)\supseteq f(T)g(T)$ of the
$S$-functional calculus [3, Theorem 5.6] and the commutation of $g(T)$ and
$D(e_{1}e_{2})(T)$ as in (69), gives

$(fg)(T)D(e_{1}e_{2})(T)\supseteq f(T)g(T)D(e_{1}e_{2})(T)\supseteq
f(T)D(e_{1}e_{2})(T)g(T).$ (72)

Since $e_{1}e_{2}$ is a regularizer of the product $fg$, we can use the
identity (72) and the product rule (43a), to get

$\displaystyle D(fg)(T)$
$\displaystyle=(e_{1}e_{2})(\overline{T})^{-1}\big{(}D(e_{1}e_{2}fg)(T)-(fg)(T)D(e_{1}e_{2})(T)\big{)}$
$\displaystyle\supseteq(e_{1}e_{2})(\overline{T})^{-1}\big{(}D(e_{1}fe_{2}g)(T)-f(T)D(e_{1}e_{2})(T)g(T)\big{)}$
$\displaystyle\supseteq\big{(}e_{1}(\overline{T})e_{2}(\overline{T})\big{)}^{-1}\big{(}D(e_{1}f)(T)(e_{2}g)(T)+(e_{1}f)(\overline{T})D(e_{2}g)(T)$
$\displaystyle\hskip
91.04872pt-f(T)De_{2}(T)e_{1}(T)g(T)-f(T)e_{2}(\overline{T})De_{1}(T)g(T).$

Using now the commutation rules

  1. $\circ$

$D(e_{1}f)(T)(e_{2}g)(T)\supseteq
D(e_{1}f)(T)e_{2}(T)e_{2}(T)^{-1}(e_{2}g)(T)=e_{2}(T)D(e_{1}f)(T)g(T)$,

  2. $\circ$

$De_{2}(T)e_{1}(T)=e_{1}(T)De_{2}(T)$, from Corollary 3.8,

  3. $\circ$

$f(T)e_{1}(T)=e_{1}(T)^{-1}(e_{1}f)(T)e_{1}(T)=(e_{1}f)(T)$,

  4. $\circ$

$f(T)e_{2}(\overline{T})\supseteq e_{2}(\overline{T})f(T)$, as in (69),

we can rearrange the terms and obtain the product rule

$\displaystyle D(fg)(T)$ $\displaystyle\supseteq
e_{1}(\overline{T})^{-1}e_{2}(\overline{T})^{-1}\big{(}e_{2}(T)D(e_{1}f)(T)g(T)+(e_{1}f)(\overline{T})D(e_{2}g)(T)$
$\displaystyle\hskip
91.04872pt-(e_{1}f)(T)De_{2}(T)g(T)-e_{2}(\overline{T})f(T)De_{1}(T)g(T)\big{)}$
$\displaystyle=e_{1}(\overline{T})^{-1}e_{2}(\overline{T})^{-1}\big{(}e_{2}(\overline{T})D(e_{1}f)(T)g(T)+(e_{1}f)(\overline{T})D(e_{2}g)(T)$
$\displaystyle\hskip
91.04872pt-(e_{1}f)(\overline{T})De_{2}(T)g(T)-e_{2}(\overline{T})f(T)De_{1}(T)g(T)\big{)}$
$\displaystyle\supseteq
e_{1}(\overline{T})^{-1}\big{(}D(e_{1}f)(T)-f(T)De_{1}(T)\big{)}g(T)$
$\displaystyle\quad+e_{1}(\overline{T})^{-1}(e_{1}f)(\overline{T})e_{2}(\overline{T})^{-1}\big{(}D(e_{2}g)(T)-De_{2}(T)g(T)\big{)}$
$\displaystyle=Df(T)g(T)+f(\overline{T})Dg(T),$

where in the second line we used (64) and in the third one we commuted
$e_{2}(\overline{T})^{-1}(e_{1}f)(\overline{T})\supseteq(e_{1}f)(\overline{T})e_{2}(\overline{T})^{-1}$
by a similar argument as in (70). ∎

As the final result of this paper, we investigate the action of the harmonic
functional calculus on intrinsic rational functions. The following Proposition
4.10 contains basically the same statement as Proposition 3.12, but without
the assumptions i), ii) on the decay on the polynomials. As a consequence we
are obliged to use the harmonic $H^{\infty}$-functional calculus (63) instead
of the calculus (19a) for decaying functions.

######  Proposition 4.10.

Let $T\in\mathcal{KC}(V)$ with $T,\overline{T}$ being injective operators of
type $\omega$. Then for every intrinsic polynomials $p,q$, with $q$ not having
any zeros in the closed sector $\overline{S_{\omega}}$, we know that $q[T]$,
$q[\overline{T}]$ are bijective and

$\displaystyle D\Big{(}\frac{p}{q}\Big{)}(T)$
$\displaystyle=\big{(}Dp[T]q[T]-p[T]Dq[T]\big{)}q[T]^{-1}q[\overline{T}]^{-1}$
$\displaystyle=\big{(}Dp[T]q[T]-p[T]Dq[T]\big{)}q[T]^{-1}q[\overline{T}]^{-1}.$

Here, the left hand side of this equation is understood as the harmonic
$H^{\infty}$-functional calculus (63) and the operators on the right via the
polynomial functional calculi (47) and (56).

###### Proof.

In the first step, let $p$ and $q$ be any intrinsic polynomials. Then, in the
sense of the polynomial functional calculi (47) and (56) we obtain the product
rule

$\displaystyle Dp[T]q[T]+p[\overline{T}]Dq[T]$
$\displaystyle=-2\sum\limits_{i=0}^{n}p_{i}\sum\limits_{k=0}^{i-1}T^{k}\overline{T}^{i-1-k}\sum\limits_{j=0}^{m}q_{j}T^{j}-2\sum\limits_{i=0}^{n}p_{i}\overline{T}^{i}\sum\limits_{j=0}^{m}q_{j}\sum\limits_{k=0}^{j-1}T^{k}\overline{T}^{j-1-k}$
$\displaystyle=-2\sum\limits_{i=0}^{n}\sum\limits_{j=0}^{m}p_{i}q_{j}\bigg{(}\sum\limits_{k=0}^{i-1}T^{j+k}\overline{T}^{i-1-k}+\sum\limits_{k=0}^{j-1}T^{k}\overline{T}^{i+k-1-k}\bigg{)}$
$\displaystyle=-2\sum\limits_{i=0}^{n}\sum\limits_{j=0}^{m}p_{i}q_{j}\sum\limits_{k=0}^{i+j-1}T^{k}\overline{T}^{i+j-1-k}=D(pq)[T].$
(73)

In the second step we choose an intrinsic rational regularizer
$e(s)=\frac{r(s)}{t(s)}$, for example the one in (67) with large enough
$n\in\mathbb{N}$. Then it follows from Proposition 3.12 and (73), that

$\displaystyle D\Big{(}\frac{rp}{tq}\Big{)}(T)$
$\displaystyle=\Big{(}D(rp)[T](tq)[T]-(rp)[T]D(tq)[T]\Big{)}(tq)[T]^{-1}(tq)[\overline{T}]^{-1}$
$\displaystyle=\Big{(}Dr[T]p[T]t[T]q[T]+r[\overline{T}]Dp[T]t[T]q[T]$
$\displaystyle\hskip
28.45274pt-r[T]p[T]Dt[T]q[\overline{T}]-r[T]p[T]t[T]Dq[T]\Big{)}(tq)[T]^{-1}(tq)[\overline{T}]^{-1},$
(74)

as well as

$\displaystyle\Big{(}\frac{p}{q}\Big{)}(T)D\Big{(}\frac{r}{t}\Big{)}(T)$
$\displaystyle=p[T]q[T]^{-1}\big{(}Dr[T]t[T]-r[T]Dt[T]\big{)}t[T]^{-1}t[\overline{T}]^{-1}$
$\displaystyle=\big{(}Dr[T]p[T]t[T]q[\overline{T}]-r[T]p[T]Dt[T]q[\overline{T}]\big{)}(tq)[T]^{-1}(tq)[\overline{T}]^{-1},$
(75)

where we used that $(\frac{p}{q})(T)=p[T]q[T]^{-1}$ by [20, Lemma 7.2.9].
Plugging the two representations (74) and (75) into the definition (63), we
get the stated representation

$\displaystyle D\Big{(}\frac{p}{q}\Big{)}(T)$
$\displaystyle=\Big{(}\frac{r}{t}\Big{)}(\overline{T})^{-1}\Big{(}D\Big{(}\frac{rp}{tq}\Big{)}(T)-\Big{(}\frac{p}{q}\Big{)}(T)D\Big{(}\frac{r}{t}\Big{)}(T)\Big{)}$
$\displaystyle=\big{(}r[\overline{T}]t[\overline{T}]^{-1}\big{)}^{-1}\Big{(}Dr[T]p[T]t[T]q[T]+r[\overline{T}]Dp[T]t[T]q[T]$
$\displaystyle\hskip
99.58464pt-r[T]p[T]t[T]Dq[T]-Dr[T]p[T]t[T]q[\overline{T}]\Big{)}(tq)[T]^{-1}(tq)[\overline{T}]^{-1}$
$\displaystyle=r[\overline{T}]^{-1}\Big{(}Dr[T]p[T]q[T]+r[\overline{T}]Dp[T]q[T]$
$\displaystyle\hskip
99.58464pt-r[T]p[T]Dq[T]-Dr[T]p[T]q[\overline{T}]\Big{)}q[T]^{-1}q[\overline{T}]^{-1}$
$\displaystyle=\big{(}Dp[T]q[T]-Dq[T]p[T]\big{)}q[T]^{-1}q[\overline{T}]^{-1},$

where in the last equation we use the identity

$Dr[T]q[\overline{T}]+r(T)Dq[T]=Dr[T]q[T]+r[\overline{T}]Dq[T],$

which can be checked by plugging in the definitions (47) and (56). ∎

## References

  * [1] D. Alpay, F. Colombo, D.P. Kimsey: The spectral theorem for quaternionic unbounded normal operators based on the $S$-spectrum. J. Math. Phys. 57(2):023503, 27 (2016)
  * [2] D. Alpay, F. Colombo, D. Kimsey, I. Sabadini: The spectral theorem for unitary operators based in the $S$-spectrum. Milan J. Math. 84, no.1, 41–61 (2016)
  * [3] 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(6):1544–1584 (2016)
  * [4] P. Auscher, A. Axelsson, A. McIntosh: On a quadratic estimate related to the Kato conjecture and boundary value problems. Harmonic analysis and partial differential equations, Contemp. Math., Amer. Math. Soc., Providence, RI, 105–129 (2010)
  * [5] P. Auscher, A. McIntosh, A. Nahmod: Holomorphic functional calculi of operators, quadratic estimates and interpolation. Indiana Univ. Math. J. 46, no. 2, 375–403 (1997)
  * [6] A. Axelsson, S. Keith, A. McIntosh: Quadratic estimates and functional calculi of perturbed Dirac operators. Invent. Math. 163, no. 3, 455–497 (2006)
  * [7] H. Begeher: Iterated integral operators in Clifford analysis. J. Anal. Appl. 18, 361-377 (1999)
  * [8] G. Birkhoff, J. von Neumann: The logic of quantum mechanics. Ann. of Math. (2), 37(4), 823–843 (1936)
  * [9] P. Cerejeiras, F. Colombo, U. Kähler, I. Sabadini: Perturbation of normal quaternionic operators. Trans. Amer. Math. Soc., 372(5),3257–3281 (2019)
  * [10] F. Colombo, A. De Martino, S. Pinton: Harmonic and polyanalytic functional calculi on the $S$-spectrum for unbounded operators, Banach J. Math. Anal. 17, 84 (2023)
  * [11] F. Colombo, A. De Martino, S. Pinton, I. Sabadini: Axially harmonic functions and the harmonic functional calculus on the $S$-spectrum. J. Geom. Anal. 33(2), 54 pp. (2023)
  * [12] F. Colombo, A. De Martino, I. Sabadini: The $\mathcal{F}$-resolvent equation and Riesz projectors for the $\mathcal{F}$-functional calculus. Complex Anal. Oper. Theory 17, no. 2, Paper No. 26, 42 pp. (2023)
  * [13] F. Colombo, A. De Martino, I. Sabadini: Towards a general $F$-resolvent equation and Riesz projectors. J. Math. Anal. Appl. 517(2), 126652 (2023)
  * [14] F. Colombo, A. De Martino, S. Pinton, I. Sabadini: The fine structure of the spectral theory on the $S$-spectrum in dimension five. arXiv:2303.00253 (2022)
  * [15] F. Colombo, D. Deniz González, S. Pinton: Fractional powers of vector operators with first order boundary conditions. J. Geom. Phys. 151:103618, 18 (2020)
  * [16] F. Colombo, D. Deniz González, S. Pinton: The noncommutative fractional Fourier law in bounded and unbounded domains. Complex Anal. Oper. Theory 15(7): Paper No. 114, 27 (2021)
  * [17] F. Colombo, J. Gantner: Formulations of the $F$-functional calculus and some consequences. Proc. Roy. Soc. Edinburgh Sect. A, 146(3):509–545 (2016)
  * [18] F. Colombo, J. Gantner: Fractional powers of quaternionic operators and Katos formula using slice hyperholomorphicity. Trans. Amer. Math. Soc. 370, no. 2, 1045–1100 (2018)
  * [19] F. Colombo, J. Gantner: An application of the $S$-functional calculus to fractional diffusion processes. Milan J. Math. 86(2):225–303 (2018)
  * [20] F. Colombo, J. Gantner: Quaternionic closed operators, fractional powers and fractional diffusion processes. Operator Theory: Advances and Applications 274. Birkhäuser/Springer, Cham, viii+322 pp. (2019)
  * [21] 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, ix+356 pp. (2018)
  * [22] F. Colombo, J. Gantner, D.P. Kimsey, I. Sabadini: Universality property of the $S$-functional calculus, noncommuting matrix variables and Clifford operators. Adv. Math. 410, Paper No. 108719, 39 pp. (2022)
  * [23] F. Colombo, D.P. Kimsey: The spectral theorem for normal operators on a Clifford module. Anal. Math. Phys. 12(1):Paper No. 25, 92 (2022)
  * [24] F. Colombo, D.P. Kimsey, S. Pinton, I. Sabadini: Slice monogenic functions of a Clifford variable via the $S$-functional calculus. Proc. Amer. Math. Soc. Ser. B 8, 281–296 (2021)
  * [25] F. Colombo, M.M. Peloso, S. Pinton: The structure of the fractional powers of the noncommutative Fourier law. Math. Methods Appl. Sci. 42(18):6259–6276 (2019)
  * [26] F. Colombo, I. Sabadini: The $\mathcal{F}$-spectrum and the $\mathcal{SC}$-functional calculus. Proc. Roy. Soc. Edinburgh Sect. A, 142(3):479–500 (2012)
  * [27] F. Colombo, I. Sabadini: The $F$-functional calculus for unbounded operators. J. Geom. Phys. 86: 392–407 (2014)
  * [28] F. Colombo, I. Sabadini, F. Sommen: The Fueter mapping theorem in integral form and the $F$-functional calculus. Math. Methods Appl. Sci. 33(17):2050–2066 (2010)
  * [29] F. Colombo, I. Sabadini, D.C. Struppa: Michele Sce’s works in hypercomplex analysis – a translation with commentaries. Birkhäuser/Springer, Cham (2020)
  * [30] F. Colombo, I. Sabadini, D.C. Struppa: Noncommutative functional calculus. Theory and applications of slice hyperholomorphic functions, volume 289 of Progress in Mathematics. Birkhäuser/Springer Basel AG, Basel, vi+221 pp. (2011)
  * [31] R. Delanghe, F. Sommen, V. Souček: Clifford algebra and spinor-valued functions. Mathematics and its Applications 53, Kluwer Academic Publishers Group, Dordrecht (1992)
  * [32] A. De Martino, K. Diki, A. Guzmán Adán: On the connection between the Fueter-Sce-Qian theorem and the generalized CK-extension, Results Math., 78 (2), 55, 35pp (2023)
  * [33] A. De Martino, K. Diki, A. Guzmán Adán: The Fueter-Sce mapping and the Clifford–Appell polynomials, Proc. Edinb. Math. Soc., (2) 66, no. 3, 642–688, (2023)
  * [34] A. De Martino, S. Pinton: A polyanalytic functional calculus of order 2 on the $S$-spectrum. Proc. Amer. Math. Soc. 151, no. 6, 2471–2488 (2023)
  * [35] A. De Martino, S. Pinton: Properties of a polyanalytic functional calculus on the $S$-spectrum. Math. Nachr., 1–37, https://doi.org/10.1002/mana.202200318 (2023)
  * [36] J. Diestel, J.J. Uhl: Vector Measures. Mathematical Surveys and Monographs 15 (1977)
  * [37] N. Dunford, J. Schwartz: Linear operators, part I: general theory. J. Wiley and Sons (1988)
  * [38] R. Fueter: Die Funktionentheorie der Differentialgleichungen $\Delta u=0$ und $\Delta\Delta u=0$ mit vier reellen Variablen. Comment. Math. Helv. 7(1):307–330 (1934)
  * [39] J. Gantner: A direct approach to the $S$-functional calculus for closed operators. J. Operator Theory 77, no. 2, 287–331 (2017)
  * [40] M. Haase: The functional calculus for sectorial operators. Operator Theory: Advances and Applications 169. Birkhäuser Verlag, Basel (2006)
  * [41] T. Hytönen, J. van Neerven, M. Veraar, L. Weis, Analysis in Banach spaces. Vol. I. Martingales and Littlewood-Paley theory. Ergebnisse der Mathematik und ihrer Grenzgebiete, 63, Springer, Cham, xvi+614 pp. (2016)
  * [42] T. Hytönen, J. van Neerven, M. Veraar, L. Weis: Analysis in Banach spaces. Vol. II. Probabilistic methods and operator theory. Ergebnisse der Mathematik und ihrer Grenzgebiete, 67, Springer, Cham, xxi+616 pp. (2017)
  * [43] B. Jefferies: Spectral properties of noncommuting operators. Lecture Notes in Mathematics, 1843, Springer-Verlag, Berlin (2004)
  * [44] A. McIntosh: Operators which have an $H^{\infty}$-functional calculus. Miniconference on operator theory and partial differential equations (North Ryde, 1986), 210–231, Proc. Centre Math. Anal. Austral. Nat. Univ., 14, Austral. Nat. Univ., Canberra (1986)
  * [45] T. Qian: Generalization of Fueter’s result to $\mathbb{R}^{n+1}$. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 8(2):111–117 (1997)
  * [46] T. Qian, P. Li: Singular integrals and Fourier theory on Lipschitz boundaries. Science Press Beijing, Beijing, Springer, Singapore, xv+306 pp (2019)
  * [47] M. Sce: Osservazioni sulle serie di potenze nei moduli quadratici. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. 23(8):220–225 (1957)
  * [48] L. Weis, _The $H^{\infty}$ holomorphic functional calculus for sectorial operators - a survey. Partial differential equations and functional analysis_, 263294, Oper. Theory Adv. Appl., 168, Birkhäuser, Basel (2006)