#  Riesz projection and essential $S$-spectrum in quaternionic setting

Hatem Baloudi  Hatem Baloudi, Department of Mathematics, Faculty of Sciences
of Gafsa, University of Gafsa, 2112 Zarroug, Tunisia.
E-mail adress<EMAIL_ADDRESS><EMAIL_ADDRESS>,  Sayda
Belgacem  Sayda Belgacem, Departement of Mathematics Faculty of Sciences of
Sfax, University of Sfax, Route de Soukra km 3.5, B.P. 1171, 3000 Sfax,
Tunisia.
E-mail adress<EMAIL_ADDRESS>and  Aref Jeribi  Aref Jeribi,
Departement of Mathematics Faculty of Sciences of Sfax, University of Sfax,
Route de Soukra km 3.5, B.P. 1171, 3000 Sfax, Tunisia.
<EMAIL_ADDRESS>
###### Abstract.

This paper is devoted to the investigation of the Weyl and the essential
$S-$spectra of a bounded right quaternionic linear operator in a right
quaternionic Hilbert space. Using the quaternionic Riesz projection, the
$S-$eigenvalue of finite type is both introduced and studied. In particular,
we have shown that the Weyl and the essential $S-$spectra do not contain
eigenvalues of finite type. We have also described the boundary of the Weyl
$S-$spectrum and the particular case of the spectral theorem of the essential
$S-$spectrum.

###### Key words and phrases:

Quaternions, Quaternionic Riesz projection, Essential $S-$spectrum, Weyl
$S-$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 Right quaternionic Hilbert space and operator
    3. 2.3 The quaternionic functional calculus
  3. 3 Riesz projection and essential spectrum
  4. 4 Some results on the Weyl $S$-spectrum

##  1\. Introduction

Over the recent years, the spectral theory for quaternionic operators has
piqued the interest and attracted the attention of multiple researchers, see
for instance [1, 4, 12, 13, 14, 15, 16, 17, 29] and references therein.
Research in this topic is motivated by application in various fields,
including quantum mechanics, fractional evolution problems [14], and
quaternionic Schur analysis [3]. The concept of spectrum is one of the main
objectives in the theory of quaternionic operators acting on quaternionic
Hilbert spaces. The obscurity that seemed to surround the precise definition
of the quaternionic spectrum of a linear operator was deemed a stumbling
block. However, in 2006, F. Colombo and I. Sabadini succeeded in devising a
new notion conducive to the complete development of the quaternionic operator
theory, namely the $S$-spectrum. We refer to [13, Subsection 1.2.1] for a
precise history. After several more years of formulating the spectral theorem
that stemmed from the $S$-spectrum, D. Alpay, F. Colombo, D.P. Kimsey, see
[6], supplied ample evidence in 2016 to further establish this fundamental
theorem for both bounded and unbounded operators. In the book [13], see also
[14], the authors briefly explain the concept of $S-$spectrum and give the
systematic basis of quaternionic spectral theory. We refer to the book [3] and
the references therein for the spectral theory on the $S$-spectrum for
Clifford operators and to [9] for some results on operators perturbation.

Motivated by the new concept of $S-$spectrum in the quaternionic setting,
Muraleetharan and Thirulogasanthar, in [29, 30], introduced the Weyl and
essential $S$-spectra and gave a characterization using Fredholm operators. We
refer to [7] for the study of the general framework of the Fredholm element
with respect to a quaternionic Banach algebra homomorphism. In general, the
set of all operators acting on right Banach space is not quaternionic Banach
algebra with respect to the composition operators. By [19, Theoreme 7.1 and
Theorem 7.3], if $V_{\mathbb{H}}^{R}$ is a separable quaternionic Hilbert
space, then $\mathcal{B}(V_{\mathbb{H}}^{R})$ (the set of all right bounded
operators) is a quaternionic two-sided $C^{*}-$algebra and the set of all
compact operators $\mathcal{K}(V_{\mathbb{H}}^{R})$ is a closed two-sided
ideal of $\mathcal{B}(V_{\mathbb{H}}^{R})$ which is closed under adjunction.
In this regard, in [29], the author defined the essential $S-$spectrum as the
$S-$spectrum of quotient map image of bounded right linear operator on the
Calkin algebra
$\mathcal{B}(V_{\mathbb{H}}^{R})/\mathcal{K}(V_{\mathbb{H}}^{R})$.

In order to explain the objective of this work, we start by recalling a few
results concerning the discrete spectrum and the Riesz projection in the
complex setting. Let $T$ be a linear operator acting on a complex Banach space
$V_{\mathbb{C}}$. We denote the spectrum of $T$ by $\sigma(T)$. Let $\sigma$
be an isolated part of $\sigma(T)$. The Riesz projection of $T$ corresponding
to $\sigma$ is the operator

$\displaystyle P_{\sigma}=\displaystyle\frac{1}{2\pi
i}\int_{C_{\sigma}}(z-T)^{-1}dz$

where $C_{\sigma}$ is a smooth closed curves belonging to the resolvent set
$\mathbb{C}\backslash\sigma(T)$ such that $C_{\sigma}$ surrounds $\sigma$ and
separates $\sigma$ from $\sigma(T)\backslash\sigma$. The discrete spectrum of
$T$, denoted $\sigma_{d}(T)$, is the set of isolated point
$\lambda\in\mathbb{C}$ of $\sigma(T)$ such that the corresponding Riesz
projection $P_{\\{\lambda\\}}$ are finite dimensional, see [21, 28]. Note that
in general we have $\sigma_{e}(T)\subset\sigma(T)\backslash\sigma_{d}(T)$,
where $\sigma_{e}(T)$ denotes the set of essential spectrum of $T$. We refer
to [8, 25, 34] for more properties of $\sigma_{e}(T)$. Note that, if $A$ is a
self-adjoint operator on a Hilbert space, then
$\sigma_{e}(T)=\sigma(T)\backslash\sigma_{d}(T)$. In particular, the essential
spectrum is empty if and only if $\sigma(T)=\sigma_{d}(T)$. We point out that
this point, namely the absence of the essential spectrum, has been studied in
many works, e.g., [22, 27].

In the quaternionic setting, if $T\in\mathcal{B}(V_{\mathbb{H}}^{R})$ and
$q\in\sigma_{S}(T)\backslash\mathbb{R}$ (where $\sigma_{S}(T)$ denote the
$S-$spectrum of $T$), then $q$ is not an isolated point of $\sigma_{S}(T)$.
Indeed, $[q]:=\\{hqh^{-1}:\ h\in\mathbb{H}^{*}\\}\subset\sigma_{S}(T)$, see
[13]. By the compactness of $[q]$, $\\{q\\}$ is not an isolated part of
$\sigma_{S}(T)$. However, if we set

$\displaystyle\Omega:=\sigma_{S}(T)/\sim$

where $p\sim q$ if and only if $p\in[q]$, and $E_{T}$ the set of
representative, then if $[q]$ is an isolated part of $\sigma_{S}(T)$ then $q$
is an isolated point of $E_{T}$.

The first aim of this work is to study the isolated part of the $S-$ spectrum
of a bounded right quaternionic operators and its relation with the essential
$S$-spectrum. To begin with, we consider the Riesz projector associated with a
given quaternionic operator $T$ which was introduced in [11]. We refer to [2,
4, 9, 13] for more details on this concept. We treat the decomposition of the
essential $S-$spectrum of $T$ as a function of the Riesz projector, and more
generally as a function of a projector which commutes with $T$, see Theorem
3.1. The technique of the proof is inspired from [4]. We also discuss the
Riesz decomposition theorem [32, Theorem 6] in quaternionic setting. More
precisely, we prove that this decomposition is unique. Motivated by this, we
study the quaternionic version of the discrete $S-$spectrum. Following the
complex formalism given in [21, 28], we show that the essential $S-$spectrum
of a given right operator acting on right Hilbert space does not contain
discrete element of the $S-$spectrum. The second aim of this work is to give
new results concerning the Weyl and essential $S$-spectra in quaternionic
setting. First off, Theorem 4.5 gives a description of the boundary of the
Weyl $S-$spectrum. The proof is based on the study of the minimal modulus of
the right quaternionic operators. We also deal with the particular case of the
spectral theorem of essential $S-$spectra. The technique of the proof is
inspired from [11].

The article is organised as follows: In Section 2, we present general
definitions about operators theory in quaternionic setting. In Section 3, we
discuss the question of decomposition of the essential $S-$spectrum. Finally,
in Section 4, we provide new results of the Weyl $S-$spectrum.

##  2\. Mathematical preliminaries

In this section, we review some basic notions about quaternions, right
quaternionic Hilbert space, right linear operator (even unbounded and define
its $S-$spectrum), and slice functional calculus. For details, we refer to the
reader [1, 11, 13, 19, 23].

###  2.1. Quaternions

Let $\mathbb{H}$ be the Hamiltonian skew field of quaternions. This class of
numbers can be written as:

$\displaystyle q=q_{0}+q_{1}i+q_{2}j+q_{3}k,$

where $q_{l}\in\mathbb{R}$ for $l=0,1,2,3$ and $i\ ,j\ ,k$ are the three
quaternionic imaginary units satisfying

$\displaystyle i^{2}=j^{2}=k^{2}=ijk=-1.$

The real and the imaginary part of $q$ is defined as ${\rm Re}(q)=q_{0}$ and
${\rm Im}(q)=q_{1}i+q_{2}j+q_{3}k$, respectively. Then, the conjugate and the
usual norm of the quaternion q are given respectively by

$\displaystyle\overline{q}=q_{0}-q_{1}i-q_{2}j-q_{3}k\mbox{ and
}|q|=\sqrt{q\overline{q}}.$

The set of all imaginary unit quaternions in $\mathbb{H}$ is denoted by
$\mathbb{S}$ and defined as

$\displaystyle\mathbb{S}=\Big{\\{}q_{1}i+q_{2}j+q_{3}k:\ q_{1},\ q_{2},\
q_{3}\in\mathbb{R},q_{1}^{2}+q_{2}^{2}+q_{3}^{2}=1\Big{\\}}.$

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

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

For every $q\in\mathbb{H}\backslash\mathbb{R}$, we associate the unique
element

$\displaystyle I_{q}:=\frac{{\rm Im}(q)}{|{\rm Im}(q)|}\in\mathbb{S}$

such that

$\displaystyle q={\rm Re}(q)+I_{q}|{\rm Im}(q)|.$

This implies that

$\displaystyle\mathbb{H}=\bigcup_{I\in\mathbb{S}}\mathbb{C}_{I}.$

where

$\displaystyle\mathbb{C}_{I}:=\mathbb{R}+I\mathbb{R}.$

We can associate to $q\in\mathbb{H}$ the $2-$dimensional sphere

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

This sphere has center at the real point ${\rm Re}(q)$ and radius $|{\rm
Im}(q)|$.

###  2.2. Right quaternionic Hilbert space and operator

In this subsection, we recall the concept of right quaternionic Hilbert space
and right linear operator (see, [1, 12, 19]).

######  Definition 2.1.

[1] Let $V_{\mathbb{H}}^{R}$ be a right vector space. The map

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

is called an inner product if it satisfies the following properties:
$(i)$ $\langle f,gq+h\rangle=\langle f,g\rangle q+\langle f,h\rangle$, for all
$f,g,h\in V_{\mathbb{H}}^{R}$ and $q\in\mathbb{H}$.
$(ii)$ $\langle f,g\rangle=\overline{\langle g,f\rangle}$, for all $f,g\in
V_{\mathbb{H}}^{R}$.
$(iii)$ If $f\in V_{\mathbb{H}}^{R}$, then $\langle f,f\rangle\geq 0$ and
$f=0$ if $\langle f,f\rangle=0$.
The pair $(V_{\mathbb{H}}^{R},\langle.,.\rangle)$ is called a right
quaternionic pre-Hilbert space. Moreover, $V_{\mathbb{H}}^{R}$ is said to be
right quaternionic Hilbert space, if

$\displaystyle\|f\|=\sqrt{\langle f,f\rangle}$

defines a norm for which $V_{\mathbb{H}}^{R}$ is complete.

In the sequel, we assume that $V_{\mathbb{H}}^{R}$ is complete and separable.
We now recall the concept of Hilbert basis in the quaternionic case. First, we
review the following proposition, the proof of which is similar to its complex
version, see [19, 33].

######  Proposition 2.2.

Let $V_{\mathbb{H}}^{R}$ be a right quaternionic Hilbert space and let
$\mathcal{F}=\\{f_{k}:\ k\in\mathbb{N}\\}$ be an orthonormal subset of
$V_{\mathbb{H}}^{R}$. The following properties are equivalent:

  1. (1)

For every $f,g\in V_{\mathbb{H}}^{R}$, the series
$\sum_{k\in\mathbb{N}}\langle f,f_{k}\rangle\langle f_{k},g\rangle$ converges
absolutely and

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

  2. (2)

For every $f\in V_{\mathbb{H}}^{R}$, we have

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

  3. (3)

$\mathcal{F}^{\bot}:=\Big{\\{}f\in V_{\mathbb{H}}^{R}:\ \langle
f,g\rangle=0\mbox{ for all }g\in\mathcal{F}\Big{\\}}=\\{0\\}$.

  4. (4)

$\langle\mathcal{F}\rangle:=\Big{\\{}\displaystyle\sum_{l=1}^{m}f_{l}q_{l}:\
f_{l}\in\mathcal{F},\ q_{l}\in\mathbb{H},\ m\in\mathbb{N}\Big{\\}}$ is dense
in $V_{\mathbb{H}}^{R}$.

######  Definition 2.3.

Let $\mathcal{F}$ be an orthonormal subset of $V_{\mathbb{H}}^{R}$.
$\mathcal{F}$ is said to be Hilbert basis of $V_{\mathbb{H}}^{R}$ if
$\mathcal{F}$ verifies one of the equivalent conditions of Proposition 2.2.

The proof of the following proposition is the same as its complex version, see
[19, 33].

######  Proposition 2.4.

Let $V_{\mathbb{H}}^{R}$ be a right quaternionic Hilbert space. Then,

  1. (1)

$V_{\mathbb{H}}^{R}$ admits a Hilbert basis.

  2. (2)

Two Hilbert basis of $V_{\mathbb{H}}^{R}$ have the same cardinality.

  3. (3)

If $\mathcal{F}$ is a Hilbert basis of $V_{\mathbb{H}}^{R}$, then every $f\in
V_{\mathbb{H}}^{R}$ 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 $V_{\mathbb{H}}^{R}$.

The quaternionic multiplication is not commutative. Afterwards, we recall that
if $V_{\mathbb{H}}^{R}$ is a right separable quaternionic Hilbert space, we
can define the left scalar multiplication on $V_{\mathbb{H}}^{R}$ using an
arbitrary Hilbert basis on $V_{\mathbb{H}}^{R}$. We refer to [19] for an
explanation of this construction. Let $\mathcal{F}=\Big{\\{}f_{k}:\
k\in\mathbb{N}\Big{\\}}$ be a Hilbert basis of $V_{\mathbb{H}}^{R}$. The left
scalar multiplication on $V_{\mathbb{H}}^{R}$ induced by $\mathcal{F}$ is
defined as the map

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

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

######  Proposition 2.5.

[19, Proposition 3.1] Let $f,g\in V_{\mathbb{H}}^{R}$ 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)

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

  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 $f\in
V_{\mathbb{H}}^{R}$. In the sequel, we consider $V_{\mathbb{H}}^{R}$ as a
right quaternionic Hilbert space equipped with the left scalar multiplication.

######  Definition 2.6.

Let $V_{\mathbb{H}}^{R}$ be a right quaternionic Hilbert space. A mapping
$T:\mathcal{D}(T)\subset V_{\mathbb{H}}^{R}\longrightarrow
V_{\mathbb{H}}^{R}$, where $\mathcal{D}(T)$ denote the domain of $T$, is
called _quaternionic right linear_ if

$\displaystyle T(f+gq)=T(f)+T(g)q,\mbox{ for all }f,\ g\in\mathcal{D}(T)\mbox{
and }q\in\mathbb{H}.$

The operator $T$ is called closed, if the graph $\mathcal{G}(T):=\\{(f,Tf):\
f\in\mathcal{D}(T)\\}$ is a closed right linear subspace of
$V_{\mathbb{H}}^{R}\times V_{\mathbb{H}}^{R}$.

We call an quaternionic right operator $T$ bounded if

$\displaystyle\|T\|:=\sup\Big{\\{}\|Tf\|:\ f\in V_{\mathbb{H}}^{R},\
\|f\|=1\Big{\\}}<+\infty.$

The set of all bounded right operators on $V_{\mathbb{H}}^{R}$ is denoted by
$\mathcal{B}(V_{\mathbb{H}}^{R})$ and the identity operator on
$V_{\mathbb{H}}^{R}$ will be denoted by $\mathbb{I}_{V_{\mathbb{H}}^{R}}$. Let
$T\in\mathcal{B}(V_{\mathbb{H}}^{R})$, we denote the null space of $T$ by
$N(T)$ and its range space by $R(T)$. A closed subspace $M$ of
$V_{\mathbb{H}}^{R}$ is said to be $T-$invariant subspace if $T(M)\subset M$.
Note that the function $f\longmapsto Tf-fq$ is not right linear, we refer to
[12, 13] for this point of view. The fundamental suggestion of [12] is to
define the spectrum using the Cauchy kernel series. We recall these concepts
from the book [13].

######  Definition 2.7.

Let $T:\mathcal{D}(T)\subset V_{\mathbb{H}}^{R}\longrightarrow
V_{\mathbb{H}}^{R}$ be a right linear operator. We define the operator
$Q_{q}(T):\mathcal{D}(T^{2})\longrightarrow V_{\mathbb{H}}^{R}$ by

$\displaystyle Q_{q}(T):=T^{2}-2{\rm
Re}(q)T+|q|^{2}\mathbb{I}_{V_{\mathbb{H}}^{R}}.$

$1)$ The _$S-$ resolvent_ set of $T$ is defined as follows:

$\displaystyle\rho_{S}(T):=\Big{\\{}q\in\mathbb{H}:\
N(Q_{q}(T))=\\{0\\},\overline{R(T)}=\mathbb{H}\mbox{ and
}Q_{q}(T)^{-1}\in\mathcal{B}(V_{\mathbb{H}}^{R})\Big{\\}}.$

$2)$ The _$S-$ spectrum_ of $T$ is defined as:

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

$3)$ The _point $S-$spectrum_ of $T$ is given by

$\displaystyle\sigma_{pS}(T):=\Big{\\{}q\in\mathbb{H}:\
N(Q_{q}(T))\neq\\{0\\}\Big{\\}}.$

For $T\in\mathcal{B}(V_{\mathbb{H}}^{R})$, the $S-$spectrum $\sigma_{S}(T)$ is
a non-empty compact set, see [13]. We recall that if
$T\in\mathcal{B}(V_{\mathbb{H}}^{R})$ and $q\in\sigma_{S}(T)$, then all the
elements of the sphere $[q]$ belong to $\sigma_{S}(T)$, see [3, Theorem
7.2.8].

Let $v\in V_{\mathbb{H}}^{R}\backslash\\{0\\}$, then $v$ is a right eigenvalue
of $T$ if $T(v)$ is a right quaternionic multiple of $v$. That, is

$\displaystyle T(v)=vq$

where $q\in\mathbb{H}$, known as the right eigenvalue. The set of right
eigenvalues coincides with the point $S-$spectrum, see [19, Proposition 4.5].

Now, we recall the definition of essential $S-$spectrum, we refer to [29, 30]
for more details. Let $V_{\mathbb{H}}^{R}$ be a separable right quaternionic
Hilbert space equipped with a left scalar multiplication. Using [19],
$\mathcal{B}(V_{\mathbb{H}}^{R})$ is a quaternionic two-sided Banach
$C^{*}-$algebra with unity, and the set of all compact operators
$\mathcal{K}(V_{\mathbb{H}}^{R})$ is a closed two-sided ideal of
$\mathcal{B}(V_{\mathbb{H}}^{R})$. We consider the natural quotient map:

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

Note that $\pi$ is a unital homomorphism, see [29]. The norm on
$\mathcal{C}(V_{\mathbb{H}}^{R})$ is given by

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

######  Definition 2.8.

[29] The essential $S-$spectrum of $T\in\mathcal{B}(V_{\mathbb{H}}^{R})$ is
the $S-$spectrum of $\pi(A)$ in the Calkin algebra
$\mathcal{C}(V_{\mathbb{H}}^{R})$. That is,

$\displaystyle\sigma_{e}^{S}(T):=\sigma_{S}(\pi(A)).$

###  2.3. The quaternionic functional calculus

The quaternionic functional calculus is defined on the class of slice regular
function $f:U\longrightarrow\mathbb{H}$ for some set $U\subset\mathbb{H}$. We
recall this concept and refer to [9, 11, 13] and the references therein on the
matter.

######  Definition 2.9.

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 in $\mathbb{C}_{I}$, for any
$I\in\mathbb{S}$.

######  Definition 2.10.

Let $U\subset\mathbb{H}$ be an open set. A real differentiable function
$f:U\longrightarrow\mathbb{H}$ is said to be left $s-$regular $($resp. right
$s-$regular$)$ if for every $I\in\mathbb{S}$, the function $f$ satisfy

$\displaystyle\displaystyle\frac{1}{2}[\frac{\partial}{\partial
x}f(x+Iy)+I\frac{\partial}{\partial y}f(x+Iy)]=0\
\Big{(}resp.\frac{1}{2}[\frac{\partial}{\partial x}f(x+Iy)$
$\displaystyle\quad\quad\quad\quad\quad+\frac{\partial}{\partial
y}f(x+Iy)I]=0\Big{)}.$

We denote the class of left $s-$regular $($resp. right $s-$regular$)$ by
$\mathcal{R}^{L}(U)$ $($resp. $\mathcal{R}^{R}(U))$. We recall that
$\mathcal{R}^{L}(U)$ is a right $\mathbb{H}-$module and $\mathcal{R}^{R}(U)$
is a left $\mathbb{H}-$module. Let $V_{\mathbb{H}}^{R}$ be a separable right
quaternionic Hilbert space equipped with a Hilbert basic $\mathcal{N}$ and
with a left scalar multiplication. We recall that
$\mathcal{B}(V_{\mathbb{H}}^{R})$ is a two-sided ideal quaternionic Banach
algebra with respect to the left multiplication given by

$\displaystyle(q.T)f=\displaystyle\sum_{g\in\mathcal{N}}gq\langle
g,Tf\rangle\mbox{ and }(Tq)f=\displaystyle\sum_{g\in\mathcal{N}}T(g)q\langle
g,f\rangle.$

######  Definition 2.11.

Let $T\in\mathcal{B}(V_{\mathbb{H}}^{R})$ and $q\in\rho_{S}(T)$. The _left
$S-$resolvent operator_ is given by

$\displaystyle
S_{L}^{-1}(q,T):=-Q_{q}(T)^{-1}(T-\overline{q}\mathbb{I}_{V_{\mathbb{H}}^{R}}),$

and the _right $S-$resolvent operator_ is defined by

$\displaystyle
S_{R}^{-1}(q,T):=-(T-\overline{q}\mathbb{I}_{V_{\mathbb{H}}^{R}})Q_{q}(T)^{-1}.$

######  Definition 2.12.

[4, Definition 3.4] Let $T\in\mathcal{B}(V_{\mathbb{H}}^{R})$ and let
$U\subset\mathbb{H}$ be an axially symmetric $s-$domain that contains the
$S-$spectrum $\sigma_{S}(T)$ and such that $\partial(U\cap\mathbb{C}_{I})$ is
union of a finite number of continuously differentiable Jordan curves for
every $I\in\mathbb{S}$. We say that $U$ is a $T-$admissible open set.

######  Definition 2.13.

Let $T\in\mathcal{B}(V_{\mathbb{H}}^{R})$, $W\subset\mathbb{H}$ be open set. A
function $f\in\mathcal{R}^{L}(W)($resp. $\mathcal{R}^{R}(W))$is said to be
locally left regular $($resp. right regular$)$ function on $\sigma_{S}(T)$, if
there is $T-$admissible domain $U\subset\mathbb{H}$ such that
$\overline{U}\subset W$.

We denote by $\mathcal{R}^{L}_{\sigma_{S}(T)}$ $($resp.
$\mathcal{R}^{R}_{\sigma_{S}(T)})$ the set of locally left $($ resp. right$)$
regular functions on $\sigma_{S}(T)$. Now, we recall the two versions of the
quaternionic functional calculus.

######  Definition 2.14.

[11, Definition 4.10.4] Let $T\in\mathcal{B}(V_{\mathbb{H}}^{R})$ and
$U\subset\mathbb{H}$ be a $T-$admissible domain. Then,

(1)
$f(T)=\displaystyle\frac{1}{2\pi}\int_{\partial(U\cap\mathbb{C}_{I})}S_{L}^{-1}(q,T)dq_{I}f(q)\
\forall f\in\mathcal{R}^{L}_{\sigma_{S}(T)}$

and

(2)
$f(T)=\displaystyle\frac{1}{2\pi}\int_{\partial(U\cap\mathbb{C}_{I})}f(q)dq_{I}S_{R}^{-1}(q,T)\
\forall f\in\mathcal{R}^{R}_{\sigma_{S}(T)}$

where $dq_{I}=-dqI$.

The two integrals that appear in Eqs $(\ref{e:4})$ and $(\ref{e:5})$ are
independent of the choice of imaginary unit $I\in\mathbb{S}$ and
$T-$admissible domain, see [11, Theorem 4.10.3].

A set $\sigma$ is called an isolated part of $\sigma_{S}(T)$ if both $\sigma$
and $\sigma_{S}(T)\setminus\sigma$ are closed subsets of $\sigma_{S}(T)$.

######  Definition 2.15.

[5, 9] Let $T\in\mathcal{B}(V_{\mathbb{H}}^{R})$. Denote by $U_{\sigma}$ an
axially symmetric $s-$domain that contains the axially symmetric isolated part
$\sigma\subset\sigma_{S}(T)$ but not any other point of $\sigma_{S}(T)$.
Suppose that the Jordan curves $\partial(U_{\sigma}\cap\mathbb{C}_{I})$ belong
to the $S-$resolvent set $\rho_{S}(T)$, for any $I\in\mathbb{S}$. We define
the quaternionic Riesz projection by

$\displaystyle
P_{\sigma}=\displaystyle\frac{1}{2\pi}\int_{\partial(U_{\sigma}\cap\mathbb{C}_{I})}S_{L}^{-1}(q,T)dq_{I}.$

######  Remark 2.16.

[5, 9] The concept of $P_{\sigma}$ can be given by using the right
$S-$resolvent operator $S_{R}^{-1}(q,T)$, that is

$\displaystyle
P_{\sigma}=\displaystyle\frac{1}{2\pi}\int_{\partial(U_{\sigma}\cap\mathbb{C}_{I})}dq_{I}S_{R}^{-1}(q,T).$

Note that $P_{\sigma}$ is a projection that commutes with $T$, see [5, Theorem
2.8].

In the sequel, we assume that $V_{\mathbb{H}}^{R}$ is a separable right
quaternionic Hilbert space with infinite dimensional.

##  3\. Riesz projection and essential spectrum

We recall that in [29, 30], the study of the essential $S-$spectrum is
established using the theory of Fredholm operators. The aim of this section is
to show that the essential $S-$spectrum does not contain discrete element of
the $S-$spectrum. In this regard, let $V_{\mathbb{H}}^{R}$ be a separable
right quaternionic Hilbert space. We note that if
$K\in\mathcal{K}(V_{\mathbb{H}}^{R})$, then
$\sigma_{e}^{S}(A+K)=\sigma_{e}^{S}(A)$ for all
$A\in\mathcal{B}(V_{\mathbb{H}}^{R})$. We start by showing that in general the
$S$-spectrum does not satisfy this property. We define
$V_{\mathbb{H}}^{{}^{\prime}}=\mathcal{B}(V_{\mathbb{H}}^{R},\mathbb{H})$ and
call $V_{\mathbb{H}}^{{}^{\prime}}$ the right dual space of
$V_{\mathbb{H}}^{R}$.

######  Theorem 3.1.

Let $T\in\mathcal{B}(V_{\mathbb{H}}^{R})$. Then,
$\sigma_{S}(T+A)\subset\sigma_{S}(A)$ for all
$A\in\mathcal{B}(V_{\mathbb{H}}^{R})$ if and only if $T\equiv 0$.

###### Proof.

Assume that $T$ is non-zero operator and $\sigma_{S}(T+A)\subset\sigma_{S}(A)$
for all $A\in\mathcal{B}(V_{\mathbb{H}}^{R})$. Let $0_{V_{\mathbb{H}}^{R}}\neq
x\in V_{\mathbb{H}}^{R}$ such that

$\displaystyle Tx=y\neq 0_{V_{\mathbb{H}}^{R}}.$

First step: There exists $f\in V_{\mathbb{H}}^{{}^{\prime}}$ such that

$\displaystyle f(x)=1\mbox{ and }f(y)\neq 0.$

Indeed, if $x=yq$ for some $q\in\mathbb{H}$, the result follows from Hahn-
Banach theorem. Now assume that $x$ and $y$ are linearly independent. Then,
there exists a right basis $Z$ of $V_{\mathbb{H}}^{R}$ such that
$\\{x,y\\}\subset Z$. We consider the following map

$\displaystyle f:u=\displaystyle\sum_{z\in Z}zq_{z}\longmapsto q_{x}+q_{y}.$

It is clear that $f$ is right linear and $f(x)=1$ and $f(y)\neq 0$.

Second step: Take

$\displaystyle A:=(x-y)\otimes f$

the rank one operator on $V_{\mathbb{H}}^{R}$ given by

$\displaystyle u\longmapsto(x-y)f(u).$

We have:

$\displaystyle(T+A)^{2}x-2(T+A)x$ $\displaystyle=(T+A)x-2x=-x.$

So, $1\in\sigma_{S}(A+T)$. Since $\dim V_{\mathbb{H}}^{R}>1$, then
$0\in\sigma_{S}(A)$. In the sequel, we assume that $x\neq y$. Let
$q\in\sigma_{S}(A)$. By [3, Lemma 4.2.3], we have

$\displaystyle(f(x-y))^{2}-2{\rm Re}(q)f(x-y)+|q|^{2}=0$

if and only if

$\displaystyle q\in\Big{\\{}hf(x-y)h^{-1}:\
h\in\mathbb{H}\backslash\\{0\\}\Big{\\}}.$

This implies that $1\not\in\sigma_{S}(A)$. Indeed, if $1$ is an $S-$eigenvalue
of $A$, there exists $h\in\mathbb{H}^{*}$ such that

$\displaystyle hf(x-y)h^{-1}=1$

and so $f(y)=0$, contradiction.∎

######  Definition 3.2.

A bounded operator $S\in\mathcal{B}(V_{\mathbb{H}}^{R})$ is called a _quasi-
inverse_ of the operator $T\in\mathcal{B}(V_{\mathbb{H}}^{R})$ if there exists
$K_{1},K_{2}\in\mathcal{K}(V_{\mathbb{H}}^{R})$ such that

$\displaystyle ST=\mathbb{I}_{V_{\mathbb{H}}^{R}}-K_{1}\mbox{ and
}TS=\mathbb{I}_{V_{\mathbb{H}}^{R}}-K_{2}.$

######  Lemma 3.3.

Let $T\in\mathcal{B}(V_{\mathbb{H}}^{R})$,
$q\in\mathbb{H}\backslash\sigma_{e}^{S}(T)$. Let $R_{q}(T)$ be a quasi-inverse
of $Q_{q}(T)$ and $A$ be an operator that commute with $T$. Then, there exist
$K\in\mathcal{K}(V_{\mathbb{H}}^{R})$ such that

$\displaystyle AR_{q}(T)=R_{q}(T)A+K.$

###### Proof.

Since $AT=TA$, then $Q_{q}(T)A=AQ_{q}(T)$. Let $K_{1},\
K_{2}\in\mathcal{K}(V_{\mathbb{H}}^{R})$ such that

$\displaystyle R_{q}(T)Q_{q}(T)=\mathbb{I}_{V_{\mathbb{H}}^{R}}-K_{1}\mbox{
and }Q_{q}(T)R_{q}(T)=\mathbb{I}_{V_{\mathbb{H}}^{R}}-K_{2}.$

Then,

$\displaystyle AR_{q}(T)=R_{q}(T)A+K$

where $K=K_{1}AR_{q}(T)-R_{q}(T)AK_{2}\in\mathcal{K}(V_{\mathbb{H}}^{R})$.∎

######  Theorem 3.4.

Let $V_{\mathbb{H}}^{R}$ be a quaternionic Hilbert space and
$T\in\mathcal{B}(V_{\mathbb{H}}^{R})$. Let $P_{1}$ be a projector in
$\mathcal{B}(V_{\mathbb{H}}^{R})$ commuting with $T$ and let
$P_{2}=\mathbb{I}_{V_{\mathbb{H}}^{R}}-P_{1}$. Take $T_{j}:=TP_{j}=P_{j}T,\
j=1,2$. Then,

$\displaystyle\sigma_{e}^{S}(T)=\sigma_{e}^{S}(T_{1}|_{R(P_{1})})\cup\sigma_{e}^{S}(T_{2}|_{R(P_{2})}),$

where $T_{i}|_{M}$ denotes the restriction of $T_{i}$ to $M$.

###### Proof.

Let $q\not\in\sigma_{e}^{S}(T)$. Then, there exist
$A_{q}(T)\in\mathcal{B}(V_{\mathbb{H}}^{R})$ and
$K_{1},K_{2}\in\mathcal{K}(V_{\mathbb{H}}^{R})$ such that

$\displaystyle A_{q}(T)Q_{q}(T)=\mathbb{I}_{V_{\mathbb{H}}^{R}}-K_{1}\mbox{
and }Q_{q}(T)A_{q}(T)=\mathbb{I}_{V_{\mathbb{H}}^{R}}-K_{2}.$

Using Lemma 3.3, we infer that there exists
$K_{3}\in\mathcal{K}(V_{\mathbb{H}}^{R})$ such that

$\displaystyle A_{q}(T)P_{1}=P_{1}A_{q}(T)+K_{3}\mbox{ and
}A_{q}(T)P_{2}=P_{2}A_{q}(T)-K_{3}.$

Therefore,

(3) $A_{q}(T)=P_{1}A_{q}(T)P_{1}+P_{2}A_{q}(T)P_{2}+K_{4},$

where
$K_{4}=(\mathbb{I}_{V_{\mathbb{H}}^{R}}-2P_{1})K_{3}\in\mathcal{K}(V_{\mathbb{H}}^{R})$.
Now, we consider the following identity

(4) $Q_{q}(T)=(T_{1}^{2}-2{\rm Re}(q)T_{1}+|q|^{2}P_{1})+(T_{2}^{2}-2{\rm
Re}(q)T_{2}+|q|^{2}P_{2}).$

We multiply the identity $(\ref{e:2})$ by $A_{q}(T)$ on the left and on the
right, we obtain

$\displaystyle\mathbb{I}_{V_{\mathbb{H}}^{R}}-K_{1}$
$\displaystyle=(P_{1}A_{q}(T)P_{1}+P_{2}A_{q}(T)P_{2}+K_{4})((T_{1}^{2}-2{\rm
Re}(q)T_{1}+|q|^{2}P_{1}))$
$\displaystyle+(P_{1}A_{q}(T)P_{1}+P_{2}A_{q}(T)P_{2}+K_{4})(T_{2}^{2}-2{\rm
Re}(q)T_{2}+|q|^{2}P_{2})]$

and

$\displaystyle\mathbb{I}_{V_{\mathbb{H}}^{R}}-K_{2}$
$\displaystyle=((T_{1}^{2}-2{\rm
Re}(q)T_{1}+|q|^{2}P_{1}))(P_{1}A_{q}(T)P_{1}+P_{2}A_{q}(T)P_{2}+K_{4})$
$\displaystyle+((T_{2}^{2}-2{\rm
Re}(q)T_{2}+|q|^{2}P_{2})(P_{1}A_{q}(T)P_{1}+P_{2}A_{q}(T)P_{2}+K_{4}).$

This leads us to conclude that

$\displaystyle\mathbb{I}_{V_{\mathbb{H}}^{R}}-K_{1}$
$\displaystyle=P_{1}A_{q}(T)P_{1}(T_{1}^{2}-2{\rm Re}(q)T_{1}+|q|^{2}P_{1})$
$\displaystyle+P_{2}A_{q}(T)P_{2}(T_{2}^{2}-2{\rm
Re}(q)T_{2}+|q|^{2}P_{2})+K_{5}$

and

$\displaystyle\mathbb{I}_{V_{\mathbb{H}}^{R}}-K_{2}$
$\displaystyle=(T_{1}^{2}-2{\rm Re}(q)T_{1}+|q|^{2}P_{1})P_{1}A_{q}(T)P_{1}$
$\displaystyle+(T_{2}^{2}-2{\rm
Re}(q)T_{2}+|q|^{2}P_{2})P_{2}A_{q}(T)P_{2}+K_{6},$

where

$\displaystyle K_{5}=K_{4}Q_{q}(T)\in\mathcal{K}(V_{\mathbb{H}}^{R})\mbox{ and
}K_{6}=Q_{q}(T)K_{4}\in\mathcal{K}(V_{\mathbb{H}}^{R}).$

Take

$\displaystyle A_{q,j}(T)=P_{j}A_{q}(T)P_{j},\mbox{ for }j=1,2.$

Then,

$\displaystyle
P_{1}(\mathbb{I}_{V_{\mathbb{H}}^{R}}-K_{1}-K_{5})P_{1}=A_{q,1}(T)(T_{1}^{2}-2{\rm
Re}(q)T_{1}+|q|^{2}P_{1})$

and

$\displaystyle
P_{1}(\mathbb{I}_{V_{\mathbb{H}}^{R}}-K_{2}-K_{6})P_{1}=(T_{1}^{2}-2{\rm
Re}(q)T_{1}+|q|^{2}P_{1})A_{q,1}(T).$

As a consequence, $A_{q,1}(T)|_{R(P_{1})}$ is a quasi-inverse of
$(T_{1}^{2}-2{\rm Re}(q)T_{1}+|q|^{2}P_{1})|_{R(P_{1})}$. Similarly, we have
$A_{q,2}(T)|_{N(P_{1})}$ is a quasi-inverse $(T_{2}^{2}-2{\rm
Re}(q)T_{2}+|q|^{2}P_{2})|_{N(P_{1})}$. Therefore, we conclude that

$\displaystyle
q\in\mathbb{H}\backslash(\sigma^{S}_{e}(T_{1}\mid_{R(P_{1})})\cup\sigma^{S}_{e}(T_{2}\mid_{R(P_{2})})).$

Conversely, let
$q\not\in\sigma_{e}^{S}(T_{1}|_{R(P_{1})})\cup\sigma_{e}^{S}(T_{2}|_{R(P_{2})})$.
Then, there exists $A_{q,i}(T)\in\mathcal{B}(R(P_{i}))$ and
$K_{i,j}\in\mathcal{K}(R(P_{i}))$, for $i,j=1,2$, such that

$\displaystyle
A_{q,i}Q_{q}(T_{i}\mid_{R(P_{i})})=\mathbb{I}_{R(P_{i})}-K_{1,i}\mbox{ and
}Q_{q}(T_{i})A_{q,i}(T)=\mathbb{I}_{R(P_{i})}-K_{2,i}.$

Let us define the operator

$\displaystyle B_{q}(T):=P_{1}A_{q,1}(T)P_{1}+P_{2}A_{q,2}(T)P_{2}.$

We get:

$\displaystyle B_{q}(T)Q_{q}(T)$
$\displaystyle=P_{1}A_{q,1}(T)Q_{q}(T)P_{1}+P_{2}A_{q,2}(T)Q_{q}(T)P_{2}$
$\displaystyle=P_{1}(\mathbb{I}_{R(P_{1})}-K_{1,1})P_{1}+P_{2}(\mathbb{I}_{R(P_{2})}-K_{1,2})P_{2}$
$\displaystyle=P_{1}+P_{2}-P_{1}K_{1,1}P_{1}-P_{2}K_{1,2}P_{2}$
$\displaystyle=\mathbb{I}_{V_{\mathbb{H}}^{R}}-K_{6}$

where
$K_{6}=P_{1}K_{1,1}P_{1}+P_{2}K_{1,2}P_{2}\in\mathcal{K}(V_{\mathbb{H}}^{R})$.
By a similar argument, we obtain

$\displaystyle Q_{q}(T)B_{q}(T)=\mathbb{I}_{V_{\mathbb{H}}^{R}}-K_{7},$

where
$K_{7}=P_{1}K_{2,1}P_{1}+P_{2}K_{2,2}P_{2}\in\mathcal{K}(V_{\mathbb{H}}^{R})$.
Therefore, $B_{q}(T)$ is a quasi-inverse of $Q_{q}(T)$. We deduce that
$q\not\in\sigma_{e}^{S}(T)$.∎

The result of [32, Theorem 6] can be reformulate as follows:

######  Theorem 3.5.

Let $T\in\mathcal{B}(V_{\mathbb{H}}^{R})$ and let $\sigma$ be an isolated part
of $\sigma_{S}(T)$. Put $V_{1,\mathbb{H}}^{R}=R(P_{\sigma})$ and
$V_{2,\mathbb{H}}^{R}=N(P_{\sigma})$. Then,
$V_{\mathbb{H}}^{R}=V_{1,\mathbb{H}}^{R}\oplus V_{2,\mathbb{H}}^{R}$, the
spaces $V_{1,\mathbb{H}}^{R}$ and $V_{2,\mathbb{H}}^{R}$ are $T-$invariant
subspaces and

(5) $\sigma_{S}(T|_{V_{1;\mathbb{H}}^{R}})=\sigma\mbox{ and
}\sigma_{S}(T|_{V_{2,\mathbb{H}}^{R}})=\sigma_{S}(T)\backslash\sigma.$

In the next result, we discuss the uniqueness of the decomposition
$(\ref{e:3})$.

######  Theorem 3.6.

Let $T\in\mathcal{B}(V_{\mathbb{H}}^{R})$ and $P$ be a projection in
$\mathcal{B}(V_{\mathbb{H}}^{R})$ such that $TP=PT$. Set

$\displaystyle T_{1}:=TP|_{R(P)}\mbox{ and
}T_{2}:=T(\mathbb{I}_{V_{\mathbb{H}}^{R}}-P)|_{N(P)}.$

If ${\rm dist}(\sigma_{S}(T_{1}),\sigma_{S}(T_{2}))>0$, then

$\displaystyle R(P)=R(P_{\sigma_{S}(T_{1})})\mbox{ and
}N(P)=N(P_{\sigma_{S}(T_{1})})$

###### Proof.

First and foremost, we have by [4, Theorem 4.4]

$\displaystyle\sigma_{S}(T)=\sigma_{S}(T_{1})\cup\sigma_{S}(T_{2})$

Take $\sigma_{1}=\sigma_{S}(T_{1})$ and $\sigma_{2}=\sigma_{S}(T_{2})$ and
assume that ${\rm dist}(\sigma_{1},\sigma_{2})>0$. According to the proof of
[32, Theorem 6], there exist a pair of a non-empty disjoint axially symmetric
domains $(U_{\sigma_{1}},U_{\sigma_{2}})$ such that

$\displaystyle\sigma_{j}\subset U_{\sigma_{j}},\
\overline{U}_{\sigma_{1}}\cap\overline{U}_{\sigma_{2}}=\emptyset$

and the boundary $\partial(U_{\sigma_{j}}\cap\mathbb{C}_{I})$ is the union of
finite number of continuously differentiable Jordan curves for $j=1,2$ and for
all $I\in\mathbb{S}$. In this way, we see that $U_{\sigma_{j}}$ is
$T_{j}$-admissible open set for $j=1,2$. So by the quaternionic functional
calculus, we deduce that

$\displaystyle\displaystyle\frac{1}{2\pi}\int_{\partial(U_{\sigma_{j}}\cap\mathbb{C}_{I})}S_{L}^{-1}(s,T_{j})ds_{I}=\mathbb{I}_{V_{j,\mathbb{H}}^{R}}\mbox{
for }j=1,2$

and

$\displaystyle\displaystyle\frac{1}{2\pi}\int_{\partial(U_{\sigma_{i}}\cap\mathbb{C}_{I})}S_{L}^{-1}(s,T_{j})ds_{I}=0\mbox{
for }i\neq j$

where

$\displaystyle V_{1,\mathbb{H}}^{R}:=R(P)\mbox{ and
}V_{2,\mathbb{H}}^{R}:=N(P).$

Define the operators:

$\displaystyle R_{1}=\left[\begin{array}[]{ccc}\mathbb{I}_{R(P)}&0\\\ 0&0\\\
\end{array}\right]\mbox{ and }R_{2}=\left[\begin{array}[]{ccc}0&0\\\
0&\mathbb{I}_{N(P)}\\\ \end{array}\right].$

We have

$\displaystyle P_{\sigma_{1}}$
$\displaystyle=\displaystyle\frac{1}{2\pi}\int_{\partial(U_{\sigma_{1}}\cap\mathbb{C}_{I})}S_{L}^{-1}(s,T)ds_{I}$
$\displaystyle=\displaystyle\frac{1}{2\pi}\int_{\partial(U_{\sigma_{1}}\cap\mathbb{C}_{I})}ds_{I}S_{R}^{-1}(s,T)$
$\displaystyle=\displaystyle\frac{1}{2\pi}\int_{\partial(U_{\sigma_{1}}\cap\mathbb{C}_{I})}ds_{I}S_{R}^{-1}(s,T)P+\displaystyle\frac{1}{2\pi}\int_{\partial(U_{\sigma_{1}}\cap\mathbb{C}_{I})}ds_{I}S_{R}^{-1}(s,T)(\mathbb{I}_{V_{\mathbb{H}}^{R}}-P)$
$\displaystyle=\displaystyle\frac{1}{2\pi}\int_{\partial(U_{\sigma_{1}}\cap\mathbb{C}_{I})}ds_{I}S_{R}^{-1}(s,T_{1})P+\displaystyle\frac{1}{2\pi}\int_{\partial(U_{\sigma_{1}}\cap\mathbb{C}_{I})}ds_{I}S_{R}^{-1}(s,T_{2})(\mathbb{I}_{V_{\mathbb{H}}^{R}}-P)$
$\displaystyle=\displaystyle\frac{1}{2\pi}\int_{\partial(U_{\sigma_{1}}\cap\mathbb{C}_{I})}ds_{I}S_{R}^{-1}(s,T_{1})P$
$\displaystyle=P_{\sigma_{1}}P$ $\displaystyle=(I-P_{\sigma_{2}})P$
$\displaystyle=R_{1}.$

We conclude that

$\displaystyle R(P)=R(P_{\sigma_{1}}).$

By the similar arguments, we achieve that

$\displaystyle P_{\sigma_{2}}=R_{2}$

and so, $N(P)=R(P_{\sigma_{2}})=N(P_{\sigma_{1}})$.∎

We now analyze the Riesz projection associated with the isolated spheres. To
start off, we give the following definition:

######  Definition 3.7.

$T\in\mathcal{B}(V_{\mathbb{H}}^{R})$. A point $q\in\sigma_{S}(T)$ is called
an eigenvalue of finite type if $V_{\mathbb{H}}^{R}$ is a direct sum of
$T-$invariant subspaces $V_{1,\mathbb{H}}^{R}$ and $V_{2,\mathbb{H}}^{R}$ such
that

$(H1)$ $\dim(V_{1,\mathbb{H}}^{R})<\infty$,

$(H2)$
$\sigma_{S}(T|_{V_{1,\mathbb{H}}^{R}})\cap\sigma_{S}(T|_{V_{2,\mathbb{H}}^{R}})=\emptyset$,

$(H3)$ $\sigma_{S}(T|_{V_{1,\mathbb{H}}^{R}})=[q]$.

######  Remark 3.8.

  1. (1)

In complex spectral theory, in a complex Hilbert space $V_{\mathbb{C}}$, for a
continuous linear operator, $T$, the condition $(H3)$ is replaced by
$\sigma(T|_{V_{1,\mathbb{C}}})=\\{q\\}$ $($where $V_{\mathbb{C}}$ is a direct
sum of $T-$invariant subspaces $V_{1,\mathbb{C}}$ and $V_{2,\mathbb{C}})$, see
[21]. In the quaternionic setting, we must take the whole $2-$sphere $[q]$
because if $q\in\sigma_{S}(T|_{V_{1,\mathbb{H}}^{R}})$, then
$[q]\subset\sigma_{S}(T|_{V_{1,\mathbb{H}}^{R}})$.

  2. (2)

Let $T\in\mathcal{B}(V_{\mathbb{H}}^{R})$. If
$q\in\sigma_{S}(T)\backslash\mathbb{R}$, then $q$ is not an isolated point of
$\sigma_{S}(T)$. Take

$\displaystyle\Omega:=\sigma_{S}(T)/\sim$

where $p\sim q$ if and only if $p\in[q]$. Let $E_{T}$ denote the set of
representatives of $\Omega$.

  3. (3)

By [24, Proposition 4.44 and Theorem 4.47], If
$\dim(V_{\mathbb{H}}^{R})<\infty$ , then the $S-$spectrum of $T$ consists of
right eigenvalues only and $\\#E_{T}<\infty$. In particular, if $T$ satisfied
the assumptions $(H1)$ and $(H3)$, then $q\in\sigma_{pS}(T)$.

######  Lemma 3.9.

Let $T\in\mathcal{B}(V_{\mathbb{H}}^{R})$. Let $E_{T}$ denote the set of
representatives as above. Then, $q$ is an isolated point of $E_{T}$ if and
only if $[q]$ is an isolated part of $\sigma_{S}(T)$.

###### Proof.

Assume that $[q]$ is isolated part of $\sigma_{S}(T)$ and let
$U_{q}\subset\mathbb{H}$ be an open set such that

$\displaystyle[q]=\sigma_{S}(T)\cap U_{q}.$

If there exists $q\neq p\in E_{T}\cap U_{q}$, then $p\not\in[q]$. This implies
that

$\displaystyle E_{T}\cap U_{q}=\\{q\\}.$

Conversely, if $q$ is an isolated point of $E_{T}$, then $[q]$ is an open
subset of $\sigma_{S}(T)$. Since $[q]$ is compact, then $[q]$ is isolated part
of $\sigma_{S}(T)$.∎

Let $V_{\mathbb{C}}$ be a complex Hilbert space and $T$ be a continuous linear
operator acting on $V_{\mathbb{C}}$. It follows from [21, Theorem 1.1] that if
$q\in\sigma(T)$ $($where $\sigma(T)$ denotes the complex spectrum of $T)$,
then $q$ is an eigenvalue of finite type if and only if $\dim
R(P_{\\{q\\}})<\infty$, where $P_{\\{q\\}}$ is the Riesz projection
corresponding to the isolated point $q$. In the next theorem, we show that the
same is true for the right eigenvalue of finite type in the quaternionic
setting.

######  Theorem 3.10.

Let $T\in\mathcal{B}(V_{\mathbb{H}}^{R})$ and $q\in\sigma_{pS}(T)$. Then, $q$
is a right eigenvalue of finite type if and only if $\\{q\\}$ is an isolated
part in $E_{T}$ and $\dim R(P_{[q]})<\infty$.

###### Proof.

Assume that $q$ is a right eigenvalue of finite type and consider the direct
sum

$\displaystyle V_{\mathbb{H}}^{R}=V_{1,\mathbb{H}}^{R}\oplus
V_{2,\mathbb{H}}^{R}$

with the properties $(H1)-(H3)$. By Theorem 3.6, this decomposition is unique
and so

$\displaystyle V_{1,\mathbb{H}}^{R}=R(P_{[q]}).$

The converse comes from Theorem 3.5.

######  Corollary 3.11.

Let $T\in\mathcal{B}(V_{\mathbb{H}}^{R})$. Then,

$\displaystyle\sigma_{e}^{S}(T)\subset\sigma_{S}(T)\backslash\sigma_{d}^{S}(T)$

where $\sigma_{d}^{S}(T)$ denotes the set of all right eigenvalue of finite
type.

###### Proof.

Assume that $q$ is a right eigenvalue of finite type. Then, $[q]$ is an
isolated part of $\sigma_{S}(T)$ and $\dim R(P_{[q]})<\infty$. Therefore,

$\displaystyle\dim R(TP_{[q]})<\infty\mbox{ and so
}\sigma_{e}^{S}(T|_{R(P_{[q]})})=\emptyset.$

Using Theorem 3.4, we infer that

$\displaystyle\sigma_{e}^{S}(T)=\sigma_{e}^{S}(T|_{R(\mathbb{I}_{V_{\mathbb{H}}^{R}}-P_{[q]})}).$

On the other hand,

$\displaystyle\sigma_{e}^{S}(T|_{R(\mathbb{I}_{V_{\mathbb{H}}^{R}}-P_{[q]})})\subset\sigma_{S}(T|_{R(\mathbb{I}_{V_{\mathbb{H}}^{R}}-P_{[q]})})=\sigma_{S}(T)\backslash[q].$

This leads to conclude that $q\not\in\sigma_{e}^{S}(T)$.∎

######  Example 3.12.

We consider the right quaternionic Hilbert 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 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

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

The right shift is 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 $y_{i}=x_{i+1}$ if $i\neq-1$ and $0$ if $i=-1$. We have

$\displaystyle\|T(x)\|^{2}=\sum_{i\neq-1}|x_{i}|^{2}\leq\|x\|^{2}.$

The $S-$spectrum of $T$ was studied in [7, 31]. In particular, we have

$\displaystyle\sigma_{S}(T)=\sigma_{pS}(T)=\nabla_{\mathbb{H}}(0,1),$

where $\nabla_{\mathbb{H}}(0,1)$ is the closed quaternionic unit ball. By
Theorem 3.10, none of these $S-$eigenvalues are of finite type.

We recall:

######  Lemma 3.13.

[18, Corollary 2.22] Let $V_{\mathbb{H}}^{R}$ be a quaternionic right vector
space and $E$ be a right linear independent subspace of $V_{\mathbb{H}}^{R}$.
Then, there exists a right basis $B$ of $V_{\mathbb{H}}^{R}$ such that
$E\subset B$. In particular, every quaternionic right vector space has a right
basis.

######  Proposition 3.14.

[24, Proposition 4.44] Let $V_{\mathbb{H}}^{R}$ be a quaternionic Hilbert
space and $T\in\mathcal{B}(V_{\mathbb{H}}^{R})$. If
$q_{1},....,q_{n}\in\mathbb{H}$ are right eigenvalues of $T$ such that
$[q_{i}]\neq[q_{j}]$, $\forall 1\leq i<j\leq n\ (n\geq 2)$, and
$Q_{q_{j}}(T)x_{j}=0,\ 0\neq x_{j}\in V_{\mathbb{H}}^{R},\forall 1\leq j\leq
n$, then $x_{1},x_{2},...,x_{n}$ are right-linearly independent in
$V_{\mathbb{H}}^{R}$.

We have the following lemma:

######  Lemma 3.15.

Assume that $\dim(V_{\mathbb{H}}^{R})<\infty$ and let
$T\in\mathcal{B}(V_{\mathbb{H}}^{R})$. Then, $\\#E_{T}<\infty$. In this case,
set

$\displaystyle E_{T}=\Big{\\{}q_{1},q_{2},...,q_{n}\Big{\\}},$

then $V_{\mathbb{H}}^{R}$ is a direct sum of $T-$invariant right subspaces
$V_{1,\mathbb{H}}^{R},\ V_{2,\mathbb{H}}^{R},\ ....,\ V_{n,\mathbb{H}}^{R}$.
Moreover, if $T_{i}:=T|_{V_{i,\mathbb{H}}^{R}}:\
V_{i,\mathbb{H}}^{R}\longrightarrow V_{i,\mathbb{H}}^{R}$, for $i=1,...,n$,
then

$\displaystyle\sigma_{S}(T_{i})=\Big{\\{}hq_{i}h^{-1}:\
h\in\mathbb{H}^{*}\Big{\\}}.$

###### Proof.

Combine Lemma 3.13, Proposition 3.14 and Theorem 3.10∎

Let $V_{\mathbb{H}}^{R}$ be a quaternionic right Hilbert space,
$T\in\mathcal{B}(V_{\mathbb{H}}^{R})$ and $q$ be a right eigenvalue of $T$ of
finite type. In this way, we see that

$\displaystyle V_{\mathbb{H}}^{R}=R(P_{[q]})\oplus
R(P_{\sigma_{S}(T)\backslash[q]}).$

######  Definition 3.16.

The algebraic multiplicity of the right eigenvalue $q$ is, by definition, the
dimension of the space $R(P_{[q]})$.

In the next, we write:

$\displaystyle m_{T}(q):=\dim R(P_{[q]}).$

We refer to [21, 25] for the definition in complex setting. Now, inspired by
the description of the Riesz projection in the complex setting, see [21], we
give a quaternionic version of a result describing the finite part of right
eigenvalues of finite type.

######  Theorem 3.17.

Let $V_{\mathbb{H}}^{R}$ be a quaternionic Hilbert space,
$T\in\mathcal{B}(V_{\mathbb{H}}^{R})$ and $\sigma$ be an axially symmetric
isolated part of $\sigma_{S}(T)$. Set $P_{\sigma}$ the Riesz projection
correspond to $\sigma$ and take

$\displaystyle E_{T}^{\sigma}=\sigma/\cong$

where $q\cong p$ if and only if $p\in[q]$. Then, $\dim R(P_{\sigma})<\infty$
if and only if $\\#E_{T}^{\sigma}<\infty$ and $q$ is a right eigenvalue of
finite type for all $q\in E_{T}^{\sigma}$. Besides, if so, then

$\displaystyle\dim R(P_{\sigma})=\displaystyle\sum_{q\in E_{T}^{\sigma}}\dim
R(P_{[q]}).$

###### Proof.

If $\dim R(P_{\sigma})<\infty$, then $P_{\sigma}T$ is a finite rank operator.
In this way, we see that

$\displaystyle\\#E_{T|_{R(P_{\sigma})}}^{\sigma}=n<\infty.$

By Lemma 3.15,

$\displaystyle R(P_{\sigma})=V_{1,\mathbb{H}}^{R}\oplus
V_{2,\mathbb{H}}^{R}\oplus...\oplus V_{n,\mathbb{H}}^{R}$

where $V_{j,\mathbb{H}}^{R}$ is $T-$invariant with the properties

$\displaystyle\sigma_{S}(T|_{V_{j,\mathbb{H}}^{R}})=[q_{j}],\ j=1,...,n.$

Let $i\in\\{1,...,n\\}$. Since $P_{\sigma}$ is a projection, then

$\displaystyle V_{\mathbb{H}}^{R}=N(P_{\sigma})\oplus R(P_{\sigma}).$

In this fashion, we have

$\displaystyle V_{\mathbb{H}}^{R}=V_{i,\mathbb{H}}^{R}\oplus
W_{i,\mathbb{H}}^{R}$

where

$\displaystyle W_{i,\mathbb{H}}^{R}=V_{1,\mathbb{H}}^{R}\oplus...\oplus
V_{i-1,\mathbb{H}}^{R}\oplus V_{i+1,\mathbb{H}}^{R}\oplus...\oplus
V_{n,\mathbb{H}}^{R}\oplus N(P_{\sigma}).$

As consequence, $V_{i,\mathbb{H}}^{R}$ and $W_{i,\mathbb{H}}^{R}$ are
$T-$invariant and $\sigma_{S}(T|_{V_{i,\mathbb{H}}^{R}})=[q_{i}]$. So, $q_{i}$
is a right eigenvalue of finite type.

Conversely, set $E_{T}^{\sigma}=\\{q_{1},\ q_{2}...,q_{n}\\}$, where $q_{i}$
is a right eigenvalue of $T$ of finite type for all $i\in\\{1,2,...,n\\}$.
Applying [12, Theorem 5.6], we have

$\displaystyle\mathbb{I}_{R(P_{\sigma})}=\displaystyle\sum_{i=1}^{n}P_{[q_{i}]}|_{R(P_{\sigma})}.$

Since $P_{[q_{i}]}P_{[q_{j}]}=0$ for all $i\neq j$, then

$\displaystyle R(P_{\sigma})=R(P_{[q_{1}]})\oplus
R(P_{[q_{2}]})\oplus...\oplus R(P_{[q_{n}]}).$

This implies that

$\displaystyle\dim(R(P_{\sigma}))=\displaystyle\sum_{i=1}^{n}\dim(R(P_{[q_{i}]})).$

In particular, we have $P_{\sigma}$ which is a finite rank operator.∎

######  Remark 3.18.

In the complex spectral theory, much attention has ben paid to eigenvalue of
finite type, see [10, 21, 25, 26, 28]. It is useful for the study of the
essential spectrum of certain operators-matrices. We refer to [10] for this
point on the two-groupe transport operators. More precisely, let
$V_{\mathbb{C}}$ be a complex Banach space and let $T$ be a closed operator in
$V_{\mathbb{C}}$. The Browder resolvent set of $T$ is given by

$\displaystyle\rho_{B}(T):=\rho(T)\cup\sigma_{d}(T),$

where we use the notation $\rho(.)$ for the resolvent set of $T$ and
$\sigma_{d}(.)$ the set of eigenvalues of finite type of $T$. In fact, the
usual resolvent

$\displaystyle R_{\lambda}(A):=(A-\lambda)^{-1}$

can be extended to $\rho_{B}(T)$, e.g. [28]. Motivated by this, [10] gives a
version of the Frobenius-Schur factorization using the Browder resolvent. This
makes it possible to study the essential spectrum of serval types operators-
matrices. In this paper, we have described the discrete $S-$spectrum in
quaternionic setting. In this regard, as in complex case, we can define the
spherical Browder resolvent. Although, we avoided studying it in this paper,
we will cover that in a future article.

##  4\. Some results on the Weyl $S$-spectrum

In this section, we develop a deeper understanding of the concept of the Weyl
$S$-spectrum of the bounded right linear operator. More precisely, we describe
the boundary of the $S$-spectrum. Likewise, we deal with the particular case
of the spectral theorem. To begin with, we recall:

######  Definition 4.1.

[30]Let $T\in\mathcal{B}(V_{\mathbb{H}}^{R})$. The Weyl $S-$spectrum is the
set

$\displaystyle\sigma_{W}^{S}(T)=\displaystyle\bigcap_{K\in\mathcal{K}(V_{\mathbb{H}}^{R})}\sigma_{S}(T+K).$

The study of the essential and the Weyl $S-$spectra are established using the
Fredholm theory, see [29, 30]. We refer to [7] for the investigation of the
Fredholm and Weyl elements with respect to a quaternionic Banach algebra
homomorphism.

######  Definition 4.2.

A Fredholm operator is an operator $T\in\mathcal{B}(V_{\mathbb{H}}^{R})$ such
that $N(T)$ and $V_{\mathbb{H}}^{R}/R(T)$ are finite dimensional. We will
denote by $\Phi(V_{\mathbb{H}}^{R})$ the set of all Fredholm operators.

From [29, 30], we have

$\displaystyle\Phi(V_{\mathbb{H}}^{R})=\Phi_{l}(V_{\mathbb{H}}^{R})\cap\Phi_{r}(V_{\mathbb{H}}^{R})$

where

$\displaystyle\Phi_{l}(V_{\mathbb{H}}^{R})=\Big{\\{}T\in\mathcal{B}(V_{\mathbb{H}}^{R}):\mbox{
R(T) is closed and }\dim(N(T))<\infty\Big{\\}}$

and

$\displaystyle\Phi_{r}(V_{\mathbb{H}}^{R})=\Big{\\{}T\in\mathcal{B}(V_{\mathbb{H}}^{R}):\mbox{
R(T) is closed and }\dim(N(T^{{\dagger}}))<\infty\Big{\\}}.$

Let $T\in\Phi_{l}(V_{\mathbb{H}}^{R})\cup\Phi_{r}(V_{\mathbb{H}}^{R})$. Then,
the index of $T$ is given by

$\displaystyle i(T):=\dim N(T)-\dim(V_{\mathbb{H}}^{R}/R(T)).$

######  Theorem 4.3.

[29, 30] Let $T\in\mathcal{B}(V_{\mathbb{H}}^{R})$. Then,

$\displaystyle\sigma_{e}^{S}(T)=\mathbb{H}\backslash\Phi_{T}\mbox{ and
}\sigma_{W}^{S}(T)=\mathbb{H}\backslash W_{T}$

where

$\displaystyle\Phi_{T}:=\Big{\\{}q\in\mathbb{H}:\
Q_{q}(T)\in\Phi(V_{\mathbb{H}}^{R})\Big{\\}}$

and

$\displaystyle W_{T}:=\Big{\\{}q\in\mathbb{H}:\
Q_{q}(T)\in\Phi(V_{\mathbb{H}}^{R})\mbox{ and }i(Q_{q}(T))=0\Big{\\}}.$

######  Remark 4.4.

Let $V_{\mathbb{H}}^{R}$ be a quaternionic space and
$T\in\mathcal{B}(V_{\mathbb{H}}^{R})$.

  1. (1)

Note that, in general, we have

$\displaystyle\sigma_{e}^{S}(T)\subset\sigma_{W}^{S}(T)=\sigma_{1,W}^{S}(T)\cup\sigma_{2,W}^{S}(T)\subset\sigma_{S}(T)\backslash\sigma_{d}(T).$

where

$\displaystyle\sigma_{1,W}^{S}(T):=\mathbb{H}\backslash\Big{\\{}q\in\mathbb{H}:\
Q_{q}(T)\in\Phi_{l}(V_{\mathbb{H}}^{R})\mbox{ and }i(Q_{q}(T))\leq 0\Big{\\}}$

and

$\displaystyle\sigma_{2,W}^{S}(T):=\mathbb{H}\backslash\Big{\\{}q\in\mathbb{H}:\
Q_{q}(T)\in\Phi_{r}(V_{\mathbb{H}}^{R})\mbox{ and }i(Q_{q}(T))\geq
0\Big{\\}}.$

In particular, $\sigma_{W}^{S}(T)$ does not contain eigenvalues of finite
type.

  2. (2)

In [7], one proves that $q\longmapsto i(T)$ is constant on any component of
$\Phi_{T}$. In this way, we see that if $\Phi_{T}$ is connected, then

$\displaystyle\sigma_{e}^{S}(T)=\sigma_{W}^{S}(T).$

The first result in this section is the next theorem.

######  Theorem 4.5.

Let $T\in\mathcal{B}(V_{\mathbb{H}}^{R})$. Then,

$\displaystyle\partial\sigma_{W}^{S}(T)\subset\sigma_{1,W}^{S}(T).$

In particular, if $\Phi_{T}$ is connected, then

$\displaystyle\partial\sigma_{e}^{S}(T)=\partial\sigma_{W}^{S}(T)\subset\Big{\\{}q\in\mathbb{H}:\
Q_{q}(T)\not\in\Phi_{l}(V_{\mathbb{H}}^{R})\Big{\\}}.$

To prove Theorem 4.5, we first study the concept of the minimum modulus. Let
$V_{\mathbb{H}}^{R}$ be a separable right Hilbert space and
$T\in\mathcal{B}(V_{\mathbb{H}}^{R})$. The minimum modulus of $T$ is given by

$\displaystyle\mu(T):=\displaystyle\inf_{\|x\|=1}\|Tx\|.$

To begin with, we give the following lemma.

######  Lemma 4.6.

Let $T$ and $S$ be two bounded right linear operators on a right quaternionic
Hilbert space. Then,

  1. (1)

If $\|T-S\|<\mu(T)$, then $\mu(S)>0$ and $\overline{R(S)}$ is not a proper
subset of $\overline{R(T)}$.

  2. (2)

If $\|T-S\|<\frac{\mu(T)}{2}$, then $\overline{R(S)}$ is not a proper subset
of $\overline{R(T)}$ and $\overline{R(T)}$ is not a proper subset of
$\overline{R(S)}$.

###### Proof.

The proof is the same as for the complex Banach space, see Lemma 2.3 and lemma
2.4 in [20] for a complex proof.∎

For $T\in\mathcal{B}(V_{\mathbb{H}}^{R})$, $q\in\mathbb{H}$ and
$\varepsilon>0$ we set:

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

It is clear that $\mathcal{O}(T,q,\varepsilon)$ is an open set in
$\mathbb{H}$.

######  Corollary 4.7.

Let $T\in\mathcal{B}(V_{\mathbb{H}}^{R})$ and $q_{0}\in\rho_{S}(T)$. Then,
$q\in\rho_{S}(T)$ for each $q\in\mathcal{O}(T,q_{0},\mu(Q_{q_{0}}(T)))$.

###### Proof.

Let $q\in\mathcal{O}(T,q_{0},\mu(Q_{q_{0}}(T)))$. Then,

$\displaystyle\|Q_{q}(T)-Q_{q_{0}}(T)\|$ $\displaystyle=\|2({\rm
Re}(q_{0})-{\rm Re}(q))T+|q|^{2}-|q_{0}|^{2}\|$ $\displaystyle\leq 2|({\rm
Re}(q_{0})-{\rm Re}(q)|\|T\|+||q|^{2}-|q_{0}|^{2}|$
$\displaystyle<\mu(Q_{q_{0}}(T)).$

We can apply Lemma 4.6 to conclude that

$\displaystyle\mu(Q_{q}(T))>0\mbox{ and
}\overline{R(Q_{q}(T))}=\overline{R(Q_{q_{0}}(T))}=V_{\mathbb{H}}^{R}.$

By [29, Proposition 3.5], $R(Q_{q}(T))$ is closed. Hence, $q\in\rho_{S}(T)$.∎

We recall:

######  Lemma 4.8.

[3, Lemma 7.3.9] Let $n\in\mathbb{N}$ and $q,s\in\mathbb{H}$. Set

$\displaystyle P_{2n}(q)=q^{2n}-2{\rm Re}(s^{n})q^{n}+|s^{n}|^{2}.$

Then,

$\displaystyle P_{2n}(q)$ $\displaystyle=\mathcal{Q}_{2n-2}(q)(q^{2}-2{\rm
Re}(s)q+|s|^{2})$ $\displaystyle=(q^{2}-2{\rm
Re}(s)q+|s|^{2})\mathcal{Q}_{2n-2}(q),$

where $\mathcal{Q}_{2n-2}(q)$ is a polynomial of degree $2n-2$ in $q$.

_Proof of Theorem 4.5_ Set:

$\displaystyle
f(T):=\displaystyle\sup_{K\in\mathcal{K}(V_{\mathbb{H}}^{R})}\mu(T+K).$

Similar proof in the complex case, we have $f(T)>0$ if and only if

$\displaystyle T\in\Phi_{l}(V_{\mathbb{H}}^{R})\mbox{ and }\dim
N(T)\leq\dim(V_{\mathbb{H}}^{R}/R(T)).$

Now, Since $\sigma_{e}^{S}(T)$ is not empty (e.g., [29, Proposition 7.14]) and
$\sigma_{e}^{S}(T)\subset\sigma_{W}^{S}(T)$, then $\partial\sigma_{W}^{S}(T)$
is not empty. Let us then take an element $p$ in $\partial\sigma_{W}^{S}(T)$.
Assume that $p\not\in\sigma_{1,w}^{S}(T)$. Then,

$\displaystyle f(Q_{p}(T))>0.$

So, there exists $K_{0}\in\mathcal{K}(V_{\mathbb{H}}^{R})$ such that

$\displaystyle\mu(Q_{p}(T)+K_{0})>0.$

Since $\mathcal{O}(T,p,\frac{\mu(Q_{p}(T)+K_{0})}{2})$ is an open neighborhood
of $p$ and

$\displaystyle p\in\overline{\Big{\\{}q\in\mathbb{H}:\
Q_{q}(T)\in\Phi(V_{\mathbb{H}}^{R})\mbox{ and }i(Q_{q}(T))=0\Big{\\}}},$

then there exists $p_{0}\in\mathcal{O}(T,p,\frac{\mu(Q_{p}(T)+K_{0})}{2})$
such that

$\displaystyle p_{0}\in W_{T}.$

On the other hand,

$\displaystyle\|Q_{p_{0}}(T)+K_{0}-Q_{p}(T)-K_{0}\|$
$\displaystyle\leq||p_{0}|^{2}-|p|^{2}|+2|Re(p)-Re(p_{0})|\|T\|$
$\displaystyle<\displaystyle\frac{\mu(Q_{p}(T)+K_{0})}{2}.$

Applying Lemma 4.6, we obtain

$\displaystyle R(Q_{p}(T)+K_{0})=V_{\mathbb{H}}^{R}.$

Indeed, since $\mu(Q_{p_{0}}+K_{0})>0$ and $p_{0}\in W_{T}$, then

$\displaystyle\dim(V_{\mathbb{H}}^{R}/R(Q_{p_{0}}(T)+K_{0}))=0.$

In this way, we see that

$\displaystyle Q_{p}(T)+K_{0}\in\Phi(V_{\mathbb{H}}^{R})\mbox{ and
}i(Q_{p}(T)+K_{0})=0.$

This implies that, $p\not\in\sigma_{W}^{S}(T)$. The rest of the proof follows
immediately from [7, Theorem 5.13].∎

We will now deal with the particular spectral theorem for the essential
S-spectra.

######  Theorem 4.9.

Let $T\in\mathcal{B}(V_{\mathbb{H}}^{R})$. Then,

$\displaystyle\sigma_{e}^{S}(T^{n})=\Big{\\{}q^{n}\in\mathbb{H}:\
q\in\sigma_{e}^{S}(T)\Big{\\}}=(\sigma_{e}^{S}(T))^{n}.$

###### Proof.

According to [3, Lemma 3.10] and the proof of [3, Theorem 7.3.11] we have

$\displaystyle T^{2n}-2{\rm
Re}(q)T^{n}+|q|^{2}\mathbb{I}_{V_{\mathbb{H}}^{R}}=\displaystyle\prod_{j=0}^{n-1}(T^{2}-2Re(q_{j})T+|q_{j}|^{2}\mathbb{I}_{V_{\mathbb{H}}^{R}}),$

where $q_{j},j=0,...,n-1$ are the solutions of $p^{n}=q$ in the complex plane
$\mathbb{C}_{I_{q}}$. Let $q\in\sigma_{e}^{S}(T^{n})$. Then,
$Q_{q}(T^{n})\not\in\Phi(V_{\mathbb{H}}^{R})$. We can apply [29, Theorem
6.13], we infer that there exists $i\in\\{0,1,...,n-1\\}$ such that

$\displaystyle Q_{q_{i}}(T)\not\in\Phi(V_{\mathbb{H}}^{R}).$

Therefore, $q_{i}\in\sigma_{e}^{S}(T)$. In this way, we see that
$q=q_{i}^{n}\in(\sigma_{e}^{S}(T))^{n}$. To prove the inverse inclusion, we
consider $p=q^{n}$, where $q\in\sigma_{e}^{S}(T)$. By Lemma 4.8 and [3,
Theorem 7.3.7], we get

$\displaystyle T^{2n}-2{\rm
Re}(q^{n})T^{n}+|q^{n}|^{2}\mathbb{I}_{V_{\mathbb{H}}^{R}}$
$\displaystyle=\mathcal{Q}_{2n-2}(T)(T^{2}-2{\rm
Re}(q)T+|q|^{2}\mathbb{I}_{V_{\mathbb{H}}^{R}})$ $\displaystyle=(T^{2}-2{\rm
Re}(q)T+|q|^{2}\mathbb{I}_{V_{\mathbb{H}}^{R}})\mathcal{Q}_{2n-2}(T).$

Since $Q_{q}(T)\not\in\Phi(V_{\mathbb{H}}^{R})$, we can apply [29, Corollary
6.14], we deduce that

$\displaystyle T^{2n}-2{\rm
Re}(q^{n})T^{n}+|q^{n}|^{2}\mathbb{I}_{V_{\mathbb{H}}^{R}}\not\in\Phi(V_{\mathbb{H}}^{R}).$

So, $p\in\sigma_{e}^{S}(T^{n})$. ∎

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# Boosting Fluxons for Improved Digital Logic using an Aharonov-Casher Ring

W. Wustmann  The Laboratory for Physical Sciences at the University of
Maryland, College Park, MD 20740, USA  K.D. Osborn<EMAIL_ADDRESS>(corresponding author) The Laboratory for Physical Sciences at the University
of Maryland, College Park, MD 20740, USA  The Joint Quantum Institute,
University of Maryland, College Park, MD 20742, USA

###### Abstract

Superconducting logic is efficient and fast relative to CMOS, but needs
fundamental studies on how to scale to very large circuits. Recently,
ballistic shift registers for single-flux quanta (SFQ) bits were shown in
simulation to allow high energy efficiency and clockless operation. However,
the gates are unpowered such that the bit inertia can only allow a finite
number of sequential gate executions because the bits slow slightly after each
gate operation. Here we show that a structure based on an Aharonov-Casher ring
can power these shift registers by boosting the bit velocity to a constant
value, in spite of unusual bit states constituted by two polarities of SFQ. We
discuss the design of the structure that we consider to be scalable, explain
its operation, and study its performance in terms of boosted velocity,
efficiency, and parameter margins.

Superconducting digital logic Holmes2013  has the potential for high-
performance computing due to its high energy efficiency and speed relative to
CMOS logic, and the large power consumed by today’s computer networks and data
centers. For this purpose, single flux quantum (SFQ) Bardeen1 ; Bardeen2 ;
Parks  are used as bits in arithmetic logic units ArithmeticLogic1 ;
ArithmeticLogic2  and processors Processor1 ; Processor2 ; Processor3 . SFQ
logic is also used in analog-to-digital converters or broadband digital
receivers Receiver1 ; Receiver2 ; Receiver3 ; Receiver4 , and Josephson
voltage standards are closely related VoltageStandard1 ; VoltageStandard2 ;
VoltageStandard3 ; VoltageStandard4 .

SFQ logic sometimes refers to the historic logic family named Rapid Single
Flux Quantum (RSFQ) logic PolonskyETAL1993 , and its variants which are made
for higher efficiency YoshikawaBias ; FujimakiBias ; KirETAL2011 ; VolETAL2013
; MukhanovETAL2014 . In RSFQ, an SFQ of a fixed polarity represents the bit
state “1”, and its absence the “0”. In this type, dc-currents power the SFQ to
propagate conditionally in a known direction during the execution of the
logic. Additionally, ac-powered logic families named Reciprocal Quantum Logic
HerrETAL2011  and Adiabatic Quantum Flux Parametron YoshikawaETAL2021  are
studied for higher efficiency. However, the most extreme methods for high
efficiency borrow concepts from thermodynamic reversibility, which in
principle allow the energy cost to scale below $\ln(2)k_{B}T$ per gate
parametricquantron1985 ; RenSemETAL2009 ; YoshikawaETAL2013 ; OsbWus2020_CNOT
.

Recent work for SFQ logic aims to scale up the size of circuits, e.g. ref.
TolpygoETAL2015, . Related to this, memory is studied for large-capacity RAM
RAM1 ; RAM2 ; RAM3  or in-logic memory named registers Register1 ; Register2 ;
Register3 . While RSFQ power is traditionally sourced to the chip ground
plane, a method to reuse bias current, called bias current recycling
CurrentRecycling , provides a scaling opportunity because the bias currents do
not necessarily increase with the number of gates, and this may avoid a
problem analogous to power-density limitations in CMOS Mukhanov2011 .
Scalability is also helped by clockless gates in SFQ logic, which is novel in
SFQ logic because most gates are synchronous and thus require clocking for
almost every computing step Rylov2019 ; KryFri2020 .

For the reversible logic type named Reversible Fluxon Logic (RFL)
WusOsb2020_RFL ; LiuqiDesign ; OsbWus2020_CNOT ; OsbWus2023_BSR__arxiv , we
have recently studied ballistic shift registers (BSRs) OsbWus2023_BSR__arxiv
which allow fully asynchronous inputs as long as a minimum time delay exists
between input bits, and thus they have potential to be clockless. RFL uses
degenerate (equal energy) bit states, where each polarity represents a state.
Some of its gates use ballistically moving SFQ in long Josephson junctions
(LJJs) such that the gates are solely powered by the input bit inertia.
However, to make BSRs practical, power must be supplied to the SFQ every so
often to scale up to a useful register size.

In this work, we take inspiration from the Aharonov-Casher effect (ACE)
ACeffect0 ; ACeffect1 ; ACeffect2 ; ACeffect3  for its ring structure and its
relation to fluxons. Various fluxon types exist in research Cybart ;
Topological1 ; Topological2 ; Topological3 ; Topological4 ; Vlad ; Wallraff ;
McDerETAL2018 ; AstrDetector , however, the ACE is fundamentally a self-
interference effect of a single fluxon and it is extremely rare in LJJ
structures due to various difficult simultaneous requirements including the
ability of charge control at the ring center.

The challenge for our digital logic is the use of different-polarity bit
states, and that gates require a minimum input velocity. RFL logic uses
fluxons in a classical regime for reliability, but more interestingly we can
borrow the above ring structure with its two connecting LJJs. Fortunately, we
find that we can power the bipolar bits with only two separate current biases,
modified LJJs, and damping resistors. Furthermore, we argue that our specific
implementation can enable a large array of these power sources and this allows
a scalable shift register memory.

The design principles of the booster follow from the science of LJJs and SFQ
of two polarities. RFL gates use LJJs, and in the input and output LJJs of the
gates the SFQ, or fluxons, constitute instances of dispersionless solitons of
sine-Gordon type BaronePaterno ; Barone1971 . It is known that a dc-current
bias through an LJJ will exert a force on a fluxon, where the force direction
is determined by the relative sign of the current direction and fluxon
polarity. A given bias current, therefore, accelerates (boosts) a fluxon, but
decelerates an antifluxon. In our RFL logic the bit states are represented by
the two fluxon polarities. Thus, a dc-current bias applied to a regular LJJ is
inadequate to boost our bits in RFL because one of the bit states will gain
energy while the other one loses energy. Here we propose a fluxon booster
which solves this problem and allows one to accelerate slow fluxons to a high
nominal velocity, irrespective of their polarity, as needed for the operation
of ballistic RFL gates in sequence.

The operation principle of the fluxon booster is illustrated in Fig. 1. The
booster consists of a ring LJJ which is connected to an input and an output
LJJ, where each connection area forms topological S-Branches. For reference,
let us first discuss the dynamics in an unpowered S-branch: in this situation,
a fluxon coming in on one of the three LJJs will penetrate evanescently into
both of the arms of the ring LJJ. However, it can not enter both of them
fully, since this would require the local creation of a second fluxon, which
could only occur through the local creation of a fluxon-antifluxon pair. At
moderate fluxon energies that are relevant for RFL, this is energetically
forbidden. Classically, it can also not enter one of the arms exclusively
since they are equivalent and neither provides an energetic advantage over the
other. Therefore, a typical S-branch OgawaETAL1976 ; YamashitaETAL1995
presents a potential barrier on which a fluxon reflects back into the input
LJJ.

Figure 1:  Illustration of booster operation for fluxons of (a) positive or
(b) negative (antifluxon) polarity. An input LJJ branches into two LJJ arms
(which constitute the ring) which then merge into an output LJJ. Bias dc-
currents $I_{b}$ are applied in both arms near the branch point, and the bias
generates anticlockwise (clockwise) force on an incoming fluxon (antifluxon).
This force allows the fluxon to enter one of the arms, as determined by the
force, and accelerate it. The critical current of the ring LJJ is increased
near the merge point (not shown) to allow the fluxon to exit the booster ring.

However, in our fluxon booster, the dynamics at the input branch is modified
by dc-currents, which bias the two arms of the LJJ ring, as shown in Fig. 1.
The two bias currents create two (overlapping) potential steps in the ring
LJJ, each of the magnitude $\Phi_{0}I_{b}$. This results in a force on a
fluxon (antifluxon) that is directed counterclockwise (clockwise). These
forces act on the front of the fluxon coming in from the input LJJ and thus
draw it into the ring LJJ. Depending on the force direction (and hence the
fluxon polarity) this will be either the upper or the lower arm, as
illustrated by panels (a) and (b). Inside the ring LJJ the fluxon is further
accelerated while in the vicinity of the bias ports. Eventually, the
accelerated fluxon reaches the merge point. As this is another S-branch and is
unpowered, one would expect the fluxon to normally back-reflect in the manner
described above. In order to avoid this behavior and to enable the fluxon to
smoothly enter the output LJJ, the arms of the ring LJJ are set thereabout to
have increased critical currents. In the discrete booster circuit, this is
implemented by larger JJ areas of the two merge-side JJs of the S-branch cell,
cf. Fig. 2(c). For a fluxon approaching the merge point from the upper arm of
the ring LJJ, the large JJ on the lower arm essentially acts as a small
inductance that connects the inner booster ring and the lower rail of the
output LJJ, and allows the fluxon to enter into the output LJJ. A short delay
in this transmission through the merge cell is caused by the potential barrier
of the large JJ on the upper arm.

Our proposed designs for booster circuits are shown in Fig. 2. Standard LJJs
used in this work are composed of discrete cells, each containing an undamped
JJ with characteristics $I_{c}$ and $C_{J}$, and a geometric inductance $L$.
In the input and output LJJs of both rails contribute $L/2$ to the total cell
inductance, whereas the inner and outer inductor rail of the ring LJJ may
contribute with a different fraction, $L=L_{i}+L_{o}$. The relative
discreteness of all LJJs is chosen equal, and this is given by the parameter
$a/\lambda_{J}=\sqrt{L/L_{J}}=(2\pi LI_{c}/\Phi_{0})^{1/2}$, where $a$
corresponds to the cell width, and $\lambda_{J}$ is the Josephson penetration
depth. Our simulations of the schematics use sufficient lengths of input and
output LJJs such that boundary effects are negligible and
$L_{J}/L=\Phi_{0}/(2\pi I_{c}L)=7$ for low but practical discreteness. The
fluxons have a characteristic length of $\lambda_{J}$ and an inverse plasma
frequency of $\omega_{J}^{-1}=\left(2\pi I_{c}/(\Phi_{0}C_{J})\right)^{-1/2}$.
The energy of a fluxon is $E_{\text{fl}}(v)=8E_{0}/\sqrt{1-(v/c)^{2}}$, where
$E_{0}=\Phi_{0}I_{c}\lambda_{J}/(2\pi a)$ and the velocity is bounded by
$|v|<c=\lambda_{J}\omega_{J}$.

While most JJs in the booster circuit follow nominal parameters, exceptions
are made for JJs at the branch and merge cells, where they differ by area, but
the ratio of critical current to capacitance, $I_{c}/C_{J}$, is the same. The
upper and lower arms of the ring LJJ are symmetric such that the fluxon
booster will boost equally for either polarity of the input fluxon. In each
arm, the 2nd JJ from the branch cell is current-biased and has a shunt
resistor $R_{J}^{b}$ in parallel which serves to dissipate fluctuations
generated during the passage of the fluxon through the bias point or the
branch and merge cell. In our designs the added damping is subcritical,
$R_{J}^{b}=5R_{J,\text{crit}}^{b}$, where
$R_{J,\text{crit}}^{b}=\sqrt{\Phi_{0}/(2\pi C_{J}I_{c})}$ is the resistance
for critical JJ damping. However, we find that the booster also works with
$R_{J}^{b}\lesssim R_{J,\text{crit}}^{b}$, albeit with somewhat reduced boost
efficiency. Table 1 shows the parameter values of our boosters.

Figure 2:  (a) Circuit schematics of the booster, in three different designs:
without series resistors (D1), with series resistors for galvanic isolation in
branch cell alone (D2), or also near merge cell (D3). (b,c) Schematics near
branch and merge cell indicating regular LJJ (black) and special (gray)
circuit elements. The top and bottom arm of the booster are fully symmetric.
(d) A scalable sequence of boosters and its possible use in an array of
ballistic gates, e.g., shift register gates, especially using D3.

The three booster designs shown in panel (a) are very similar, but differ by
small series resistors on the ring-LJJ rails: these are either absent (D1), or
are present only in the branch cell (D2), or are present also near the merge
cell (D3). While our circuit simulations show that all designs can boost an
input fluxon, we argue that D3 is scalable, as discussed next.

It has previously been shown WusOsb2020_RFL ; OsbWus2020_CNOT ;
OsbWus2023_BSR__arxiv  that unpowered (ballistic) RFL gates can operate with
high efficiency, e.g., the output fluxon energy can be above 90 % of the input
fluxons. This is very high compared to irreversible gates which usually
dissipate $\sim I_{c}\Phi_{0}$ per switching of a powered JJ. Nevertheless,
the gradual energy loss of a fluxon traversing through a sequence of RFL gates
will eventually be sufficiently large that a subsequent ballistic gate will
fail. Therefore, we need to periodically include boosters, which restore the
fluxon energy to a nominal value, e.g., to $E_{\text{fl}}(v)=10E_{0}$,
corresponding to fluxon velocity $v=0.6c$. As an example, Fig. 2(d) shows a
$2\times 2$ register with boosters. In this example, the gates could be BSRs
with two input and output ports, where input fluxons are powered by the left
boosters from left to right, and the memory could also be accessed from top to
bottom. A key feature of the scalable design is that the boosters are all
powered from the same two current sources.

Table 1:  Special circuit parameters for booster of Fig. 2, with $L^{B}=L^{M}=0.1L$. The margins are calculated for D3, using $10$ JJs per ring arm and $I_{\text{b}}/I_{c}=2.8$, and with the criterion $v_{f}\geq 0.5c$ for $v_{i}=0.1c$. They are determined by varying a single circuit parameter, keeping all others fixed. JJ critical current and capacitance are varied separately. Most parameter margins range from $-100\%$ to (far) above $+200\%$, except those given below.  $\displaystyle\frac{L_{o}}{L}$ | $\displaystyle\frac{L_{i}}{L}$ | $\displaystyle\frac{I_{c}^{B1}}{I_{c}}$ | $\displaystyle\frac{I_{c}^{B2}}{I_{c}}$ | $\displaystyle\frac{I_{c}^{M1}}{I_{c}}$ | $\displaystyle\frac{I_{c}^{M2}}{I_{c}}$ | $\displaystyle\frac{R_{J,\text{crit}}^{b}}{R_{J}^{b}}$ | $\displaystyle\frac{R_{s}}{\sqrt{L/C_{J}}}$
---|---|---|---|---|---|---|---
0.7 | 0.3 | 1.5 | 1.1 | 1.7 | 4.2 | 5.0 | (0.019)111$\sqrt{L/C_{J}}$ is the characteristic impedance of the LJJ.
$\begin{array}[]{l}-52\\\\[-3.44444pt] +83\end{array}$% |  |  $\begin{array}[]{l}-100\\\\[-3.44444pt] +50\end{array}$% |  |  |  $\begin{array}[]{l}-27\\\\[-3.44444pt] +39\end{array}$% |  |

For several boosters to share the same current sources, it is necessary that
the dc-bias supercurrents through the device are at least isolated from one
another, even in a circuit with slight asymmetries resulting from fabrication
uncertainties. To that end, we introduce serial resistors into the rails of
the booster’s ring LJJs as shown in the designs D2 and D3 of Fig. 2(a). In
principle, the location of series resistors may be chosen differently, as long
as they isolate the superconducting path between the bias current ports.
However, in our circuit simulations we find that the booster performance
(output velocity, booster efficiency, margins) is negatively affected when
series resistors are introduced in the merge cell (not shown). That is why,
our design D3 for a fully isolated booster introduces them in the adjacent
ring cells.

Numerical simulations of the fluxon dynamics in the booster circuits are
executed to obtain performance data. Figure 3 illustrates the performance of
the three booster designs of Fig. 2(a), which all are found to have similar
characteristics.

Panel (a) shows the boosted output velocity $v_{f}$ of the fluxon as a
function of the bias current $I_{\text{b}}$. The booster has a threshold in
$I_{\text{b}}$, below which an incoming fluxon is not transmitted through the
entire booster. Below the threshold, the fluxon may become pinned when first
reaching the branch cell (due to nearby resistors $R_{J}^{b}$), or when
reaching the biased and damped JJ, or otherwise, it can reflect off the merge
cell and eventually pin at the bias JJ on its second encounter. Once
$I_{\text{b}}$ exceeds the threshold, the boosted velocity $v_{f}$ first
increases with $I_{\text{b}}$, as one would expect. The boosted velocity
$v_{f}$ reaches a maximum in the current range $I_{\text{b}}/I_{c}\approx
2.5-3.0$, depending on the design, before starting to lower for larger
$I_{\text{b}}$. From our simulations, we conclude that this lowering is a
combined effect of the finite (short) length of the booster arms on the one
hand and the discreteness of the LJJ cells on the other hand 222The bias
current creates a localized mode of finite width. In interaction with this
mode, the fluxon typically undergoes two distinct boost phases, respectively
caused by the attractive (repulsive) force before (after) the mode center. In
a discrete LJJ the fluxon typically slows down temporarily between these two
boost phases, causing a short time delay after crossing the bias point. This
effect disappears in the limit of small $a/\lambda_{J}$, and remains to be
studied in detail. We observed that if the arms of the ring LJJ are relatively
short and the fluxon velocity is already high after the first boost phase (due
to large $I_{b}$), the fluxon will reach the end of the arm already during
this short time delay such that it misses the 2nd boost phase and exits the
ring with a relatively low output velocity. .

The formation of a $v_{f}$-maximum at moderate $I_{\text{b}}$ is one of the
reasons to keep the booster arms relatively short. In Fig. 3 and Table 1 we
have chosen $10$ JJs in each arm of the booster ring, and we note that the
arms then form “short LJJs” which are longer than but comparable to
$\lambda_{J}$. Making the arms even shorter appears to be detrimental to the
performance at large initial velocities: in our simulations we observe a rapid
decrease of $v_{f}$ as a function of $v_{i}$, for $v_{i}\gtrsim 0.5$. Whereas
for $N=10$, as Fig. 3(c) illustrates, the output velocity of the booster is
relatively uniform up till the break-even point $v_{f}\approx v_{i}$ (gray
dashed line). This covers the entire range of relevant input velocities
$v_{i}<0.6c$, given that the nominal operation velocity in most RFL gates is
set to $v\approx 0.6c$. Making the booster arms longer ($N>10$), allows
$v_{f}$ to saturate before exiting the ring, but at a larger value of
$I_{\text{b}}$ where the booster efficiency is much reduced (see below).

The threshold for booster operation typically lies above $I_{\text{b}}>I_{c}$,
cf. Fig. 3(a). This is quite different from RSFQ circuits, where the
critically damped JJs are typically biased with $I_{\text{b}}\approx
0.7I_{c}$. In our case, however, the circuit forms (weakly) discrete LJJs,
where the small cell inductance $L$ causes the bias current to spread over
several JJs. The undamped (underdamped) JJs are thus biased sub-critically
when the bias current $I_{\text{b}}$ is in a steady state. Additionally, the
$I_{c}$ has been previously designed LiuqiDesign  to be much smaller than in
RSFQ, such that a single subcritical bias current should also be smaller than
in RSFQ.

We calculate the booster efficiency $\eta=\Delta E_{\text{fl}}/W$ as the ratio
of the fluxon’s energy gain $\Delta
E_{\text{fl}}=E_{\text{fl}}(v_{f})-E_{\text{fl}}(v_{i})$ to the energy cost of
the current source $W$, where $W$ is calculated numerically from all of the
circuit elements. In cases of a successful fluxon boost, where one of the two
biased JJs undergoes a $2\pi$-phase change during the fluxon’s passage, the
total energy cost is consistent with the formula $W=I_{\text{b}}\Phi_{0}$.
Thus the energy cost formula of the booster is the same as for JJ switching in
an RSFQ gate. However, in the proposed architecture of RFL, the SFQ receive
only occasional boosts and thus much fewer biases are required per logic gate
sequence than in RSFQ.

Using sine-Gordon perturbation theory McLaughlinScott1978 , we also find
$I_{\text{b}}\Phi_{0}$ as the height of the potential energy step experienced
by a fluxon in a continuous LJJ with a local current bias. The booster
efficiency $\eta$ is shown in Fig. 3(b) as function of $I_{\text{b}}$. Over a
range of moderate $I_{\text{b}}$, it increases together with $v_{f}$ and
$\Delta E_{\text{fl}}$. However, $\eta$ assumes a maximum when $v_{f}$ and
$\Delta E_{\text{fl}}$ no longer substantially grows with $I_{\text{b}}$ and
afterwards it decreases because of the $I_{\text{b}}$-proportional energy cost
$W$.

As a function of the initial velocity $v_{i}$, the efficiency $\eta$
decreases, as seen in Fig. 3(d). This follows since $W=I_{\text{b}}\Phi_{0}$
is independent of $v_{i}$ while the $v_{i}$-insensitive output velocity
$v_{f}$ implies that $\Delta E_{\text{fl}}$ is largest for small $v_{i}$. The
black dashed line shows an approximation for D3, using $\Delta
E_{\text{fl}}=E_{\text{fl}}(\bar{v}_{f})-E_{\text{fl}}(v_{i})$ with the mean
output velocity $\bar{v}_{f}=0.68$ in the range $v_{i}/c\leq 0.6$. For the
data shown in panel (b), which is for $v_{i}=0.3$, and at the operation point
$I_{\text{b}}/I_{c}=2.8$, the output velocity is $v_{f}=0.67c$ and the
efficiency is $\eta=0.36$ ($\Delta E_{\text{fl}}=2.5E_{0}$, $W=6.7E_{0}$,
$E_{\text{diss}}=3.4E_{0}$). This can be compared with the performance of the
(polarity-dependent) fluxon boost in a regular (discrete) LJJ. Using the same
values of $a/\lambda_{J}$, $R_{J}^{b}$, $I_{\text{b}}=2.8I_{c}$, and
$v_{i}=0.3c$ we find that a fluxon in a regular LJJ is boosted to
$v_{f}=0.79c$ with $\eta=0.72$ ($\Delta E_{\text{fl}}=4.8E_{0}$, $W=6.7E_{0}$,
$E_{\text{diss}}=0.7E_{0}$). Thus, the efficiency in our scalable ring booster
is reduced by a factor of $\approx 1/2$ relative to the regular LJJ (for known
fluxon polarity), owing to the dynamics at the branch and merge cells, the
boundary conditions imposed by these cells during the fluxon boost, and the
added series resistors.

Figure 3:  Booster performance: $I_{\text{b}}$-dependence of (a) fluxon output
velocity $v_{f}$ and (b) booster efficiency $\eta$, and (c,d) their dependence
on fluxon input velocity $v_{i}$, for the three booster designs of Fig. 2. In
(a,b) the input velocity of the fluxon is set to $v_{i}=0.3c$, for which a
maximum in $v_{f}$ appears at $I_{\text{b}}=2.8I_{c}$ in the scalable design
D3. In (c,d) the bias is fixed at that value, $I_{\text{b}}=2.8I_{c}$, and we
note that $v_{f}$ is $v_{i}$-insensitive over a wide $v_{i}$-range, before
decreasing near where $v_{f}\equiv v_{i}$, marked by the dashed line in (c).
The dashed line in (d) shows an estimate for $\eta$ assuming constant $v_{f}$.

In conclusion, SFQ logic needs better scaling of logic and memory. Also, a
previously studied shift-register gate for SFQ computing did not require
synchronous input bits and had the potential to impact SFQ broadly in terms of
clocking and scaling. In the gate, and other gates from the same logic family,
the bit states are unconventional. In this letter, boosters are proposed to
power and scale-up register memory from the individual ballistic shift
registers. Here we report on the first boosters; they were built up from a
classical version of an Aharonov-Casher ring structure. We explain the
operation and performance of the boosters related to two fluxon polarities,
fluxon dynamics, JJ modifications, and resistors added to enable scalability.
The data show uniformly high output velocity with good efficiency, reasonable
margins, and simplicity of design. Briefly put, the boosters are efficient in
the proposed scalable register architecture because the SFQ bit only needs to
pass through one biased JJ per gate sequence, which contrasts standard logic
in which an SFQ passes through many biased JJs per gate.

###### Acknowledgements.

KDO is grateful for scientific discussions on digital logic with Q. Herr, A.
Herr, M. Frank, I. Sutherland, B. Sarabi, C. Richardson, G. Herrera, and N.
Yoshikawa, and also broader scientific discussions with B. Palmer, B. Butera,
F. Gaitan, R. Lewis, and A. Murphy. Both KDO and WW would like to thank H. Cai
for recent discussions of experiments. WW would like to thank the University
of Otago Physics Department for hosting her.

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