# Fredholm theory in quaternionic Banach algebras

Hatem Baloudi  Hatem Baloudi, Department of Mathematics, Faculty of Sciences
of Gafsa, University of Gafsa, 2112 Zarroug, Tunisia

###### Abstract.

Muraleetharan and Thirulogasanthan in (J. Math. phys. 59, No. 10, 103506, 27p.
(2018)) introduced the concept of Calkin S-spectrum of a bounded quaternionic
linear operators. The study of this spectrum is establisched using the
Fredholm operators theory. Motivated by this, we study the general framework
of the Fredholm element with respect to a quaternionic Banach algebra
homomorphism. First, we investigate the Fredholm S-spectrum of the sum of two
elements in quaternionic Banach algebra by means of the Fredholm S-spectrum of
the two elements. Next, we prove a perturbation result on this spectrum. We
also study the boundary S-spectrum. As application, we investigate the
Fredholm and Weyl S-spectra of bounded right quaternionic linear operators.

###### Key words and phrases:

Quaternions, Quaternionic Banach algebra, Weyl spectrum, Fredholm spectrum.

###### 2010 Mathematics Subject Classification:

46S10, 47A60, 47A10, 47A53, 47B07

###### Contents

  1. 1 Introduction
  2. 2 Mathematical preliminaries
    1. 2.1 Quaternions
    2. 2.2 Slice hyperholomorphic functions
    3. 2.3 Quaternionic two-sided Banach algebra
    4. 2.4 S-spectrum
  3. 3 Fredholm S-spectrum
  4. 4 Boundary S-spectrum
  5. 5 Application to right quaternionic linear operators
    1. 5.1 Preliminary
    2. 5.2 Fredholm and Weyl S-spectra of quaternionic operators
    3. 5.3 Right shift

##  1\. Introduction

In the complex setting, much attention has been paid to Fredholm theory of
linear operators on a complex Banach spaces [5, 7, 19, 20]. It is useful for
the study of non-parabolicity at infinity of Dirac type operators on non-
compact Riemannian manifuld [18]. We refer to [4] for the study of this point
on the discrete Gauss-Bonnet operator on an infinite graph. Moreover, Fredholm
operators theory plays an important role in the study of non-discrete
spectrum, and the study of the stability of some parts on the spectrum under
some perturbation [6, 21, 22, 31].

In complex spectral theory, it is well known (Atkinson’s theorem) that a
bounded linear operator on a complex Banach space is Fredholm if, and only if,
it is invertible in the Calkin algebra. In particular, the Fredholm spectrum
of a bounded operator $A$ is given by the spectrum of $\pi(A)$, where $\pi$ is
the natural quotient map associated with the Calkin algebra. As an extension,
here Fredholm and perturbation theory relative to a unital homomorphism
$\mathcal{A}:\ \mathcal{V}\longrightarrow\mathcal{W}$ between unital complex
Banach algebras $\mathcal{V}$ and $\mathcal{W}$. This class introduced firstly
by R. Harte in [19]. A generalized Fredholm theory is recently discuss in
[13].

In the quaternionic setting, the quaternionic multiplication is not
commutative. This leads to three types of Hilbert spaces: left, right, and
two-sided, depending on how vectors are multiplied by scalars. Therefore, an
apparent problem in defining the spectrum of a quaternionic operator. In fact,
if $T$ is a right linear operator on a quaternionic vector space
$\mathcal{V}$, the function $x\longmapsto Tx-xq$ is not right linear, we refer
to [8, 9] for this point of view. The fundamental suggestion of [8] is to
define the spectrum using the Cauchy kernel series
$S^{-1}(q,T)=\sum_{n}T^{n}q^{-n-1}$, where $q$ is a quaternionic number and
$T$ is a right linear operator on a right quaternionic vector space. More
precisely, they identified the operator whose inverse is $S^{-1}(q,T)$. This
leads to define a new spectrum called S-spectrum. Based on this concept, [25]
introduced and studied the Fredholm and Weyl S-spectra.

The aim of this article is to complete the knowledge of the concept of the
Fredholm S-spectrum and give a general framework of Fredholm theory relative
to a quaternionic Banach algebra homomorphism. In general, the set of right
linear operators on the quaternionic Banach space is not quaternionic Banach
algebra with respect to the composition of operators. If $\mathcal{V}$ is a
right separable Hilbert space, it is possible to define the left scalar
multiplication on $\mathcal{V}$ by using an arbitrary Hilbert basis on
$\mathcal{V}$. We refer to [17] for an explanation of this construction.
Thanks to this construction, the set of right linear operators becomes a
quaternionic two-sided Banach $C^{*}-$algebra with unity, see [17, Theorem
3.4]. In this regard, [25] introduce the Calkin spherical spectrum as the
spherical spectrum of the quotient map image of a bounded right linear
operator on the Calkin algebra. They gave a characterization of this new
spectrum by using the concept of Fredholm operators. Motivated by this, we
introduce and we study the general framework of the Fredholm element with
respect to a quaternionic Banach algebra homomorphism. In particular, we give
some new results for the Calkin and Weyl S-spectra of the bounded right
quaternionic linear operator.

To start off, we introduce the quaternionic version of the Fredholm elements
relative to the homomorphism $\mathcal{A}:\
\mathcal{V}\longrightarrow\mathcal{W}$, where $\mathcal{V}$ and $\mathcal{W}$
are two quaternionic two-sided Banach algebra with unit $1\neq 0$. In
addition, we define the Fredholm and Weyl S-spectra with respect to
$\mathcal{A}$. One of the main results is: The Fredholm S-spectrum of the sum
of two elements in a Banach algebra is written by means of the Fredholm
S-spectra of the two elements when their products are in the kernel of
$\mathcal{A}$, see Theorem 3.5. This result is applicable for the Calkin (or
Fredholm) S-spectrum of the sum of two bounded quaternionic right linear
operators when their products are compact, see Theorem 5.9. On the other hand,
we show the invariance of the Fredholm and Weyl S-spectra under certain
perturbations. Furthermore, we investigate the boundary S-spectrum of an
element in quaternionic Banach algebra.

The article is organized as follows. In Section 2, we gather some definitions
and notations, the facts connected to our work. In Section 3, the concept of
the Fredholm and Weyl S-spectra in a quaternionc Banach algebra is introduced.
Some properties and perturbation results of these spectra are investigated. In
Section 4, we introduce and we study the boundary S-spectrum. Finally, in
Section 5, we apply the results of Section 3 to describe the Fredholm and Weyl
S-spectra of the sum of two bounded right linear operators. We also discuss
the coincidence between the Fredholm and Weyl S-spectrum.

##  2\. Mathematical preliminaries

In this section we recall some standard definitions and we gives some results
that we will need in the sequel. For details we refer to the reader to [1, 9,
12, 17].

###  2.1. Quaternions

The algebra of quaternion $\mathbb{H}$ is defined as

$\displaystyle\mathbb{H}=\Big{\\{}q=q_{0}+iq_{1}+jq_{2}+kq_{3}:(q_{0};q_{1},q_{2},q_{3})\in\mathbb{R}^{4}\Big{\\}},$

where the imaginary units $i,\ j,\ k$ satisfy

$\displaystyle i^{2}=j^{2}=k^{2}=-1,\ ij=-ji=k,\ jk=-kj=i,\ ki=-ik=j.$

The quaternionic conjugate of $q=q_{0}+iq_{1}+jq_{2}+kq_{3}$ is

$\displaystyle\overline{q}=q_{0}-iq_{1}-jq_{2}-kq_{3},$

while $|q|=(q\overline{q})^{\frac{1}{2}}$ denotes the usual norm of the
quaternion $q$. If $q$ is non-zero element, it has inverse

$\displaystyle\displaystyle q^{-1}=\frac{\overline{q}}{|q|^{2}}.$

A purely imaginary quaternion with absolute value $1$ is called an imaginary
unit. We denote the set of all imaginary units by $\mathbb{S}$, that is,

$\displaystyle\mathbb{S}=\left\\{q\in\mathbb{H}:\ \overline{q}=-q,\
|q|=1\right\\}.$

The name imaginary unit is justified by the fact that, for any
$I\in\mathbb{S}$, we have

$\displaystyle I^{2}=-\overline{I}I=-|I|^{2}=-1.$

We set

$\displaystyle\mathbb{C}_{I}=\Big{\\{}x+Iy\in\mathbb{H}:\
x,y\in\mathbb{H}\Big{\\}},$

i.e, $\mathbb{C}_{I}$ is the real vector subspace of $\mathbb{H}$ generated by
$1$ and $I\in\mathbb{S}$. Therefore, the plane $\mathbb{C}_{I}$ is isomorphic
to the field of complex number $\mathbb{C}$. For a quaternion
$q\in\mathbb{H}$, we set

$\displaystyle
I_{q}:=\left\\{\begin{array}[]{rl}\noindent\frac{q}{|Im(q)|},&\mbox{ if
}Im(q)\neq 0,\\\ \nobreak\leavevmode\nobreak\leavevmode\hfil\\\
\displaystyle\mbox{any }I\in\mathbb{S},&\mbox{ if }Im(q)=0.\end{array}\right.$

Then $I_{q}\in\mathbb{S}$ and $q\in\mathbb{C}_{I_{q}}$. More precisely,
$q=Re(q)+I_{q}|Im(q)|$, where $Re(q)$ is the real part of the quaternion $q$
and $Im(q)$ is the vector part of $q$. The set

$\displaystyle[q]:=\\{Re(q)+I|Im(q)|:\ I\in\mathbb{S}\\},$

is a 2-sphere of radius $|Im(q)|$ centred at the real point $Re(q)$.

###  2.2. Slice hyperholomorphic functions

In this section, we recall the concept of slice hyperholomorphic function and
refer to [14, 10, 11, 12] for surveys on the matter.

######  Definition 2.1.

A set $U\subset\mathbb{H}$ is called
$(i)$ axially symmetric if $[x]\subset U$ for any $x\in U$ and
$(ii)$ a slice domain if $U$ is open, $U\cap\mathbb{R}\neq\emptyset$ and
$U\cap\mathbb{C}_{I}$ is a domain for any $I\in\mathbb{S}$.

Let $I$ be an imaginary unit. We denote the Wirtinger derivatives with respect
to the complex and the conjugate variable on the plane $\mathbb{C}_{I}$ by
$\partial_{I}$ and $\overline{\partial_{I}}$, that is, the operators

$\displaystyle\partial_{I}=\frac{1}{2}\left(\frac{\partial}{\partial_{q_{0}}}-I\frac{\partial}{\partial_{q_{1}}}\right)\mbox{
and
}\overline{\partial}_{I}=\frac{1}{2}\left(\frac{\partial}{\partial_{q_{0}}}+I\frac{\partial}{\partial_{q_{1}}}\right)$

Let $U\subset\mathbb{C}_{I}$ be an open set and let $f:\
U\longrightarrow\mathbb{H}$ be a real differentiable function. If
$\partial_{I}$ act on the right, then

$\displaystyle\overline{\partial}_{I}f(q)=\displaystyle\frac{1}{2}\left(\frac{\partial}{\partial_{q_{0}}}f_{I}(q_{0}+Iq_{1})+\frac{\partial}{\partial_{q_{1}}}f_{I}(q_{0}+Iq_{1})I\right)$

for $q=q_{0}+Iq_{1}\in U$, where $f_{I}=f|_{U\cap\mathbb{C}_{I}}$

######  Proposition 2.2.

[9] For any non-real quaternion $q\in\mathbb{H}\setminus\mathbb{R}$, there
exist, and are unique, $q_{0},q_{1}\in\mathbb{R}$ with $q_{1}>0$, and
$I\in\mathbb{S}$ such that $q=q_{0}+q_{1}I$.

######  Definition 2.3.

$($Slice-regular function [9, 15, 32]$)$ Let $U$ be a domain in $\mathbb{H}$
and let $\mathcal{V}_{\mathbb{H}}^{R}$ be a vector space under right
multiplication by quaternions. A real differentiable operator-valued function
$f:U\longrightarrow\mathcal{V}_{\mathbb{H}}^{R}$ is said to be slice right
regular if, for every quaternion $I\in\mathbb{S}$, the restriction of $f$ to
the complex plane $\mathbb{C}_{I}=\mathbb{R}+I\mathbb{R}$ passing through the
origin, and containing $1$ and $I$, has continuous partial derivatives (with
respect to $q_{0}$ and $q_{1}$, every element in $\mathbb{C}_{I}$ being
uniquely expressible as $q_{0}+q_{1}$I) and satisfies

$\displaystyle\overline{\partial}f(q_{0}+q_{1}I)=0$

where $f_{I}=f|_{U\cap\mathbb{C}_{I}}$.

###  2.3. Quaternionic two-sided Banach algebra

We recall the notion of quaternionic two-sided Banach algebra and refer to
[16, 17, 33] for surveys on the matter. Let $\mathcal{V}$ be a quaternionic
two-sided vector space. We say that $\mathcal{V}$ is a (associative)
quaternionic two-sided Banach algebra if it is endowed with an associative
product $\mathcal{V}\times\mathcal{V}\longrightarrow\mathcal{V}:\
(x,y)\longmapsto xy$ such that

  1. (1)

$x(y+z)=xy+xz$ $\forall x,y,z\in\mathcal{V}$,

  2. (2)

$(x+y)z=xz+yz$ $\forall x,y,z\in\mathcal{V}$,

  3. (3)

$q(xy)=(qx)y$ $\forall x,y\in\mathcal{V}$, $\forall q\in\mathbb{H}$

  4. (4)

$(xy)q=x(yq)$ $\forall x,y\in\mathcal{V}$, $\forall q\in\mathbb{H}$

We say that $\mathcal{V}$ is a normed two-sided algebra provide $\mathcal{V}$
is endowed with an $\mathbb{H}-$norm such that $\|xy\|\leq\|x\|\|y\|$ for
every $x,y\in\mathcal{V}$. If $\mathcal{V}$ is complete we say that
$\mathcal{V}$ is a quaternionic two-sided Banach algebra. If in addition
$\mathcal{V}$ is non-trivial and $\|1_{\mathcal{V}}\|=1$, then $\mathcal{V}$
is called a quaternionic two-sided Banach algebra with unit.
A map $\mathcal{A}:\mathcal{V}\longrightarrow\mathcal{W}$ between quaternionic
two-sided Banach algebras with unity is called homomorphism if

  1. (1)

$\mathcal{A}(u+v)=\mathcal{A}(u)+\mathcal{A}(v)$ and
$\mathcal{A}(uv)=\mathcal{A}(u)\mathcal{A}(v)$,

  2. (2)

$\mathcal{A}(qu)=q\mathcal{A}(u)$ and $\mathcal{A}(uq)=\mathcal{A}(u)q$,

  3. (3)

$\mathcal{A}(1_{\mathcal{V}})=1_{\mathcal{W}}$

for every $u,v\in\mathcal{V}$ and $q\in\mathbb{H}$. A mapping
$v\longrightarrow v^{*}$ $(v\in\mathcal{V})$ is called involution if it
possesses the following properties:

  1. (1)

$(u+v)^{*}=u^{*}+v^{*}$,

  2. (2)

$(uv)^{*}=v^{*}u^{*}$,

  3. (3)

$(u^{*})^{*}=u$,

  4. (4)

$(puq)^{*}=\overline{q}u^{*}\overline{p}$,

for every $u,v\in\mathcal{V}$ and $p,q\in\mathbb{H}$. Finally, a quaternionic
two-sided Banach algebra is called a quaternionic two-sided $C^{*}-$algebra if
$\|u^{*}u\|=\|u\|^{2}$ for all $u\in\mathcal{V}$

######  Remark 2.4.

[17, Remark.3.3]
$1)$ The concept of real two-sided Banach unital algebra is equivalent to the
usual one of real Banach unital algebras.
$2)$ Let $I\in\mathbb{S}$. The concept of complex Banach unital algebras is
equivalent to the notion of two-sided $\mathbb{C}_{I}-$Banach unital algebras
$\mathcal{V}$ having the following property

$\displaystyle su=us\mbox{ for every }u\in\mathcal{V}\mbox{ and
}s\in\mathbb{C}_{I}.$

Let $\mathcal{A}:\mathcal{V}\longrightarrow\mathcal{W}$ be a homomorphism of
Banach algebra, where $\mathcal{V}$ and $\mathcal{W}$ are two quaternionic
two-sided Banach algebra with unit $1\neq 0$. We use $\mathcal{V}^{-1}$ and
$\mathcal{W}^{-1}$ for the groups of invertible elements in $\mathcal{V}$ and
$\mathcal{W}$, it follows that
$\mathcal{A}(\mathcal{V}^{-1})\subset\mathcal{W}^{-1}$. It is easy to see that
$\mathcal{V}^{-1}$ is an open set. We consider the following definition of
Fredholm element relative to the homomorphism $\mathcal{A}$. For the classical
definition in complexe case we refer to [6, 19, 23].

######  Definition 2.5.

  1. (1)

An element $v\in\mathcal{V}$ is called a Fredholm element relative to the
homomorphism $\mathcal{A}$, if $\mathcal{A}(v)\subset\mathcal{W}^{-1}$.

  2. (2)

An element $v\in\mathcal{V}$ is called a Weyl element relative to
$\mathcal{A}$, if $v\in\mathcal{V}^{-1}+\mathcal{A}^{-1}(0)$, the sum of an
invertible element and one whose image is zero.

We denote the set of Fredholm elements relative to the homomorphism
$\mathcal{A}$ buy $\Phi_{\mathcal{A}}$. Let $\Phi_{\mathcal{A}}^{0}$ the set
of Weyl elements relative to $\mathcal{A}$.

######  Remark 2.6.

  1. (1)

If $\mathcal{A}$ is continuous, then $\Phi_{\mathcal{A}}$ is an open set in
$\mathcal{V}$. Indeed,
$\Phi_{\mathcal{A}}=\mathcal{A}^{-1}(\mathcal{W}^{-1})$, where
$\mathcal{W}^{-1}$ is an open set in $\mathcal{W}$.

  2. (2)

It is clear that

$\displaystyle\mathcal{V}^{-1}\subset\Phi_{\mathcal{A}}\subset\Phi_{\mathcal{A}}^{0}.$

  3. (3)

Since $\mathcal{A}$ is a homomorphism, then
$\mathcal{A}(xy)=\mathcal{A}(x)\mathcal{A}(y)$, thus, for all
$x,y\in\Phi_{\mathcal{A}}$ the product $xy\in\Phi_{\mathcal{A}}$.

  4. (4)

If $ab\in\Phi_{\mathcal{A}}$ and $ba\in\Phi_{\mathcal{A}}$, then
$a\in\Phi_{a}$ and $b\in\Phi_{\mathcal{A}}$. Indeed, let $i,j\in\mathcal{W}$
such that $i\mathcal{A}(ab)=\mathcal{A}(ba)j=1$. Then
$i\mathcal{A}(a)=i\mathcal{A}(a)\mathcal{A}(b)\mathcal{A}(a)j=\mathcal{A}(a)j.$

###  2.4. S-spectrum

Colombo et all in [11] introduced the concept of S-spectrum of a bounded
quaternionic operator. This notion is related to the Cauchy kernel series in
the quaternionic setting. In this subsection, we introduce the general
framework of the S-spectrum of an element in a quaternionic Banach algebra.

######  Definition 2.7.

Let $v\in\mathcal{V}$ and $q\in\mathbb{H}$. The left Cauchy kernel series is
given by

$\displaystyle\sum_{n\in\mathbb{N}}v^{n}q^{-1-n}$

for $\|v\|<|q|$.

By repeating the same proof of [8, Theorem 4.2] we have: If
$v-\overline{q}1_{\mathcal{V}}\in\mathcal{V}^{-1}$, then

$\displaystyle-(v-\overline{q}1_{\mathcal{V}})^{-1}(v^{2}-2Re(q)v+|q|^{2}1_{\mathcal{V}})$

is the inverse of $\sum_{n\in\mathbb{N}}v^{n}q^{-1-n}$. Moreover, we have

$\displaystyle\sum_{n\in\mathbb{N}}v^{n}q^{-1-n}=-(v^{2}-2Re(q)v+|q|^{2}1_{\mathcal{V}})^{-1}(v-\overline{q}1_{\mathcal{V}}).$

Therefore, it is natural to define the S-spectrum:

$\displaystyle\sigma_{S}(v)=\\{q\in\mathbb{H}:\
v^{2}-2Re(q)v+|q|^{2}1_{\mathcal{V}}\not\in\mathcal{V}^{-1}\\}.$

######  Definition 2.8.

Let $v\in\mathcal{V}$. For $q\in\mathbb{H}\setminus\sigma_{S}(v)$, we define
the left S-resolvent as

$\displaystyle
S_{L}^{-1}(q,v):=-(v^{2}-2Re(q)v+|q|^{2}1_{\mathcal{V}})^{-1}(v-\overline{q}1_{\mathcal{V}})$

######  Remark 2.9.

Let $v\in\mathcal{V}$ and $q\in\mathbb{H}\setminus\sigma_{S}(v)$.

  1. (1)

If $qv=vq$, then the left S-resolvent coincide with the classical resolvent;
i.e,

$\displaystyle S_{L}^{-1}(q,v)=(v-q1_{\mathcal{V}})^{-1}$

  2. (2)

In analogy to [2, Proposition 6.1.27], $S_{L}^{-1}(q,v)$ is a
$\mathcal{V}-$valued right slice regular function of the variable
$q\in\mathbb{H}\setminus\sigma_{S}(v)$, that is,
$\overline{\partial}S_{L}^{-1}(q_{0}+Iq_{1},v)=0$ for all $I\in\mathbb{S}$.

######  Example 2.10.

We note that the set of matrices of type $n\times n$ with quaternion entries
is quaternionic two-sided Banach algebra with the usual addition of matrices,
the standard product of matrices, the left and the right multiplication
$Aq=(a_{i,j}q)$ and $qA=(qa_{i,j})$.

  1. (1)

[8, Example 4.15, 1)] Consider the following $2\times 2$ quaternionic matrix

$\displaystyle A=\left(\begin{array}[]{cc}i&0\\\ 0&j\\\ \end{array}\right).$

Then,

$\displaystyle
A^{2}-2Re(s)A+|s|^{2}1_{M_{2}(\mathbb{H})}=\left(\begin{array}[]{cc}-1-2iRe(s)+|s|^{2}&0\\\
0&-1-2jRe(s)+|s|^{2}\\\ \end{array}\right)$

and therefore, $\sigma_{S}(A)=\mathbb{S}$.

  2. (2)

We now consider the following $3\times 3$ quaternionic matrix

$\displaystyle B=\left(\begin{array}[]{ccc}0&0&i\\\ 0&j&0\\\ k&0&0\\\
\end{array}\right)$

Then $s\in\sigma_{S}(B)$ if and only if

$\displaystyle\left(\begin{array}[]{ccc}-j+|s|^{2}&0&-2Re(s)i\\\
0&-1-2Re(s)j+|s|^{2}&0\\\ -2Re(s)k&0&j+|s|^{2}\\\
\end{array}\right)\left(\begin{array}[]{ccc}q_{1}\\\ q_{2}\\\ q_{3}\\\
\end{array}\right)=\left(\begin{array}[]{ccc}0\\\ 0\\\ 0\\\
\end{array}\right)$

for some $q=(q_{1},q_{2},q_{3})\in\mathbb{H}^{3}\backslash\\{0\\}$. We observe
that $q_{1}=0$ if and only if $q_{2}=0$, then we have

(1) $1+2Re(s)j-|s|^{2}=0$

or

$\displaystyle(S)\displaystyle\left\\{\begin{array}[]{ll}(-j+|s|^{2})q_{1}-2Re(s)iq_{3}=0\\\
\\\ -2Re(s)kq_{1}+(j+|s|^{2})q_{3}=0\end{array}\right.$

The equation Eq.(1) gives

$\displaystyle 1+2Re(s)j-|s|^{2}=0\mbox{ if and only if}\ s\ \in\mathbb{S}.$

We now consider the system $(S)$: we have

$\displaystyle q_{1}=2Re(s)\displaystyle\frac{|s|^{2}+j}{|s|^{4}+1}iq_{3}$

and replacing it in the second equation of $(S)$ we get

$\displaystyle\displaystyle\frac{4Re^{2}(s)}{|s|^{4}+1}(1+j|s|^{2})-(|s|^{2}+j)=0$

which gives

$\displaystyle|s|=1\mbox{ and }Re(s)\in\\{\pm\frac{\sqrt{2}}{2}\\}.$

We conclude that

$\displaystyle\sigma_{S}(B)=\mathbb{S}\cup\Big{\\{}q\in\mathbb{H}:\
|q|=1\mbox{ and }Re(s)\in\\{\pm\frac{\sqrt{2}}{2}\\}\Big{\\}}.$

Notation We denote by $\mathbb{N}$ the non-negative integer. In particular,
$0\in\mathbb{N}$.

##  3\. Fredholm S-spectrum

The Fredholm S-spectrum of $v\in\mathcal{V}$ relative to the homomorphism
$\mathcal{A}$, is given by

$\displaystyle\sigma_{\mathcal{S},\mathcal{A}}^{\Phi}(v):=\sigma_{S}(\mathcal{A}(v))=\\{q\in\mathbb{H}:\
R_{q}(v)\not\in\Phi_{\mathcal{A}}\\}.$

We define the Weyl S-spectrum of $v\in\mathcal{V}$ relative to the
homomorphism $\mathcal{A}$ as

$\displaystyle\sigma_{\mathcal{S},\mathcal{A}}^{\Phi^{0}}(v):=\\{q\in\mathbb{H}:\
R_{q}(v)\not\in\Phi_{\mathcal{A}}^{0}\\}.$

######  Remark 3.1.

Let $v\in\mathcal{V}$.

  1. (1)

If $q\in\sigma_{\mathcal{S},\mathcal{A}}^{\Phi}(v)$ $($resp.
$\sigma_{\mathcal{S},\mathcal{A}}^{\Phi^{0}}(v))$ then
$[q]\subset\sigma_{\mathcal{S},\mathcal{A}}^{\Phi}(v)$ $($resp.
$\sigma_{\mathcal{S},\mathcal{A}}^{\Phi^{0}}(v))$. Therefore,
$\sigma_{S,\mathcal{A}}^{\Phi}(v)$ and $\sigma_{S,\mathcal{A}}^{\Phi^{0}}(v)$
are axially symmetric.

  2. (2)

In general, we have the following inclusion

$\displaystyle\sigma_{S,\mathcal{A}}^{\Phi}(v)\subset\sigma_{S,\mathcal{A}}^{\Phi^{0}}(v)\subset\sigma_{S}(v).$

######  Definition 3.2.

An element $p\in\mathcal{V}$ is called Fredholm perturbation if
$p+t\in\Phi_{\mathcal{A}}$ for all $t\in\Phi_{\mathcal{A}}$. We denote this
set by $Pr(\Phi_{\mathcal{A}})$.

######  Lemma 3.3.

$Pr(\Phi_{\mathcal{A}})$ is closed two-sided ideal of $\mathcal{A}$.

###### Proof.

The proof is the same as the proof in a complex Banach space, see[7, 6].∎

######  Remark 3.4.

Let $\mathcal{A}:\mathcal{V}\longrightarrow\mathcal{W}$ be a homomorphism of
Banach algebra.

  1. (1)

If $p\in Pr(\Phi_{\mathcal{A}})$, then

$\displaystyle\sigma_{\mathcal{S},\mathcal{A}}^{\Phi}(p)\backslash\\{0\\}=\emptyset.$

Indeed, let $q\in\mathbb{H}^{*}$, then
$|q|^{2}1_{\mathcal{V}}\in\Phi_{\mathcal{A}}$. Since $Pr(\Phi_{\mathcal{A}})$
is closed two-sided ideal of $\mathcal{V}$, then
$R_{q}(p)\in\Phi_{\mathcal{A}}$.

  2. (2)

If $p\in Pr(\Phi_{\mathcal{A}})$, then $p\not\in\mathcal{V}^{-1}$. Indeed,
$1_{\mathcal{V}}\not\in Pr(\Phi_{\mathcal{A}})$.

  3. (3)

In general, we have the following inclusion

$\displaystyle\mathcal{A}^{-1}(0)\subset Pr(\Phi_{\mathcal{A}}).$

  4. (4)

Let $v\in\mathcal{V}$ and $q\in\mathbb{H}$ with $|q|>\|\mathcal{A}(v)\|$. Then

$\displaystyle v^{2}-2Re(q)v+|q|^{2}1_{\mathcal{V}}\in\Phi_{\mathcal{A}}.$

Indeed, for each $n\in\mathbb{N}$, set

$\displaystyle a_{n}=|q|^{-2n-2}\sum_{h=0}^{n}q^{h}\overline{q}^{n-h}.$

According to the proof of [17, Theorem 4.3(a)], $a_{0}=|q|^{-2}$,
$-2Re(q)a_{0}+|q|^{2}a_{1}=0$ and $a_{n-2}-2Re(q)a_{n-1}+|q|^{2}a_{n}=0$ for
every $n>1$. Moreover, we have

$\displaystyle\sum_{n=0}^{\infty}\|\mathcal{A}(v^{n})\||a_{n}|\leq\sum_{n=0}^{\infty}\|\mathcal{A}(v)\|^{n}(n+1)|q|^{-n-2}.$

Hence, $\sum_{n\in\mathbb{N}}\mathcal{A}(v^{n})$ converge absolutely. In this
way, we have

$\displaystyle\mathcal{A}(R_{q}(v))(\sum_{n\in\mathbb{N}}\mathcal{A}(v^{n})a_{n})=(\sum_{n\in\mathbb{N}}\mathcal{A}(v^{n})a_{n})\mathcal{A}(R_{q}(v))=1_{\mathcal{W}}.$

We first prove the following result:

######  Theorem 3.5.

Let $\mathcal{A}:\mathcal{V}\longrightarrow\mathcal{W}$ be a homomorphism of
Banach algebra, where $\mathcal{V}$ and $\mathcal{W}$ are two quaternionic
two-sided Banach algebra with unit $1\neq 0$. Take $a,b\in\mathcal{V}$.
1) If $ab\in Pr(\Phi_{\mathcal{A}})$ and $ba\in Pr(\Phi_{\mathcal{A}})$, then

(2)
$\sigma_{\mathcal{S},\mathcal{A}}^{\Phi}(a+b)\backslash\\{0\\}=\left[\sigma_{\mathcal{S},\mathcal{A}}^{\Phi}(a)\cup\sigma_{\mathcal{S},\mathcal{A}}^{\Phi}(b)\right]\backslash\\{0\\}.$

2) If $ab\in\mathcal{A}^{-1}(0)$ and $ba\in\mathcal{A}^{-1}(0)$, then

(3)
$\sigma_{\mathcal{S},\mathcal{A}}^{\Phi^{0}}(a+b)\backslash\\{0\\}\subset\left[\sigma_{\mathcal{S},\mathcal{A}}^{\Phi^{0}}(a)\cup\sigma_{\mathcal{S},\mathcal{A}}^{\Phi^{0}}(b)\right]\backslash\\{0\\}.$

If, further,
$\sigma_{\mathcal{S},\mathcal{A}}^{\Phi^{0}}(a)=\sigma_{\mathcal{S},\mathcal{A}}^{\Phi}(a)$,
then

(4)
$\sigma_{\mathcal{S},\mathcal{A}}^{\Phi^{0}}(a+b)\backslash\\{0\\}=\left[\sigma_{\mathcal{S},\mathcal{A}}^{\Phi^{0}}(a)\cup\sigma_{\mathcal{S},\mathcal{A}}^{\Phi^{0}}(b)\right]\backslash\\{0\\}.$

###### Proof.

For $q\in\mathbb{H}^{*}$, we can write

(5) $\displaystyle\frac{R_{q}(b)R_{q}(a)}{|q|^{2}}=R_{q}(a+b)-ab-
ba+\displaystyle\frac{b^{2}a^{2}-2Re(q)(b^{2}a+ba^{2}-2Re(q)ba)}{|q|^{2}}$

and

(6) $\displaystyle\frac{R_{q}(a)R_{q}(b)}{|q|^{2}}=R_{q}(a+b)-ab-
ba+\displaystyle\frac{a^{2}b^{2}-2Re(q)(a^{2}b+ab^{2}-2Re(q)ab)}{|q|^{2}}.$

Let
$q\not\in\sigma_{\mathcal{S},\mathcal{A}}^{\Phi}(a)\cup\sigma_{\mathcal{S},\mathcal{A}}^{\Phi}(b)\cup\\{0\\}$,
then

$\displaystyle R_{q}(a)\in\Phi_{\mathcal{A}}\mbox{ and
}R_{q}(b)\in\Phi_{\mathcal{A}}.$

Using Remark 2.6 $(3)$, we obtain $R_{q}(a)R_{q}(b)\in\Phi_{\mathcal{A}}$.
Since $ab$ and $ba$ are two Fredholm perturbation, we can apply Eq. (6), we
infer that

$\displaystyle|q|^{2}R_{q}(a+b)\in\Phi_{\mathcal{A}}.$

Also, since $q\neq 0,$ then $R_{q}(a+b)\in\Phi_{\mathcal{A}}$. Therefore,

(7)
$\sigma_{\mathcal{S},\mathcal{A}}^{\Phi}(a+b)\backslash\\{0\\}\subset\left[\sigma_{\mathcal{S},\mathcal{A}}^{\Phi}(a)\cup\sigma_{\mathcal{S},\mathcal{A}}^{\Phi}(b)\right]\backslash\\{0\\}.$

To prove the inverse inclusion of Eq. (7). Suppose that
$q\not\in\sigma_{\mathcal{S},\mathcal{A}}^{\Phi}(a+b)\cup\\{0\\}$, then
$R_{q}(a+b)\in\Phi_{\mathcal{A}}$. Since $ab\in Pr(\Phi_{\mathcal{A}})$ and
$ba\in Pr(\Phi_{\mathcal{A}})$, then by Eqs. (5) and (6), we have

$\displaystyle R_{q}(a)R_{q}(b)\in\Phi_{\mathcal{A}}\mbox{ and
}R_{q}(b)R_{q}(a)\in\Phi_{\mathcal{A}}.$

Again, using Remark 2.6 $(4)$, we have $R_{q}(a)\in\Phi_{\mathcal{A}}$ and
$R_{q}(b)\in\Phi_{\mathcal{A}}$. Hence
$q\not\in\sigma_{\mathcal{S},\mathcal{A}}^{\Phi}(a)\cup\sigma_{\mathcal{S},\mathcal{A}}^{\Phi}(b)$.
This proves that Eq. (2).
The proof of $(\ref{e88})$ may be cheked in the same way as the proof of
$(\ref{e0})$. To prove the inverse inclusion of $(\ref{e88})$. Let
$q\not\in\sigma_{\mathcal{S},\mathcal{A}}^{\Phi^{0}}(a+b)\cup\\{0\\}$. Since
$ab$ and $ba$ are in $\mathcal{A}^{-1}(0)$, then by Eq. (5), we have

$\displaystyle R_{q}(a)R_{q}(b)=v_{0}+c_{0}$

for some $v_{0}\in\mathcal{V}^{-1}$ and $c_{0}\in\mathcal{A}^{-1}(0)$. Also,
since
$\sigma_{\mathcal{S},\mathcal{A}}^{\Phi^{0}}(a)=\sigma_{\mathcal{S},\mathcal{A}}^{\Phi}(a)$,
then there exists $v_{1}\in\mathcal{V}^{-1}$ and $c_{1}\in\mathcal{A}^{-1}(0)$
such that

$\displaystyle R_{q}(b)=v_{1}(c_{1}R_{q}(b)+v_{0}+c_{0}).$

In this way, we see that $R_{q}(a)$ and $R_{q}(b)$ are in
$\Phi_{\mathcal{A}}^{0}$. This proof is complete.∎

######  Theorem 3.6.

Let $a\in\mathcal{V}^{-1}$. Then

$\displaystyle\sigma_{\mathcal{S},\mathcal{A}}^{\Phi}(a^{-1})=\left\\{\displaystyle\frac{\overline{q}}{|q|^{2}}:\
q\in\sigma_{\mathcal{S},\mathcal{A}}^{\Phi}(a)\right\\}$

and

$\displaystyle\sigma_{\mathcal{S},\mathcal{A}}^{\Phi^{0}}(a^{-1})=\left\\{\displaystyle\frac{\overline{q}}{|q|^{2}}:\
q\in\sigma_{\mathcal{S},\mathcal{A}}^{\Phi^{0}}(a)\right\\}.$

###### Proof.

We can write for $q\in\mathbb{H}\backslash\\{0\\}$

$\displaystyle\displaystyle\frac{R_{q}(a)}{|q|^{2}}$
$\displaystyle=\left(a^{-2}-\frac{2Re(q)}{|q|^{2}}a^{-1}+\frac{1}{|q|^{2}}\right)a^{2}$
$\displaystyle=a^{2}\left(a^{-2}-\frac{2Re(q)}{|q|^{2}}a^{-1}+\frac{1}{|q|^{2}}\right)$

because $\mathcal{V}$ is two-sided Banach algebra. This shows that
$R_{q}(a)\in\Phi_{\mathcal{A}}$ if and only if
$(a^{-2}-\frac{2Re(q)}{|q|^{2}}a^{-1}+\frac{1}{|q|^{2}})a^{2}\in\Phi_{\mathcal{A}}$.
Since $a^{2}\in\mathcal{V}^{-1}$, then

$\displaystyle q\in\sigma_{\mathcal{S},\mathcal{A}}^{\Phi}(a)\mbox{ if and
only if
}\frac{\overline{q}}{|q|^{2}}\in\sigma_{\mathcal{S},\mathcal{A}}^{\Phi}(a^{-1}).$

Similary, we have that $R_{q}(a)\in\Phi_{\mathcal{A}}^{0}$ if and only if
$R_{\frac{\overline{q}}{|q|^{2}}}\in\Phi_{\mathcal{A}}^{0}$.∎

######  Corollary 3.7.

Let $a,b\in\mathcal{V}^{-1}$.

  1. (1)

If $a^{-1}-b^{-1}\in\Phi_{\mathcal{A}}$, then

$\displaystyle\sigma_{\mathcal{S},\mathcal{A}}^{\Phi}(a)=\sigma_{\mathcal{S},\mathcal{A}}^{\Phi}(b).$

  2. (2)

If $a^{-1}-b^{-1}\in\mathcal{A}^{-1}(0)$, then

$\displaystyle\sigma_{\mathcal{S},\mathcal{A}}^{\Phi^{0}}(a)=\sigma_{\mathcal{S},\mathcal{A}}^{\Phi^{0}}(b).$

###### Proof.

Since $a^{-1}-b^{-1}\in\Phi_{\mathcal{A}}$, then
$\sigma_{\mathcal{S},\mathcal{A}}^{\Phi}(a^{-1})=\sigma_{\mathcal{S},\mathcal{A}}^{\Phi}(b^{-1})$.
Applying Theorem 3.6, we obtain
$\sigma_{\mathcal{S},\mathcal{A}}^{\Phi}(a)=\sigma_{\mathcal{S},\mathcal{A}}^{\Phi}(b)$.∎

######  Theorem 3.8.

Let $\mathcal{A}:\mathcal{V}\longrightarrow\mathcal{W}$ be a bounded below
homomorphism and $a\in\mathcal{V}$. Then,

$\displaystyle\partial\sigma_{S}(a)\subset\partial\sigma_{S,\mathcal{A}}^{\Phi}(a).$

###### Proof.

Let $q\in\partial\sigma_{S}(a)$. Then $q\not\in
int(\sigma_{S,\mathcal{A}}^{\Phi}(a))$. So, it suffices to prove that
$q\in\sigma_{S,\mathcal{A}}^{\Phi}(a)$. Suppose that
$R_{q}(a)\in\Phi_{\mathcal{A}}$. Then,

$\displaystyle(\mathcal{A}(a))^{2}-2Re(q)\mathcal{A}(a)+|q|^{2}1_{\mathcal{W}}\in\mathcal{W}^{-1}.$

Now, let $(q_{n})_{n}$ be a sequence in $\mathbb{H}\backslash\sigma_{S}(a)$
such that $q_{n}\longrightarrow q$. So,

$\displaystyle(\mathcal{A}(a))^{2}-2Re(q_{n})\mathcal{A}(a)+|q_{n}|^{2}1_{\mathcal{W}}\longrightarrow(\mathcal{A}(a))^{2}-2Re(q)\mathcal{A}(a)+|q|^{2}1_{\mathcal{W}}$

By the continuity of the map $u\longmapsto u^{-1}$, we get

$\displaystyle
R_{q_{n}}^{-1}(\mathcal{A}(a))=\mathcal{A}(R_{q_{n}}^{-1}(a))\longrightarrow
R_{q}^{-1}(\mathcal{A}(a)).$

Therefore, the sequence $\mathcal{A}(R_{q_{n}}(a)))_{n}$ is Cauchy in
$\mathcal{W}$. Since $\mathcal{A}$ is bounded below, then
$(R_{q_{n}}^{-1}(a))_{n}$ converges, say $R_{q_{n}}^{-1}(a)\longrightarrow
b\in\mathcal{V}$. This show that

$\displaystyle bR_{q}(a)=R_{q}(a)b=1_{\mathcal{V}}$

which contradicts the fact that $q\in\sigma_{S}(a)$.∎

######  Lemma 3.9.

Let $(a_{n})$ be a sequence elements of $\mathcal{V}^{-1}$ converging to a
non-invertible element. Then $\lim_{n}\|a_{n}^{-1}\|=+\infty$.

###### Proof.

The proof is the same as the proof in a complex Banach algebra, see [3].∎

######  Theorem 3.10.

Let $a\in\mathcal{A}$ and $q\in\partial\sigma_{S}(a)$. Then there exists a
sequence $(b_{n})$ of elements of $\mathcal{A}$ such that $\|b_{n}\|=1$ and

$\displaystyle\lim_{n}(a^{2}-2Re(q)a+|q|^{2}1_{\mathcal{V}})b_{n}=\lim_{n}b_{n}(a^{2}-2Re(q)a+|q|^{2}1_{\mathcal{V}})=0.$

###### Proof.

Let $q\in\partial\sigma_{S}(a)$. Then, there exists a sequence
$(q_{n})\subset\mathbb{H}\setminus\sigma_{S}(a)$ of quaternion number
converging to $q$. Take

$\displaystyle
b_{n}:=\displaystyle\frac{(a^{2}-2Re(q_{n})a+|q_{n}|^{2}1_{\mathcal{V}})^{-1}}{\|(a^{2}-2Re(q_{n})a+|q_{n}|^{2}1_{\mathcal{V}})^{-1}\|}.$

Then we have $\|b_{n}\|=1$ and

$\displaystyle
R_{q}(a)b_{n}=(a^{2}-2Re(q_{n})a+|q_{n}|^{2}1_{\mathcal{V}})b_{n}+(2Re(q_{n}-q)a+(|q^{2}|-|q_{n}|^{2})1_{\mathcal{V}})b_{n}.$

Consequently

$\displaystyle\|R_{q}(a)b_{n}\|\leq\displaystyle\frac{1}{\|(a^{2}-2Re(q_{n})a+|q_{n}|^{2})^{-1}\|}+\|a\|\left(2|Re(q_{n}-q))|\right)+||q|^{2}-|q_{n}|^{2}|$

By Lemma 3.9, the result follow.∎

######  Theorem 3.11.

Let $\mathcal{A}:\mathcal{V}\longrightarrow\mathcal{W}$ be a homomorphism of
Banach algebra and let $v_{1},v_{2}\in\mathcal{V}$. Take

$\displaystyle\mathbb{H}_{p,0}:=\left\\{q\in\mathbb{H}^{*}:\
Re(q)=0\right\\}.$

Then,

$\displaystyle\sigma_{S,\mathcal{A}}^{\Phi}(v_{1}v_{2})\backslash\mathbb{H}_{p,0}=\sigma_{S,\mathcal{A}}^{\Phi}(v_{2}v_{1})\backslash\mathbb{H}_{p,0}.$

###### Proof.

Let $a,b\in\mathcal{W}$ and
$q\not\in\sigma_{S}(ab)\backslash\mathbb{H}_{p,0}$. Then there exist
$c\in\mathcal{W}$ such that

$\displaystyle(abab+|q|^{2}1_{\mathcal{W}})c=c(abab+|q|^{2}1_{\mathcal{W}})=1_{\mathcal{W}}.$

So, we have

$\displaystyle R_{q}(ba)(bcaba-1_{\mathcal{W}})$
$\displaystyle=b(ab)^{2}caba-(ba)^{2}+|q|^{2}bcaba-|q|^{2}1_{\mathcal{W}}$
$\displaystyle=b(1_{\mathcal{W}}-|q|^{2}c)aba-(ba)^{2}+|q|^{2}bcaba-|q|^{2}1_{\mathcal{W}}$
$\displaystyle=-|q|^{2}1_{\mathcal{W}}$

and

$\displaystyle(bcaba-1_{\mathcal{W}})R_{q}(ba)$
$\displaystyle=bc(ab)^{2}aba+|q|^{2}bcaba-(ba)^{2}-|q|^{2}1_{\mathcal{A}}$
$\displaystyle=-|q|^{2}1_{\mathcal{W}}$

Consequently, $R_{q}(ba)$ is invertible in $\mathcal{W}$. Now, let
$s=\mathcal{A}(v_{1})$ and $t=\mathcal{A}(v_{2})$, then

$\displaystyle\sigma_{S,\mathcal{A}}^{\Phi}(v_{1}v_{2})\backslash\mathbb{H}_{p,0}$
$\displaystyle=\sigma_{S}^{\Phi}(\mathcal{A}(v_{1}v_{2}))\backslash\mathbb{H}_{p,0}$
$\displaystyle=\sigma_{S}^{\Phi}(st)\backslash\mathbb{H}_{p,0}$
$\displaystyle=\sigma_{S}^{\Phi}(ts)\backslash\mathbb{H}_{p,0}$
$\displaystyle=\sigma_{S,\mathcal{A}}^{\Phi}(v_{2}v_{1})\backslash\mathbb{H}_{p,0}.$

∎

##  4\. Boundary S-spectrum

Let $\mathcal{V}$ be a quaternionic two-sided Banach algebra with unit. The
boundary S-spectrum of $v\in\mathcal{V}$ is given by

$\displaystyle B_{S,\partial}(v):=\Big{\\{}q\in\mathbb{H}:\
R_{q}(v)\in\partial(\mathcal{V}\backslash\mathcal{V}^{-1})\Big{\\}}.$

######  Proposition 4.1.

Let $\mathcal{V}$ be a quaternionic two-sided Banach algebra with unit. Then

$\displaystyle\partial\sigma_{S}(v)\subseteq
B_{S,\partial}(v)\subseteq\sigma_{S}(v).$

###### Proof.

Since $\mathcal{V}^{-1}\setminus\mathcal{V}$ is closed,
$\partial(\mathcal{V}\backslash\mathcal{V}^{-1})\subseteq\mathcal{V}\backslash\mathcal{V}^{-1}$.
So, $B_{S,\partial}(v)\subseteq\sigma_{S}(v)$. To prove
$\partial\sigma_{S}(v)\subseteq B_{S,\partial}(v)$, let
$q\in\partial\sigma_{S}(v)$. Then
$R_{q}(v)\in\mathcal{V}\backslash\mathcal{V}^{-1}$. So, it suffice to prove
that $R_{q}(v)\not\in int(\mathcal{V}\backslash\mathcal{V}^{-1})$. Since
$q\not\in int(\sigma_{S}(v))$, then there exists a sequence $(q_{n})$ of
elements of $\rho_{S}(v)$ such that $\lim_{n}q_{n}=q$. This show that

$\displaystyle\lim_{n}(v^{2}-2Re(q_{n})v+|q_{n}|)=R_{q}(v)\in\overline{\mathcal{V}^{-1}}$

and therefore, $R_{q}(v)\not\in int(\mathcal{V}\backslash\mathcal{V}^{-1})$.∎

In general, $\partial\sigma_{S}(v)$ is properly contained in
$B_{S,\partial}(v)$, see Section 5.3.

######  Remark 4.2.

Let $\mathcal{V}$ be a quaternionic two-sided Banach algebra with unit.

  1. (1)

It follows, immediately, from the previous proposition that, for every
$v\in\mathcal{V}$, the set $B_{S,\partial}(v)$ is non-empty compact set.

  2. (2)

If $a\in\mathcal{V}^{-1}$, then
$B_{S,\partial}(v^{-1})=(B_{S,\partial}(v))^{-1}$. Indeed, we can write for
$q\neq 0$,

$\displaystyle
R_{q}(v^{-1})=|q|^{2}R_{\frac{1}{q}}(v)v^{-2}=|q|^{2}v^{-2}R_{\frac{1}{q}}(v)$

because $\mathcal{V}$ is two-sided Banach algebra. So, if $q\in
B_{S,\partial}(v^{-1})$, then

$\displaystyle|q|^{2}R_{\frac{1}{q}}(v)v^{-2}\in\partial(\mathcal{V}\backslash\mathcal{V}^{-1}).$

It follows from (2) that

$\displaystyle
R_{\frac{1}{q}}(v)\in\partial(\mathcal{V}\backslash\mathcal{V}^{-1}).$

  3. (3)

Going over the same techniques of the proof of [24, Theorem 2.15], we obtain:
if $\mathcal{A}:\ \mathcal{V}\longrightarrow\mathcal{W}$ is a continuous
isomorphism then
$\mathcal{A}(\partial(\mathcal{V}\setminus\mathcal{V}^{-1}))=\partial(\mathcal{W}\setminus\mathcal{W}^{-1})$.

######  Theorem 4.3.

If $\mathcal{A}:\ \mathcal{V}\longrightarrow\mathcal{W}$ is a continuous
isomorphism then

$\displaystyle
B_{S,\partial}(v)=B_{S,\partial}(\mathcal{A}(v))=\displaystyle\cup_{c\in\mathcal{A}^{-1}(0)}B_{S,\partial}(v+c)$

for all $v\in\mathcal{V}$.

###### Proof.

Let $q\in B_{S,\partial}(v)$, then
$R_{q}(v)\in\partial(\mathcal{V}\setminus\mathcal{V}^{-1})$. therefore, by
Remark 4.2 (4), we have

$\displaystyle(\mathcal{A}(v))^{2}-2Re(q)\mathcal{A}(v)+|q|^{2}\in\partial(\mathcal{W}\setminus\mathcal{W}^{-1}).$

In this way we see that $q\in B_{S,\partial}(\mathcal{A}(v))$. Now, suppose
that $q\in B_{S,\partial}(v+c)$ for some $c\in\mathcal{A}^{-1}(0)$, then

$\displaystyle
R_{q}(\mathcal{A}(v))=\mathcal{A}((a+c)^{2}-2Re(q)(a+c)+|q|^{2})\in\partial(\mathcal{W}\setminus\mathcal{W}^{-1}).$

Therefore, $q\in B_{S,\partial}(\mathcal{A}(v))$. Now, it remains to prove
that $B_{S,\partial}(\mathcal{A}(v))\subset B_{S,\partial}(v)$. Let $q\in
B_{S,\partial}(\mathcal{A}(v))$, then

$\displaystyle\mathcal{A}(R_{q}(v))=\mathcal{A}(v_{0})\mbox{ for some
}v_{0}\in\partial(\mathcal{V}\setminus\mathcal{V}^{-1}).$

The injectivity of $\mathcal{A}$ imply that

$\displaystyle
v^{2}-2Re(q)v+|q|^{2}=v_{0}\in\partial(\mathcal{V}\setminus\mathcal{V}^{-1}).$

In this way we see that $q\in B_{S,\partial}(v)$.

##  5\. Application to right quaternionic linear operators

###  5.1. Preliminary

In this subsection, we recall some definitions and we give some lemmas that we
will need in the sequel. We refer to [1, 17, 34] for more details.

######  Definition 5.1.

Let $\mathcal{V}_{\mathbb{H}}^{R}$ be a linear vector space under right
multiplication by quaternionic scalars. For
$f,g,h\in\mathcal{V}_{\mathbb{H}}^{R}$ and $q\in\mathbb{H}$, the inner product

$\displaystyle\langle.,.\rangle:\
\mathcal{V}_{\mathbb{H}}^{R}\times\mathcal{V}_{\mathbb{H}}^{R}\longrightarrow\mathbb{H}$

satisfies the following properties:

  1. (1)

$\overline{\langle f,g\rangle}=\langle g,f\rangle$.

  2. (2)

$\|f\|^{2}=\langle f,f\rangle>0$ unless $f=0$, a real norm.

  3. (3)

$\langle f,g+h\rangle=\langle f,g\rangle+\langle f,h\rangle$.

  4. (4)

$\langle f,gq\rangle=\langle f,g\rangle q$.

  5. (5)

$\langle fq,g\rangle=\overline{q}\langle f,g\rangle$.

In the sequel, we assume that $\mathcal{V}_{\mathbb{H}}^{R}$ is complete under
the norm given above and separable. In this case, $\langle.,.\rangle$ defines
a right quaternionic Hilbert space.

######  Proposition 5.2.

[17, Proposition 2.5] Let $\mathcal{F}=\\{f_{k}:\ k\in\mathbb{N}\\}$ be an
orthonormal subset of $\mathcal{V}_{\mathbb{H}}^{R}$. Then, the following
conditions are pairwise equivalent:

  1. (1)

For every $f,g\in\mathcal{V}_{\mathbb{H}}^{R}$, the series
$\sum_{k\in\mathbb{N}}\langle f,f_{k}\rangle\langle f_{k},g\rangle$ converges
absolutely and it holds:

$\displaystyle\langle f,g\rangle=\sum_{k\in\mathbb{N}}\langle
f,f_{k}\rangle\langle f_{k},g\rangle.$

  2. (2)

For every $f\in\mathcal{V}_{\mathbb{H}}^{R}$, it holds:

$\displaystyle\|f\|^{2}=\sum_{k\in\mathbb{N}}\mid\langle
f_{k},f\rangle\mid^{2}$

  3. (3)

$\mathcal{F}^{\bot}=\\{0\\}$.

  4. (4)

$\langle\mathcal{F}\rangle$ is dense in $\mathcal{V}_{\mathbb{H}}^{R}$.

######  Definition 5.3.

The set $\mathcal{F}$ as in the previous Proposition is called a Hilbert basis
of $\mathcal{V}_{\mathbb{H}}^{R}$.

######  Proposition 5.4.

[17, Propsition 2.6] Every quaternionic Hilbert space
$\mathcal{V}_{\mathbb{H}}^{R}$ has a Hilbert basis. All the Hilbert base of
$\mathcal{V}_{\mathbb{H}}^{R}$ have the same cardinality.
Furthermore, if $\mathcal{F}$ is Hilbert basis of
$\mathcal{V}_{\mathbb{H}}^{R}$, then every $f\in\mathcal{V}_{\mathbb{H}}^{R}$
can be uniquely decomposed as follows:

$\displaystyle f=\sum_{k\in\mathbb{N}}f_{k}\langle f_{k},f\rangle$

where the series $\sum_{k\in\mathbb{N}}f_{k}\langle f_{k},f\rangle$ converges
absolutely in $\mathcal{V}_{\mathbb{H}}^{R}$.

######  Definition 5.5.

Let $\mathcal{V}_{\mathbb{H}}^{R}$ and $\mathcal{W}_{\mathbb{H}}^{R}$ be two
quaternionic right Hilbert spaces. A mapping
$T:\mathcal{D}(T)\subset\mathcal{V}_{\mathbb{H}}^{R}\longrightarrow\mathcal{W}_{\mathbb{H}}^{R}$,
where $\mathcal{D}(T)$ stand for the domain of $T$, is called quaternionic
right linear if

$\displaystyle T(fq+g)=T(f)q+T(g)$

for all $f,g\in\mathcal{V}_{\mathbb{H}}^{R}$ and $q\in\mathbb{H}$.

Denote by
$\mathcal{L}(\mathcal{V}_{\mathbb{H}}^{R},\mathcal{W}_{\mathbb{H}}^{R})$ the
set of all right linear operator from $\mathcal{V}_{\mathbb{H}}^{R}$ to
$\mathcal{W}_{\mathbb{H}}^{R}$. If
$\mathcal{V}_{\mathbb{H}}^{R}=\mathcal{W}_{\mathbb{H}}^{R}$ then
$\mathcal{L}(\mathcal{V}_{\mathbb{H}}^{R},\mathcal{V}_{\mathbb{H}}^{R})$ is
replaced by $\mathcal{L}(\mathcal{V}_{\mathbb{H}}^{R})$. We call an operator
$T\in\mathcal{L}(\mathcal{V}_{\mathbb{H}}^{R},\mathcal{W}_{\mathbb{H}}^{R})$
bounded if there exist $M\in\mathbb{R}_{+}$ such that

$\displaystyle\|Tf\|\leq M\|f\|,\mbox{ for all }f\in\mathcal{D}(T).$

Let $\mathcal{B}(\mathcal{V}_{\mathbb{H}}^{R},\mathcal{W}_{\mathbb{H}}^{R})$
denote the set of all bounded right linear operators from
$\mathcal{V}_{\mathbb{H}}^{R}$ to $\mathcal{W}_{\mathbb{H}}^{R}$. If
$\mathcal{V}_{\mathbb{H}}^{R}=\mathcal{W}_{\mathbb{H}}^{R}$, we will write
$\mathcal{B}(\mathcal{V}_{\mathbb{H}}^{R})$. The identity linear operator on
$\mathcal{V}_{\mathbb{H}}^{R}$ will be denoted by
$\mathbb{I}_{\mathcal{V}_{\mathbb{H}}^{R}}$. As in the complex case if
$T\in\mathcal{L}(\mathcal{V}_{\mathbb{H}}^{R},\mathcal{W}_{\mathbb{H}}^{R})$,
we set

(8)
$\|T\|=\sup_{f\in\mathcal{D}(T)\backslash\\{0\\}}\displaystyle\frac{\|Tf\|}{\|f\|}.$

In this case, we have
$T\in\mathcal{B}(\mathcal{V}_{\mathbb{H}}^{R},\mathcal{W}_{\mathbb{H}}^{R})$
if and only if $\|T\|<\infty$. Also, we have

$\displaystyle\|T+S\|\leq\|T\|+\|S\|\mbox{ and }\|TS\|\leq\|T\|\|S\|.$

In the sequel, we investigate the $C^{*}-$algebras structure of
$\mathcal{B}(\mathcal{V}_{\mathbb{H}}^{R})$. We refer to [17] for more
details. Set $\mathcal{F}=\\{f_{k}:\ k\in\mathbb{N}\\}$ the Hilbert basis of
$\mathcal{V}_{\mathbb{H}}^{R}$. The left scalar multiplication on
$\mathcal{V}_{\mathbb{H}}^{R}$ induced by $\mathcal{F}$ is defined as the map

$\displaystyle\mathbb{H}\times\mathcal{V}_{\mathbb{H}}^{R}\longrightarrow\mathcal{V}_{\mathbb{H}}^{R}$
$\displaystyle\ (q,f)\longmapsto\
qf=\displaystyle\sum_{k\in\mathbb{N}}f_{k}q\langle f_{k},f\rangle.$

The properties of the left scalar multiplication are described in the
following proposition

######  Proposition 5.6.

[17, Proposition 3.1] Let $f,g\in\mathcal{V}_{\mathbb{H}}^{R}$ and
$p,q\in\mathbb{H}$, then

  1. (1)

$q(f+g)=qf+qg$ and $q(fp)=(qf)p$.

  2. (2)

$\|qf\|=|q|\|f\|$.

  3. (3)

$q(pf)=(qp)f$.

  4. (4)

$\langle\overline{q}f,g\rangle=\langle f,qf\rangle$.

  5. (5)

$rf=fr$, for all $r\in\mathbb{R}$.

  6. (6)

$qf_{k}=f_{k}q$, for all $k\in\mathbb{N}$.

It is easy to see that $(p+q)f=pf+qf$, for all $p,q\in\mathbb{H}$ and
$f\in\mathcal{V}_{\mathbb{H}}^{R}$. We can also endow
$\mathcal{B}(\mathcal{V}_{\mathbb{H}}^{R})$ with the two scalar multiplication

(9) $(qT)f=q(Tf)=\sum_{k\in\mathbb{N}}f_{k}q\langle f_{k},Tf\rangle\mbox{ and
}(Tq)f=T(qf),$

for all $q\in\mathbb{H}$ and $f\in\mathcal{V}_{\mathbb{H}}^{R}$.
Notation  Denote by $T^{*}$ the adjoint of
$T\in\mathcal{B}(\mathcal{V}_{\mathbb{H}}^{R})$. We recall that the concept of
adjoint quaternionic linear operator is the same as complex Hilbert space see
[17, Definition 2.12].

######  Theorem 5.7.

[17, Theorem 3.4] Let $\mathcal{V}_{\mathbb{H}}^{R}$ be a right quaternionic
Hilbert space equipped with a left scalar multiplication. Then the set
$\mathcal{B}(\mathcal{V}_{\mathbb{H}}^{R})$, equipped pointwise sum, with the
left and right scalar multiplication defined in Eq. (9), with the composition
as product, with the adjunction $T\longrightarrow T^{*}$ as $*-$involution and
with the norm defined in Eq. (8) is a quaternionic two-sided Banach
$C^{*}-$algebras with unit $\mathbb{I}_{\mathcal{V}_{\mathbb{H}}^{R}}$.

The concept of quaternionic compact operators is the same as for complex
Banach space. Let
$\mathcal{K}(\mathcal{V}_{\mathbb{H}}^{R},\mathcal{W}_{\mathbb{H}}^{R})$
denote the set of compact operators from $\mathcal{V}_{\mathbb{H}}^{R}$ to
$\mathcal{W}_{\mathbb{H}}^{R}$. This class of operators was introduced and
investigated in [25]. In particular, it is shown that
$\mathcal{K}(\mathcal{V}_{\mathbb{H}}^{R})$ is closed two-sided ideal of
$\mathcal{B}(\mathcal{V}_{\mathbb{H}}^{R})$, see [25, Theorem 7.3].

###  5.2. Fredholm and Weyl S-spectra of quaternionic operators

We turn to the question of the S-spectrum. We will apply the results described
in Section 3 to investigate the properties of the Fredholm and Weyl S-spectra
of quaternionic operators. We recall that
$\mathcal{K}(\mathcal{V}_{\mathbb{H}}^{R})$ is closed two-sided ideal of
$\mathcal{B}(\mathcal{V}_{\mathbb{H}}^{R})$. So,
$\mathcal{B}(\mathcal{V}_{\mathbb{H}}^{R})/\mathcal{K}(\mathcal{V}_{\mathbb{H}}^{R})$
is a unital Banach algebras with unit
$\mathbb{I}_{\mathcal{V}_{\mathbb{H}}^{R}}+\mathcal{K}(\mathcal{V}_{\mathbb{H}}^{R})$.
Now, we consider the natural quotion map

$\displaystyle\pi:\mathcal{B}(\mathcal{V}_{\mathbb{H}}^{R})\longrightarrow\mathcal{C}(\mathcal{V}_{\mathbb{H}}^{R}):=\mathcal{B}(\mathcal{V}_{\mathbb{H}}^{R})/\mathcal{K}(\mathcal{V}_{\mathbb{H}}^{R})$
$\displaystyle\ \ \quad\ \quad T\longmapsto\ \
[T]=T+\mathcal{K}(\mathcal{V}_{\mathbb{H}}^{R})$

Note that $\pi$ is a unital homomorphism, see [25]. The norm on
$\mathcal{C}(\mathcal{V}_{\mathbb{H}}^{R})$ is given by

$\displaystyle\|[T]\|=\inf_{K\in\mathcal{K}(\mathcal{V}_{\mathbb{H}}^{R})}\|A+K\|.$

######  Definition 5.8.

The Calkin (or Fredholm) S-spectrum of
$T\in\mathcal{B}(\mathcal{V}_{\mathbb{H}}^{R})$ is the Fredholm S-spectrum of
$T$ relative to the homomorphism $\pi$, i.e.

$\displaystyle\sigma_{S,\pi}^{\Phi}(T)=\sigma_{S}(\pi(T))$

where $\sigma_{S}(\pi(T))$ is the S-spectrum of $\pi(T)$.

Let $\sigma_{S,\pi}^{\Phi^{0}}(T)$ denote the Weyl S-spectrum of $T$ relative
to the homorphism $\pi$. This two spectrum was introduced and studied by
Muraleetharam and Thirulogasanthar in [25, 27]. In particular, we have
$A\in\Phi_{\pi}^{0}$ if and only if $A\in\Phi_{\pi}$ and $ind(A)=0$, where

$\displaystyle ind(A)=dim(A^{-1}(0))-dim((A^{*})^{-1}(0)).$

######  Theorem 5.9.

Let $A,\ B\in\mathcal{B}(\mathcal{V}_{\mathbb{H}}^{R})$.

  1. (1)

If $AB\in\mathcal{K}(\mathcal{V}_{\mathbb{H}}^{R})$ and
$BA\in\mathcal{K}(\mathcal{V}_{\mathbb{H}}^{R})$, then

(10)
$\sigma_{\mathcal{S},\pi}^{\Phi}(A+B)\backslash\\{0\\}=\left[\sigma_{\mathcal{S},\pi}^{\Phi}(A)\cup\sigma_{\mathcal{S},\pi}^{\Phi}(B)\right]\backslash\\{0\\}.$

and

(11)
$\sigma_{\mathcal{S},\pi}^{\Phi^{0}}(A+B)\backslash\\{0\\}\subset\left[\sigma_{\mathcal{S},\pi}^{\Phi^{0}}(A)\cup\sigma_{\mathcal{S},\pi}^{\Phi^{0}}(B)\right]\backslash\\{0\\}.$

If, further,
$\sigma_{\mathcal{S},\pi}^{\Phi^{0}}(A)=\sigma_{\mathcal{S},\pi}^{\Phi}(A)$,
then

(12)
$\sigma_{\mathcal{S},\pi}^{\Phi^{0}}(A+B)\backslash\\{0\\}=\left[\sigma_{\mathcal{S},\pi}^{\Phi^{0}}(A)\cup\sigma_{\mathcal{S},\mathcal{A}}^{\Phi^{0}}(B)\right]\backslash\\{0\\}.$

  2. (2)

If $0\not\in\sigma_{S}(A)\cup\sigma_{S}(B)$ and
$A^{-1}-B^{-1}\in\mathcal{K}(\mathcal{V}_{\mathbb{H}}^{R})$, then

$\displaystyle\sigma_{S,\pi}^{\Phi}(A)=\sigma_{S,\pi}^{\Phi}(B)\mbox{ and
}\sigma_{S,\pi}^{\Phi^{0}}(A)=\sigma_{S,\pi}^{\Phi^{0}}(B).$

  3. (3)

Recall that $\mathbb{H}=\\{q\in\mathbb{H}^{*}:\ Re(q)=0\\}$. Then

$\displaystyle\sigma_{S,\pi}^{\Phi}(AB)\backslash\mathbb{H}_{p,0}=\sigma_{S,\pi}^{\Phi}(BA)\backslash\mathbb{H}_{p,0}.$

###### Proof.

Note that

$\displaystyle\mathcal{K}(\mathcal{V}_{\mathbb{H}}^{R})\subset
Pr(\Phi_{\pi})\mbox{ and
}\mathcal{K}(\mathcal{V}_{\mathbb{H}}^{R})=\pi^{-1}([0]).$

Applying Theorem 3.5 and Corollary 3.7 the results follows.∎

We mention two remarks.

######  Remark 5.10.

For the complex Banach space, in [5], we have prove that if
$T_{1},T_{2}\in\mathcal{B}(X)$ such that $\sigma_{ess}(T_{1}T_{2})=\\{0\\}$
and $T_{1}T_{2}-T_{2}T_{1}\in\mathcal{K}(X)$, then

$\displaystyle\sigma_{ess}(T_{1}+T_{2})\backslash\\{0\\}=\left[\sigma_{ess}(T_{1})\cup\sigma_{ess}(T_{2})\right]\backslash\\{0\\},$

where $\sigma_{ess}(T_{i})$ is the essential spectrum of $T_{i}$. The case of
quaternionic Hilbert space is more complicated and we were not able to refine
the hypothesis in the previous theorem. We leave this question open.

######  Remark 5.11.

If $K\in\mathcal{K}(\mathcal{V}_{\mathbb{H}}^{R})$ then
$\sigma_{\mathcal{S},\pi}^{\Phi^{0}}(K)=\sigma_{\mathcal{S},\pi}^{\Phi}(K)=\\{0\\}$.

In the wat follows we show that there exist a quaternionic operator such that
its Calkin S-spectrum coincides with its Weyl S-spectrum. We first mention the
following theorem:

######  Theorem 5.12.

[25, Theorem 6.20] Let $T\in\Phi_{\pi}$. Then there exists a constant
$\varepsilon>0$ such that for every operators
$S\in\mathcal{B}(\mathcal{V}_{\mathbb{H}}^{R})$ with $\|S\|<\varepsilon,\
A+S\in\Phi_{\pi}$ and $ind(A+S)=ind(A)$.

Set:

$\displaystyle\Phi_{T,\pi}:=\Big{\\{}q\in\mathbb{H}:\
R_{q}(T)\in\Phi_{\pi}\Big{\\}}.$

######  Theorem 5.13.

Let $T$ be a bounded right operator. Then, $q\longmapsto ind(R_{q}(T))$ is
constant on any component of $\Phi_{T,\pi}$.

###### Proof.

Let $q_{1}$ and $q_{2}$ be two elements of $\Phi_{T,\pi}$ that are connected
by a smooth curve $\Gamma$. Take $q\in\Gamma$, then $R_{q}(T)\in\Phi_{\pi}$.
By Theorem 5.12, there exists a constant $\varepsilon>0$ such that for all
$S\in\mathcal{B}(\mathcal{V}_{\mathbb{H}}^{R})$ with $\|S\|<\varepsilon$,

$\displaystyle R_{q}(T)+S\in\Phi_{\pi}\mbox{ and
}ind(R_{q}(T)+S)=ind(R_{q}(T)).$

Set

$\displaystyle\mathcal{O}(A,q,\varepsilon):=\Big{\\{}q^{\prime}\in\mathbb{H}:\
2\mid
Re(q)-Re(q^{\prime})\mid\|A\|+\mid|q^{\prime}|^{2}-|q|^{2}\mid<\varepsilon.\Big{\\}}$

Let $q^{\prime}\in\mathcal{O}(A,q,\varepsilon)$. Then

$\displaystyle
R_{q}(A)+2(Re(q)-Re(q^{\prime}))A+(|q^{\prime}|^{2}-|q|^{2})\mathbb{I}_{\mathcal{V}_{\mathbb{H}}^{R}}=R_{q^{\prime}}(A)\in\Phi_{\pi}$

and

$\displaystyle
ind(R_{q}(A)+2(Re(q)-Re(q^{\prime}))A+(|q^{\prime}|^{2}-|q|^{2})\mathbb{I}_{\mathcal{V}_{\mathbb{H}}^{R}})=ind(R_{q}(A)).$

A cover of the set $\Gamma$ by open sets was thus constructed. By the Heine-
Borel theorem, there is a finite number of such sets which cover $\Gamma$.
Since the index $q\longmapsto ind(R_{q}(A))$ is constant on each of these sets
(each crosses at least on other), then $ind(R_{q_{1}}(A))=ind(R_{q_{2}}(A))$.

######  Corollary 5.14.

Let $T$ be a bounded right operator. If $\Phi_{T,\pi}$ is connected, then

$\displaystyle\sigma_{S,\pi}^{\Phi}(T)=\sigma_{S,\pi}^{\Phi^{0}}(T).$

###### Proof.

Since the inclusion
$\sigma_{S,\pi}^{\Phi}(T)\subset\sigma_{S,\pi}^{\Phi^{0}}(T)$ is know, then it
suffices to show that $\Phi_{T,\pi}\cap\sigma_{S,\pi}^{\Phi^{0}}(T)$ is empty.
Let $q\in\Phi_{T,\pi}$. Since $\rho_{S}(T)$ is not empty, then there exists
$p\in\mathbb{H}$ such that
$R_{p}(T)\in\mathcal{B}(\mathcal{V}_{\mathbb{H}}^{R})^{-1}$ and consequently
$R_{p}(T)\in\Phi_{\pi}$ and $ind(R_{p}(T))=0$. Moreover, $\Phi_{T,\pi}$ is
connected, by the previous theorem, $ind(T)$ is constant on $\Phi_{T,\pi}$.
Therefore

$\displaystyle ind(R_{q}(T))=ind(R_{q}(T))=0.$

In this way we see that $q\not\in\sigma_{S,\pi}^{\Phi^{0}}(T)$.∎

######  Corollary 5.15.

Let $T\in\mathcal{B}(\mathcal{V}_{\mathbb{H}}^{R})$ and
$S\in\mathcal{B}(\mathcal{V}_{\mathbb{H}}^{R})$. Assume that $ST$ and $TS$ are
right linear compact operators, then

$\sigma_{\mathcal{S},\pi}^{\Phi^{0}}(T+S)\backslash\\{0\\}=\left[\sigma_{\mathcal{S},\pi}^{\Phi^{0}}(T)\cup\sigma_{\mathcal{S},\mathcal{A}}^{\Phi^{0}}(S)\right]\backslash\\{0\\}.$

###### Proof.

Combine Theorem 5.13 and Corollary 5.14.∎

######  Corollary 5.16.

Let $T$ be a bounded right operator. If $\Phi_{T,\pi}$ is connected, then

$\displaystyle\sigma_{k}^{S}(T)=\emptyset$

for all $k\in\mathbb{Z}\backslash\\{0\\}$, where

$\displaystyle\sigma_{k}(T)=\\{q\in\mathbb{H}:\ R_{q}(T)\in\Phi_{\pi}\mbox{
and }ind(R_{q}(T))=k\\}.$

###### Proof.

Combine Corollary 5.14 with [27, Remark 6.7].∎

###  5.3. Right shift

We consider the right quaternionic space:

$\displaystyle\ell^{2}_{\mathbb{H}}(\mathbb{Z})=\Big{\\{}x:\mathbb{Z}\longrightarrow\mathbb{H}\mbox{
such that }\|x\|^{2}:=\sum_{i\in\mathbb{Z}}|x_{i}|^{2}<\infty\Big{\\}}.$

with the right left scalar multiplication

$\displaystyle xa=(x_{i}a)_{i\in\mathbb{Z}}$

for $x=(x_{i})_{i\in\mathbb{Z}}$ and $a\in\mathbb{H}$. The associated scalar
product is given by

$\displaystyle\langle x,y\rangle:=\langle
x,y\rangle_{\ell^{2}_{\mathbb{H}}(\mathbb{Z})}:=\sum_{i\in\mathbb{Z}}\overline{x_{i}}y_{i}.$

Take $\\{e_{n}:\ n\in\mathbb{Z}\\}$ the Hilbert basis of
$\ell^{2}_{\mathbb{H}}(\mathbb{Z})$ where, for each $n\in\mathbb{Z}$,
$e_{n}=(e_{n}^{k})_{k\in\mathbb{Z}}$, where $e_{n}^{n}=1$ and $e_{n}^{i}=0$
for all $i\neq n$. Consider the right shift

$\displaystyle R:$
$\displaystyle\ell_{\mathbb{H}}^{2}(\mathbb{Z})\longrightarrow\ell_{\mathbb{H}}^{2}(\mathbb{Z})$
$\displaystyle x\longmapsto\ y=(y_{i})_{i\in\mathbb{Z}}$

where $y_{i}=x_{i+1}$ if $i\neq-1$ and $0$ if $i=-1$. We have
$\|R(x)\|^{2}=\sum_{i\neq-1}|x_{i}|^{2}\leq\|x\|^{2}$. Also, since
$\|R(e_{1})\|=1$, we have $\|R\|=1$. We recall the following definition:

######  Definition 5.17.

[25] Let $A\in\mathcal{B}(V_{\mathbb{H}}^{R})$. The approximate $S-$spectrum
of $A$, denoted by $\sigma_{app}^{S}(A)$, is defined as:

$\displaystyle\sigma_{app}^{S}(A)=\Big{\\{}q\in\mathbb{H}:\exists(f_{n})_{n}\mbox{
with }\|f_{n}\|=1\mbox{ and }\|R_{q}(A)f_{n}\|\rightarrow 0\Big{\\}}.$

The set of right eigenvalues coincide with the point $S-$spectrum, see [17],
Proposition 4.5. We turn to the question of the spectrum of the right shift
$R$. Consider the right eigenvalue problem of $R$. Let $q\in\mathbb{H}$ and
$x=(x_{i})_{i\in\mathbb{Z}}\in\ell^{2}_{\mathbb{H}}(\mathbb{Z})$ with
$x_{-i}=0$ for all $i\in\mathbb{N}^{*}$ such that $Rx=xq$, then
$x_{i+1}=x_{i}q$ for all $i\in\mathbb{N}$. This implies that
$B_{\mathbb{H}}(0,1)\subset\sigma_{app}^{S}(R)\subset\sigma_{S}(R)$. Going
over same technique of [29, Example 6.31] we have
$\sigma_{S}(R)=\nabla_{\mathbb{H}}(0,1)$ the closed quaternionic unit ball. In
particular $0\not\in\partial\sigma_{S}(R)$: the boundary of $\sigma_{S}(A)$.
In the following, we will show that $0\in B_{S,\partial}(R)$ the boundary
S-spectrum of $R$. To begin, let’s consider the map:

$\displaystyle T:$
$\displaystyle\ell_{\mathbb{H}}^{2}(\mathbb{Z})\longrightarrow\ell_{\mathbb{H}}^{2}(\mathbb{Z})$
$\displaystyle x\longmapsto\ y=(y_{i})_{i\in\mathbb{Z}}$

where $y_{-1}=x_{0}$ and $y_{j}=0$ for all $j\neq-1$. We note that
$T(e_{0})=e_{-1}$ and $T(e_{k})=0$ for all $k\neq 0$. Therefore,

$\displaystyle(Tq)(x)$ $\displaystyle=T(\sum_{i\in\mathbb{Z}}e_{k}q\langle
e_{k},x\rangle)$ $\displaystyle=T(e_{0}q\langle e_{0},x\rangle)$
$\displaystyle=e_{-1}qx_{0}$

for all $q\in\mathbb{H}$ and
$x=(x_{i})_{i\in\mathbb{Z}}\in\ell^{2}_{\mathbb{H}}(\mathbb{Z})$.

######  Proposition 5.18.

$\sigma_{S,\pi}^{\Phi^{0}}(R)$ is properly contained in $\sigma_{S}(R)$.

###### Proof.

We have $(R+T)(x)=y=(y_{i})_{i\in\mathbb{Z}}$, where $y_{i}=x_{i+1}$ for all
$i\in\mathbb{Z}$. Therefore,

$\displaystyle R+T\in\Phi_{\pi}\mbox{ and }ind(R+T)=0.$

Since $T\in\mathcal{K}(\ell_{\mathbb{H}}^{2}(\mathbb{Z}))$, we can apply [25,
Theorem 6.16], we infer that

$\displaystyle R\in\Phi_{\pi}\mbox{ and }ind(R)=ind(R+T)=0.$

Again, using [25, Corllary 6.15], we have

$\displaystyle R^{2}\in\Phi_{\pi}\mbox{ and }ind(R^{2})=2ind(R)=0.$

Hence $0\in\sigma_{S}(R)\backslash\sigma_{S,\pi}^{\Phi^{0}}(R)$.∎

The technique of the proof of the following Proposition are inspired from [24]
in the case of $\ell_{\mathbb{C}}^{2}(\mathbb{Z})$.

######  Proposition 5.19.

$\partial\sigma_{S}(R)$ is properly contained in $B_{S,\partial}(R)$.

###### Proof.

For $q\in\mathbb{H}\backslash\\{0\\}$,
$(R+Tq)^{2}\in(\mathcal{B}(\ell_{\mathbb{H}}^{2}(\mathbb{Z})))^{-1}$ and its
inverse is given by

$\displaystyle(R+Tq)^{-2}y=x=(x_{i})_{i\in\mathbb{Z}},$

where

$\displaystyle
x_{i}=\displaystyle\left\\{\begin{array}[]{rl}\noindent\frac{q}{|q|^{2}}y_{i-2},\mbox{
if }i\in\\{0,1\\},\\\ \nobreak\leavevmode\nobreak\leavevmode\hfil\\\ y_{i-2},\
\ \ \ \ \ \mbox{ otherwise }.\end{array}\right.$

Now, let $0<|q|<1$, then $R_{n}=(R+Tq^{n})^{2}$ is invertible for all
$n\in\mathbb{N}$ and

$\displaystyle\|R_{n}-R^{2}\|=\|RTq^{n}+Tq^{n}R+(Tq^{n})^{2}\|\leq
2|q|^{n}+|q|^{2n}.$

So, $R_{n}$ converges to $R^{2}$. This imply that
$R^{2}\in\mathcal{B}(\ell_{\mathbb{H}}^{2}(\mathbb{Z}))\backslash\overline{\mathcal{B}^{-1}(\ell_{\mathbb{H}}^{2}(\mathbb{Z}))}$.

## References

  * [1] Adler SL. _Quaternionic quantum mechanics and quantum fields_. New york: The clarendon Press., oxford university press; 1995. (international series of monographs on physics; vol 88). p. xii+586.
  * [2] Alpay D, Colombo F, Sabadini I. _Slice hyperholomorphic Schur analysis._ Springer international publisching, 2016.
  * [3] Aupetit B. _A primer on spectral theory_. New York: Springer-Verlag. xii, 193 p. DM 48.00(1991).
  * [4] Ayadi H. _Spectra of Laplacians on forms an infinite graphs_. Oper. Matrices. 2017;11(2):567-886.
  * [5] Baloudi H, Jeribi A. _Left-right Fredholm and Weyl spectra of the sum of two bounded operators and applications._ Mediterr. J. Math. 2014;11(3):939-953.
  * [6] Baklouti H. _$\mathcal{I}$ -Fredholm analysis and application to operator theory_. J.Math. Anal. App. 2017;369:283-289.
  * [7] Ben Ali A, Moualla N. _Fredholm perturbation theory and some essential spectra with respect to subalgebra_. Indag. Math., New Ser. 2017;28(2):276-286.
  * [8] Colombo F, Gentili G, Sabadini I, Struppa DC. _Non commutative functional calculus: bounded operators_. Complex Anal. Oper. Theory. 2010;4:821-843.
  * [9] Colombo F, Gantner J, Kimseyn DP. _Spectral theory on the S-spectrum for quaternionic operators._ Springer Nature, Switzerland AG, (2018).
  * [10] Colombo F, Sabaini I, Sommen F. _The Fueter mapping theorem in integral form and the $\mathcal{F}$-functional calculus._ Math. Methods Appl. Sci. 2010;33(17):2050-2066.
  * [11] Colombo F, Sabadini I, Struppa DC. _A new functional calculus for noncommuting operators._ J. Funct. Anal. 2008;254(8):2255-2274.
  * [12] Colombo F, Sabadini I, Struppa DC. _Noncommutative Functional Calculus-Theory and Applications of Slice Hyperholomorphic Functions._ Vol. 289. Progress in mathematics. Basel: Birkäuser, (2011).
  * [13] Djordjevic SV, Kantun-Montiel G. _On generalised T-Fredholm elements in Banach algebra_. Math. Proc. R. Ir. Acad. Sect. A. 2009;109(1):61-66.
  * [14] Gantner J. _A direct approach to the S-functional calculus for closed operators._ J. Oper. Theory 2017;77(2):287-331.
  * [15] Gentili G, Struppa DC. _A new theory of regular functions of a quaternionic variable._ Adv. Math. 2007;216:279-301.
  * [16] Ghilani R, Recupero V. _Slice regular semigroup_ , Trans. Am. Math. Soc. 2018;370(7):4993-5032.
  * [17] Ghiloni R, Moretti V, Perotti A. _Continuous slice functional calculus in quaternionic Hilbert spaces_. Rev. Math. Phys. 2013;25(4):1350006, 83 pp.
  * [18] Gilles C. _A relative index theorem_. Pac. J. Math. 2001;198(1):81-107.
  * [19] Harte R. _Fredholm element relative to a Banach homomorphism_. Math. Z. 1989;179:431-436.
  * [20] Jeribi A. _Spectral theory and applications of linear operators and block operator matrices_. New York: Springer-Verlag; 2015.
  * [21] Jeribi A. _Linear operators and their essential pseudospectra_. Boca Raton: CRC Press; 2018
  * [22] Kato T, _Perturbation theory for nullity, deficiency and other quantities of linear operators_. J. Anal. Math. 1958;6:261-322.
  * [23] Mouton T, Rauberheimer H. _Fredholm theory relative to two Banach algebra Homomorphism_. Quaestiones Mathematicae. 1991;14:371-382.
  * [24] Mouton S. _On the boundary spectrum in Banach algebras_. Bull. Aust. Math. Soc. 2006;74(2):239-246.
  * [25] Muraleetharam B, Thirulogasanthar K. _Fredholm operators and essential S-spectrum in the quaternionic setting_. J. Math. Phys. 2018;59(10):103506, 27p.
  * [26] Muraleetharam B, Thirulogasanthar K. _Berberian extension and its S-spectra in quaternionic Hilbert space_. Adv. Appl. Chifford Algebr. 2020;30(2):1-18.
  * [27] Muraleetharam B, Thirulogasanthar K. _Weyl and Browder S-spectra in a right quaternionic Hilbert space_. J. Geom. Phys. 2019;135:7-20.
  * [28] Muraleetharam B, Thirulogasanthar K. _Deficiency indices of some classes of unbounded $\mathbb{H}$-operators_. Complex Anal. Oper. Theory. 2019;13(1):85-113.
  * [29] Muraleetharam B, Thirulogasanthar K. _Decomposable operators, local S-spectrum and S-spectrum in the quaternionic setting_. arXiv preprint arXiv: 190505936 (2019).
  * [30] Muraleetharam B, Thirulogasanthar K. _Kato S-spectrum in the quaternionic setting_. Adv. Appl. Clifford Algebr. 2020;30(40):33p.
  * [31] Schechter M. _Principle of Functional Analysis_ , Academic Press. New York 1971.
  * [32] Stoppato C. A new series expansion for slice regular functions. Adv. Math. 2002;231:1401-1416.
  * [33] Tagasev A. _On the symmetric quaternionic Banach algebras I: Geljfand theory_. Publ. Inst. Math. Nouv. Sér. 1978;241(38):173-188.
  * [34] Viswanath K. _Normal operators on quaternionic Hilbert space_. Trans. Amer. Math. Soc. 1971;162:337-350.