#  Solution of JWST and HST problems by primordial black holes

A.D. Dolgov<EMAIL_ADDRESS>Novosibirsk State University, Novosibirsk, 630090,
Russia  Bogolyubov Laboratory of Theoretical Physics, JINR, Dubna, 141980,
Russia

###### Abstract

Observations of the early universe at redshifts of order 10, collected during
the last several years presented by the Hubble Space Telescope (HST) and
recent data by the James Webb Space telescope (JWST) created strong doubts on
the validity of the conventional $\Lambda$CDM cosmology. It is argued here
that the 30 year old conjecture by A.Dolgov & J.Silk DS  of galaxy and quasar
seeding by supermassive primordial black holes (SMBHs) naturally solves the
problem of the observed dense early population of the universe, as well as the
problem of the formation of SMBH in the contemporary universe. The idea of
seeding of cosmic structures by SMBH is supported by a good agreement of the
predicted log-normal mass spectrum of primordial black holes (PBH) with
observations and by a noticeable antimatter population of the Milky Way.

##  I Introduction

Recent observations of two orbital telescopes, HST and JWST created strong
doubts on the validity of the standard cosmology. However, the 30 year old
conjecture DS , that the galaxy formation is seeded by supermassive primordial
black holes (PBH), not only rescued the accepted cosmological model but in
fact predicted the observed rich population of the early universe by well
developed galaxies and quasars.

The suggested mechanism of PBH formation allows for creation of sufficient
density of massive PBHs that seeded structure formation not only in the early
universe at redshifs of order ten but during all cosmological epochs including
the present day universe as well. Recently the idea of seeding by black holes
was rediscovered in several papers under the pressure of HST and JWST
observations.

There is a strong confirmation of the model by ”experiment”. Namely the
predicted log-normal mass spectrum of PBH very well agrees with observations;
noticeable antimatter population of the Galaxy is predicted and confirmed by
large amount of data: positrons, antinuclei, and (possibly) antistars, and the
model could supply the sufficient number of SMBH binaries that presumably
distorts the pulsar timing.

The paper is organised as follows. In section II the observations of HST, JWST
and ALMA are discussed and a possible conflict of the data with the standard
$\lambda$CDM cosmological model is described. In sec. III the seeding of
cosmic structures by primordial black holes, originally proposed in our papers
DS ; DKK  and recently rediscovered in several works, is discussed. Next, in
sec. IV, the observations of gravitational waves generated by the coalescing
black hole binary systems is considered and the arguments in favour that these
black holes are primordial are presented. The chirp mass distribution are
presented and compared with log-normal spectrum of PBHs. In sec. V the
possibility of PBH being the cosmological dark matte is advocated. Sec. VI is
dedicated to discussion of antimatter in cosmology. In the following section
observational data indicating to noticeable amount of antimatter in the Milky
Way is presented. Lastly in sec. VIII the mechanism of PBH creation with log-
normal mass spectrum and antimatter formation in the Milky way suggested in
refs. DS ; DKK  is briefly described

##  II  Crisis in cosmology

###  II.1  Crisis in cosmology, is it real?

Dense population of the early universe, younger than one billion years at
redshifts $z\sim 10$, discovered by Hubble Space Telescope (HST) and James
Webb Space Telescope (JWST), was taken as a strong blow to the conventional
$\Lambda$CDM cosmology. However, the resolution of the problems by primordial
black holes was suggested long before these problems emerged, see refs. DS ;
DKK  and several subsequent papers by our group.

The proposed model of the PBH creation is strongly supported by the predicted
log-normal mass spectrum of the black holes:

$\displaystyle\frac{dN}{dM}=\mu^{2}\exp\left[-\gamma\log^{2}\left(M/M_{0})\right)\right],$
(1)

where $\mu$ is an unknown normalisation parameter with dimension of mass,
$\gamma$ is not determined by the model as well, but the central mass $M_{0}$
is theoretically predicted AD-KP-M0  to be close to $M_{0}\approx
10M_{\odot}$, where $M_{\odot}\approx 2\times 10^{33}$ gram is the solar mass.

###  II.2 Comparison of JWST and HST

The orbit of HST is at the distance of 570 km from the Earth. The orbit of
JWST is much larger, it is about $1.5\times 10^{6}$km. The mirror of HST has
diameter equal to 2.4 m, while JWST has 2.7 time larger one and
correspondingly the area of JWST mirror is approximately 7.4 times larger. In
fig. 1 the images of HST and JWST are presented. HST operates in optical wave
length range, for example 450 nm, corresponding to blue light. It has also a
possibility to catch the signal in the infrared range with the wave length
0.8-2.5 microns. JWST has high sensitivity to infrared radiation with the wave
length 0.6 - 28,5 micron. It allows to penetrate deep into the early universe,
up to redshifts $z\sim 15$.

Figure 1: Photos of HST and JWST

By a lucky chance of gravitational lensing HST was able to discover the galaxy
at $z\approx 12$ beyond the expected sensitivity. The same galaxy was observed
by JWST, as is depicted in Fig. 2 .

Figure 2: The same galaxy observed by HST and JWST

###  II.3 Early objects observed by JWST and ALMA

The observations of far away galaxies discovered by JWST at several high
redshifts are presented in Fig. 3 and compared with the theoretical
expectations of the $\Lambda$CDM cosmological model. The difference between
the canonical theory and the data differs by several orders of magnitude
especially at $z\approx 15$.

Figure 3: JWST registration of high $z$ galaxies versus the predicted density
from the $\Lambda$CDM model

Initially the discoveries of JWST have been met with a grain of pepper, since
only continuum in micron range was measured till February 2023. That raised
doubts about accuracy of the determination the observed galaxies redshifts.

Quite soon, however, numerous spectral determinations presented excellent
support of the early data.

In ref. Bunker  the galaxy GN-z11 has been studied and the spectroscopy
confirms that GN-z11 is a remarkable galaxy with extreme properties seen 430
Myr after the Big Bang. The galaxy redshift is derived, equal to z = 10.603
(somewhat lower than previous determinations) based on multiple emission lines
in the low and medium resolution spectra over $(0.8-5.3)\mu$m.

Another example of spectrum determination is presented by a different
instrument, the radio telescope array ALMA (Atacama Large Milimeter Array).
According to their results the age of most distant galaxy identified by JWST
is accurately established with Oxygen observation and is equal to 367 million
years after the Big Bang.

ALMA also confirmed an existence of a hyper-luminous radio-loud AGN at
$z=6.853$ endsley  in extremely massive reionization-era galaxy with
$M_{*}=1.7\times 10^{11}M_{\odot}$. Such a very high AGN luminosity suggests
that this object is powered by $1.6\times 10^{9}M_{\odot}$ black hole if
accreting near the Eddington limit. Nearly impossible, but PBH could seed such
a monster.

In ref. labbe  six candidate massive galaxies with stellar mass
$>10^{10}M_{\odot}$ at $7.4<z<9.1$ i.e 500–700 Myr after the Big Bang have
been found, including one galaxy with a possible stellar mass of $\sim
10^{11}M_{\odot}$, too massive to be created in so early universe. As is
stated in the publication, according to the ’science’ it is impossible to
create so well developed galaxies. The authors even suggest that they are
supermassive black holes of the kind never seen before. That might mean a
revision of usual understanding of black holes. NB: This conclusion very well
agrees with our predictions of PBHs.

##  III Seeding of cosmic objects by black holes

The hypothesis DS  (1993) and DKK  (2006), that SMBH seeded galaxy formation
allows to explain presence of SMBH in all large and several small galaxies
accessible to observation. This mechanism explains how the galaxies observed
by JWST in the very young universe, as well as in the present day universe,
might be created.

Though the idea of seeding of the cosmic structures by ”parent” black holes
has been accepted in several recent publications (see the following
subsection), still the BH that performed the seeding are supposed to be
created rather late in the process of the cosmic evolution. That creates in
turn the problem of their creation. Much more natural idea, that the seeds are
primordial black holes, is not accepted by the astrophysical community in
spite of serious observational evidence in favour of primordial black hole
existence pbh-evi .

###  III.1 Seeding in the early universe

This idea of seeding was rediscovered in a few recent works. For example in
ref. bromm  is stated that recent observations with JWST have identified
several bright galaxy candidates at $z\gtrsim 10$, some of which appear
unusually massive (up to $\sim 10^{11}M_{\odot}$). Such early formation of
massive galaxies is difficult to reconcile with standard $\Lambda$CDM
predictions, The observed massive galaxy candidates can be explained if
structure formation is accelerated by massive ($\gtrsim 10^{9}M_{\odot}$) PBHs
that enhance primordial density fluctuations.

As it is stated in ref. bogdan-1 : detection of an X-ray quasar in a
gravitationally-lensed z = 10.3 galaxy suggests that early supermassive black
holes originate from heavy seeds. The authors noted that ”observations of
high-redshift quasars reveal that many supermassive black holes were in place
less than 700 Million years after the Big Bang. However, the origin of the
first BHs remains a mystery.” To resolve the puzzle a population of seeds of
the first BHs are postulated. They are assumed to be either light i.e.,
$(10-100)M_{\odot}$ being the remnants of the first stars or heavy i.e.,
$(10^{4}-10^{5})M_{\odot}$, originating from direct collapse of gas clouds.
The latter looks rather unnatural. Much simpler and straightforward if the
seeds are primordial BH, as predicted in refs. DS ; DKK .

In the same paper bogdan-1  the detection of an X-ray-luminous quasar powered
by SMBH with the mass $\sim 4\times 10^{7}M_{\odot}$ in the galaxy identified
by JWST at $z\approx 10.3$ is reported. It is argued that the detection of
such quasar suggests that early supermassive black holes also originate from
heavy seeds. The mass of this QSO is comparable to the inferred stellar mass
of its host galaxy, in contrast to the usual examples (but not always) from
the local universe where mostly the BH mass is $\sim 0.1\%$ of the host
galaxy’s stellar mass. The combination of such a high BH mass and large BH-to-
galaxy stellar mass ratio $\sim 500$ Myrs after the Big Bang is consistent
with a picture wherein such BHs originated from heavy seeds.

This conclusions coincide with the predictions made in refs. DS ; DKK  but the
mechanisms of seed creation are drastically different.

Further investigation of the seeding problem applied to early SMBH
observations is performed in ref. goulding . The James Webb Space Telescope
detects early black holes (BHs) as they transform from seeds to supermassive
BHs. In ref. bogdan-1  the detection of an X-ray luminous supermassive black
hole with a photometric redshift at z ¿ 10 is reported. According to the
authors s uch an extreme source at this very high redshift provides new
insights on seeding and growth models for BHs given the short time available
for formation and growth. The resulting ratio of the black hole mass to the
stellar mass remains two to three orders of magnitude higher than local
values, thus lending support to the heavy seeding channel for the formation of
supermassive BHs within the first billion years of cosmic evolution.

###  III.2 Seeding in contemporary universe

Primordial IMBHs with masses of a few thousand solar mass could explain
formation of globular clusters (GCs) and dwarf galaxies, as is predicted in
ref. AD-KP . In the last several years such IMBH inside GSs and dwarfs are
observed.

The seeding of dwarfs by intermediate mass BHs is confirmed by the recent
data, e.g. in the dwarf galaxy SDSS J1521+1404 the BH is discovered with the
mass $M\sim 10^{5}M_{\odot}$ that is contains by far too much matter to be
created by accretion inside such tiny creature.

The observation of two Candidates for Dual AGN in Dwarf-Dwarf Galaxy Mergers
dual-AGN : for the first time, astronomers have spotted evidence of a pair of
dwarf galaxies featuring giant black holes on a collision course with each
other. In fact, they haven’t just found just one pair – they’ve found two.

In ref. IMBH-dwarf  a discovery of an intermediate-mass black hole (IMBH) with
a mass of $M_{BH}=3.6^{+5.9}_{-2.3}\times 10^{5}M_{\odot}$, is reported, that
surely cannot be created by accretion but might seed the dwarf formation.

To summarise, a large amount of observational data are at odds with the
conventional model but nicely fits the model of creation of primordial black
holes and primordial stars proposed in A. Dolgov, J.Silk, PRD 47 (1993) 4244,
(DS) and further developed in A.Dolgov, M. Kawasaki, N. Kevlishvili, Nucl.
Phys. B807 (2009) 229, (DKK) The proposed mechanism is the first where
inflation and Affleck-Dine baryogenesis are applied to PBH formation, repeated
now in a number of works. The impressive feature of the mechanism is the log-
normal mass spectrum which is the only known spectrum tested by experiment, in
perfect agreement.

##  IV Gravitational waves from BH binaries

GW discovery by LIGO strongly indicate that the sources of GW are PBHs, see
e.g. BDPP . There are several features that indicated to primordial origin of
the GW sources:
1\. Origin of heavy BHs, $M\sim 30M_{\odot}$, via conventional astrophysical
processes is problematic, though is not excluded. Recently there appeared much
more striking problem of BH observation with $M\sim 100M-\odot$. To form so
heavy BH, the progenitors should have $M>100M_{\odot}$ and a low metal
abundance to avoid too much mass loss during the evolution. Such heavy stars
might be present in young star-forming galaxies but they are not yet observed.
A possible way out is suggested in ref. Zieg-KF  by suggestion that possible
dark matter annihilation inside stars could permit formation of BHs with
$M\gtrsim 100M_{\odot}$.
On the other hand, PBHs with the observed by LIGO masses may be easily created
with sufficient density.

2\. Formation of BH binaries from the original stellar binaries has quite low
probability. Stellar binaries are formed from common interstellar gas clouds
and are quite frequent in galaxies. If BH is created through stellar collapse,
a small non-sphericity results in a huge velocity of the BH and the binary is
destroyed. BH formation from PopIII stars and subsequent formation of BH
binaries with tens of M is estimated to be small. The problem of the binary
formation is simply solved if the observed sources of GWs are the binaries of
primordial black holes. They were at rest in the comoving volume, when inside
horizon they are gravitationally attracted and may loose energy due to
dynamical friction in the early universe. The probability to become
gravitationally bound is quite significant.
The conventional scenario is not excluded but looks much less natural

3\. The low values of the BH spins in almost all except for three events
strongly constrains astrophysical BH formation from close binary systems.
Astrophysical BHs are expected to have considerable angular momentum,
nevertheless the dynamical formation of double massive low-spin BHs in dense
stellar clusters is not excluded, though difficult. On the other hand, PBH
practically do not rotate because vorticity perturbations in the early
universe are vanishingly small. However, individual PBH forming a binary
initially rotating on elliptic orbit could gain collinear spins about 0.1 -
0.3, rising with the PBH masses and eccentricity post-mit ; post-kur-mit .
This result is in agreement with the GW170729 LIGO event produced by the
binary with masses 50and 30M and GW190521.

###  IV.1 Chirp mass distribution

Two rotating gravitationally bound massive bodies are known to emit
gravitational waves. In quasi-stationary inspiral regime, the radius of the
orbit and the rotation frequency are approximately constant and the GW
frequency is twice the rotation frequency. The luminosity of the GW radiation
is:

$\displaystyle
L=\frac{32}{5}\,m_{Pl}^{2}\left(\frac{M_{c}\omega_{orb}}{M_{Pl}^{2}}\right)^{10/3},$
(2)

where $m_{Pl}=1.22\times 10^{19}$ GeV is the Planck mass and $M_{c}$ is the so
called chirp mass:

$\displaystyle M_{c}=\frac{(M_{1}M_{2})^{3/5}}{M_{1}+M2)^{1/5}}$ (3)

and $M_{1}$ and $M_{2}$ are the masses of two bodies in the binary system. The
orbital frequency is equal to:

$\displaystyle\omega_{orb}^{2}=\frac{M_{1}+M_{2}}{m_{Pl}^{2}R^{3}},$ (4)

where $R$ is the radius of the orbit.

In ref. DKMPPSS  he available data on the chirp mass distribution of the black
holes in the coalescing binaries in O1-O3 LIGO/Virgo runs are analyzed and
compared with theoretical expectations based on the hypothesis that these
black holes are primordial with log-normal mass spectrum. The inferred best-
fit mass spectrum parameters, $M_{0}=17M_{\odot}$ and $\gamma=0.9$, fall
within the theoretically expected range and shows excellent agreement with
observations, see fig. 4 .

Figure 4: Model distribution $F_{PBH}(<M)$ with parameters $M_{0}$ and
$\bm{\gamma}$ for two best Kolmogorov-Smirnov tests. EDF= empirical
distribution function.

In Fig. 5 cumulative distributions $F(<M)$ for several astrophysical models of
binary BH coalescences are presented. Evidently binary black hole formation
based on massive binary star evolution requires additional adjustments to
reproduce the observed chirp mass distribution

Figure 5: Chirp mass distribution of astrophysical BHs compared to
observations (blue)

To conclude, PBHs with log-normal mass spectrum perfectly fit the data, while
astrophysical BHs seem to be disfavoured.

An analysis of the new set of Ligo-Virgo-Kagra data was performed recently in
ref. Pos-Mit-new . The authors dfound that the chirp-mass distribution of LVK
GWTC-3 BH+BH binaries has two distinct bumps that could be explained by two
different populations of BH+BH binaries: 1) the low-mass bump with $M_{0}\sim
10M_{\odot}$ due to the astrophysical BH+BH formed in the local Universe from
the evolution of massive binaries;
2) the PBH binaries with log-normal mass spectrum with $M_{0}\sim 10M_{\odot}$
and $\gamma\sim 10$. The central mass of the PBH distribution is somewhat
larger than the expected PBH mass at the QCD phase transition, $M_{0}\sim
8M_{\odot}$ but still can be accommodated with the mass of the cosmological
horizon provided that the temperature of the QCD phase transition $T_{PT}\sim
70$ MeV, that is possible for a large chemical potential. However, the huge
value of $\gamma$, corresponding to the second bump, leads to a sharp BH mass
distribution sharply concentrated near $M=M_{0}$.

##  V Black dark matter

The first suggestion PBH might be dark matter ”particles” was made by S.
Hawking in 1971 SH-dm  and later repeated by G. Chapline in 1975 chaplin  who
assumed the flat mass spectrum in log interval: $dN=N_{0}(dM/M)$ with maximum
mass $M_{max}\approx 10^{22}$ g, which hits the allowed mass range, according
to the review Carr-Kuh-1 ; Carr-Kuh-1 . The next work DS  of 1993, entitled
”Baryon isocurvature fluctuations at small scales and baryonic dark matter”, a
more interesting mass range with much higher BH masses was proposed, in
particular with masses observed by LVC (LIGO-Virgo-Kagra).

Large values of BH masses strongly exceeding the horizon mass in the early
universe was allowed, because of the pioneering suggestion that cosmological
inflation was invoked for the PBH formation. As a result the PBHs with masses
as high as $10^{6}M_{\odot}$ and even higher can be created.

Constraints on PBHs on the cosmological fraction of the mass density as a
function of mass are reviewed in refs. Carr-Kuh-1 ; Carr-Kuh-2  and presented
in Fig. 6. The bounds are derived for the monochromatic mass spectrum of BHs
and according to the authors are model dependent and have caveats.

Figure 6: Allowed fraction of the cosmological BH density

As is argued in ref. rubin  PBHs can be formed in clusters. Dynamical
interactions in PBH clusters offers additional channel for the orbital energy
dissipation thus increasing the merging rate of PBH binaries, and the
constraints on fPBH obtained by assuming a homogeneous PBH space distribution
can be weaker. A recent analysis er-stas  based on the PBH formation model
sasaki  and nakamura  shows that the cosmological mass fraction even up to
$f_{PBH}=1$ is not excluded 111I thank K. Postnov for indicating the above
references.

##  VI Antimatter in cosmology and in the Galaxy

Astronomical data of the several recent years present strong evidence in
favour of noticeable antimatter population in our Galaxy including:
• Observation of gamma-rays with energy 0.511 MeV, which surely originate from
electron-positron annihilation at rest.
• Very large flux of anti-helium nuclei, observed at AMS.
• Several stars are found which produce excessive gamma-rays with energies of
several hundred MeV which may be interpreted as indication that these stars
consist of antimatter.

###  VI.1 Antimatter history

The search for galactic antimatter (anticomets) was pioneered by B.P.
Konstantinov and collaborators konst-1 ; konst-2  in 1968 and 1969.

Antimatter in the universe was first considered by F.W. Stecker starting for
1971 steck-1 ; steck-2  Summary of the situation at 2002 was presented at two
lectures FS ; AD .

The father of antimatter is justly considered Paul A.M. Dirac who in his Nobel
lecture in 1933 “Theory of electrons and positrons”, anticipated ”It is quite
possible that… these stars being built up mainly of positrons and negative
protons. In fact, there may be half the stars of each kind. The two kinds of
stars would both show exactly the same spectra, and there would be no way of
distinguishing them by present astronomical methods.” It is shown in ref.
citeDNV that this is not exactly so. As it was srgued in ref. DNV  the spectra
are not exactly the same, even if CPT is unbroken and the polarisation of
radiation form weak decays could be a good indicator or the type of emitted
neutrinos/antineutrinos from (anti)supernovae. However, observations of the
effects indicated in DNV  are not realistic at the present days.

In fact Dirac was the second person to talk about antimatter. In 1898, 30
years before Dirac and one year after discovery of electron (J.J. Thomson,
1897) Arthur Schuster (another British physicist) conjectured that there might
be other sign electricity, antimatter, and supposed that there might be entire
solar systems, made of antimatter, indistinguishble from ours. Schuster’s wild
guess: matter and antimatter are capable to annihilate and produce vast energy
surprisingly happened to be true. On the other hand he believed that matter
and antimatter were gravitationally repulsive since antimatter had negative
mass. Two such objects on close contact should have vanishing mass!? Quoting
his paperSchuster : “When the year’s work is over and all sense of
responsibility has left us, who has not occasionally set his fancy free to
dream about the unknown, perhaps the unknowable?… Astronomy, the oldest and
yet most juvenile of the sciences, may still have some surprises in store. May
antimatter be commended to its case”.

##  VII Antimatter in the Milky Way

Based on the conventional approach no antimatter object is expected to be in
the Galaxy. However, it was predicted in 1993 DS  and elaborated in 2009!DKK
that noticeable amount of antimatter, even antistars might be in the Galaxy
and in its halo.

Bounds on the density of galactic antimatter and in particular on antistars
are quite loose CB-AD ; DB ; BDP . The reason for that is a relatively small
energy release in the process of annihilation because it takes place only in a
thin surface layer of the star.

###  VII.1 Galactic positrons

Observation of intense 0.511 line is a proof of abundant positron population
in the Galaxy. In the central region of the Galaxy electron–positron
annihilation proceeds at a surprisingly high rate, creating the flux:

$\displaystyle\Phi_{511\;{\rm keV}}={1.07\pm 0.03\cdot 10^{-3}}\;{\rm
photons\;cm^{-2}\,s^{-1}}.$ (5)

The width of the line is about 3 keV. Emission mostly goes from the Galactic
bulge and at much lower level from the disk e+1 ; e+2 ; e+3  Until recently
the commonly accepted explanation was that $e^{+}$ are created in the strong
magnetic fields of pulsars but the recent results of AMS probably exclude this
mechanism, since the spectrum of $\bar{p}$ and $e^{+}$ at high energies are
identical Ting-2021 .

###  VII.2 Galactic antinuclei

In 2018 AMS-02 announced possible observation of six ${\bar{He}^{3}}$ and two
${\bar{He}^{4}}$ AMS-antinuc .

In 2022 more events have been presented AMS-antinuc-2 : 7 $\bar{D}$
($E\lesssim 15$ GeV) and 9 $\overline{He}$ ($E\sim 50$ GeV). The ratio of the
fluxes ${\overline{He}/He\sim 10^{-9}}$ is too high to be explained by the
secondary production of antinuclei in by cosmic rays. It is not excluded that
the flux of anti-helium is even much higher because low energy
${\overline{He}}$ may escape registration at AMS.

Antinuclei creation in cosmic rays are estimated in ref. anti-nuc-CR  Anti-
deuterium can be created in the collisions ${\bar{p}\,p}$ or ${\bar{p}\,He}$
that would produce the flux of $\bar{D}$. ${\sim
10^{-7}/m^{2}/s^{-1}}$/steradian/GeV/neutron), i.e. 5 orders of magnitude
below the observed flux of antiprotons.

The fluxes of ${\overline{He}^{3}}$ and ${\overline{He}^{4}}$, that could be
created in cosmic rays are respectively 4 and 8 orders of magnitude smaller
than the flux of anti-D.

After the AMS announcement of observations of anti-$He^{4}$ there appeared
several theoretical attempts to create anti-$He^{4}$ through dark matter
annihilation. It looks quite unnatural, especially because the main channel of
annihilation is expected to be elementary particles but not bound systems of
antiprotons and antineutrons.

There is noticeable discrepancy between the large fraction of D with respect
to He. In the case of the standard BBN this ratio should be smaller than
unity, but the observed one is practically 1.

According to the model DS ; DKK  the abundances of D and He are determined by
BBN with large baryon-to-photon ratio $\beta$ (or $\eta$). However if
$\beta\sim 1$ there is no primordial D. On the other hand in our scenario
formation of primordial elements takes place inside non-expanding compact
stellar-like objects with fixed temperature. If the temperature is
sufficiently high, this so called BBN may stop before abundant He formation
with almost equal abundances of D and He. One can see that looking at
abundances of light elements as a function of temperature. If it is so,
antistars may have equal amount of $\overline{D}$ and $\overline{He}$.

###  VII.3  Antistars in the Galaxy.

In ref. gal-anti-stars  surprising evidence for antistar observation in the
Galaxy are presented. Quoting the authors: ”We identify in the catalog 14
antistar candidates not associated with any objects belonging to established
gamma-ray source classes and with a spectrum compatible with baryon-antibaryon
annihilation.” The positions of these possible antistars are presented in Fig.
7.

Figure 7:  Positions and energy flux in the 100 MeV - 100 GeV range of
antistar candidates selected in 4FGL-DR2. Galactic coordinates. The background
image shows the Fermi 5-year all-sky photon counts above 1 GeV

An alternative way to identify antistars is proposed in paper BBBDP . In
astrophysically plausible cases of the interaction of neutral atmospheres or
winds from antistars with ionised interstellar gas, the hadronic annihilation
will be preceded by the formation of excited ${p\bar{p}}$ and $He{\bar{p}}$
atoms. These atoms rapidly cascade down to low levels prior to annihilation
giving rise to a series of narrow lines which can be associated with the
hadronic annihilation gamma-ray emission. The most significant are L (3p-2p)
1.73 keV line (yield more than 90%) from ${p\bar{p}}$ atoms, and M (4-3) 4.86
keV (yield $\sim 60$%) and L (3-2) 11.13 keV (yield about 25%) lines from
$He^{4}\bar{p}$ atoms. These lines can be probed in dedicated observations by
forthcoming sensitive X-ray spectroscopic missions XRISM and Athena and in
wide-field X-ray surveys like SRG/eROSITA all-sky survey.

Possible sources of antinuclei in cosmic rays from antistars which are
predicted in a modified Affleck-Dine baryogenesis scenario of DS ; DKK  are
studied in ref. BPBBD . The expected fluxes and isotopic content of antinuclei
in the GeV cosmic rays produced in scenarios involving antistars are
estimated. It is shown that the flux of antihelium cosmic rays reported by the
AMS-02 experiment can be explained by Galactic anti-nova outbursts,
thermonuclear anti-SN Ia explosions, a collection of flaring antistars, or an
extragalactic source with abundances not violating existing gamma-ray and
microlensing constraints on the antistar population.

##  VIII Anti-creation mechanism

The model of creation of antimatter in the galaxy is a byproduct of the DS ;
DKK  scenario of creation of massive primordial black holes. It is based on
supersymmetry (SUSY) motivated baryogenesis, proposed by Affleck and Dine (AD)
aff-dine . SUSY predicts existence of bosons with non-zero baryonic number
${B\neq 0}$. Such bosons may condense along flat directions of the quartic
potential:

$\displaystyle{{U_{\lambda}(\chi)=\lambda|\chi|^{4}\left(1-\cos
4\theta\right)}}$ (6)

and of the mass term, ${U_{m}=m^{2}\chi^{2}+m^{*\,2}\chi^{*\,2}}$:

$\displaystyle
U_{m}(\chi)=m^{2}|\chi|^{2}\left[1-\cos(2\theta+2\alpha)\right],$ (7)

where ${\chi=|\chi|\exp(i\theta)}$ and ${m=|m|e^{\alpha}}$. In AD scenatio
baryonic number is naturally non-conserved .

After inflation ${\chi}$ was away from the origin and, when inflation was
over, started to evolve down to equilibrium point, ${\chi=0}$, according to
the equation:

$\displaystyle{{\ddot{\chi}+3H\dot{\chi}+U^{\prime}(\chi)=0.}}$ (8)

Baryonic charge of $\chi$, $B_{\chi}=\dot{\theta}|\chi|^{2}$, is analogous to
mechanical angular momentum in complex $\chi$ plane. ${{\chi}}$ decays
transferred baryonic charge to that of quarks in B-conserving process.

AD baryogenesis could lead to baryon asymmetry of order of unity, much larger
than the observed ${10^{-9}}$.

If ${m\neq 0}$, the angular momentum, B, is generated by different directions
of the quartic and quadratic valleys at low ${\chi}$. If CP-odd phase
${\alpha}$ is small but non-vanishing, both baryonic and antibaryonic domains
might be formed with possible dominance of one of them.

A new input of the model DS ; DKK  is the coupling of $\chi$ to inflaton
$\Phi$ (the first term in the equation below):

$\displaystyle
U={g|\chi|^{2}(\Phi-\Phi_{1})^{2}}+\lambda|\chi|^{4}\,\ln\left(\frac{|\chi|^{2}}{\sigma^{2}}\right)$
$\displaystyle{+\lambda_{1}(\chi^{4}+h.c.)+(m^{2}\chi^{2}+h.c.).\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}$
(9)

Coupling to inflaton is a general renormalizable one. When the window to the
flat direction is open, near ${\Phi=\Phi_{1}}$, the field ${\chi}$ slowly
diffuses to large value, according to quantum diffusion equation derived by
Starobinsky, generalised to a complex field $\chi$.

If the window to flat direction, when ${\Phi\approx\Phi_{1}}$ is open only
during a short period, cosmologically small but possibly astronomically large
bubbles with high ${\beta}$ could be created, occupying a small fraction of
the universe, while the rest of the universe has normal ${{\beta\approx 6\cdot
10^{-10}}}$, created by a small ${\chi}$. This mechanism of massive PBH
formation quite different from all others. The fundament of PBH creation is
build at inflation by making large isocurvature fluctuations at relatively
small scales, with practically vanishing density perturbations. Initial
isocurvature perturbations are in chemical content of massless quarks. Density
perturbations are generated after the QCD phase transition. The emerging
universe looks like a piece of Swiss cheese, where holes are high baryonic
density objects occupying a minor fraction of the universe volume.
Results of (Anti-)Creation.
1.PBHs with log-normal mass spectrum - confirmed by the data!
2\. Compact stellar-like objects, as e.g. cores of red giants.
3\. Disperse hydrogen and helium clouds with (much) higher than average
$\bm{n_{B}}$ density. Strange stars with unusual chemistry and velocity.
4.${\beta}$ may be negative leading to creation of (compact?) antistars which
could survive annihilation with the homogeneous baryonic background.
5.Extremely old stars would exist. Several such stars are observed. Even,
”older than universe star” is found older-star ; the older age is mimicked by
the unusual initial chemistry.

## FUNDING

This work was supported by RSF Grant 23-42-00066.

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# Browder $S$-resolvent equation in quaternionic setting

Hatem Baloudi  Hatem Baloudi, Department of Mathematics, Faculty of Sciences
of Gafsa, University of Gafsa, 2112 Zarroug, Tunisia  ,  Aref JERIBI  Aref
Jeribi, Department of Mathematics, Faculty of Sciences of Sfax, University of
Sfax, 3000 Sfax, Tunisia  and  Habib Zmouli  Habib Zmouli, Departement of
Mathematics Faculty of Sciences of Sfax, University of Sfax, Route de Soukra
km 3.5, B.P. 1171, 3000 Sfax, Tunisia.

###### Abstract.

This paper is devoted to the study of the $S$-eigenvalue of finite type of a
bounded right quaternionic linear operator acting in a right quaternionic
Hilbert space. The study is based on the different properties of the Riesz
projection associated with the connected part of the $S$-spectrum.
Furthermore, we introduce the left and right Browder $S$-resolvent operators.
Inspired by the $S$-resolvent equation, we give the Browder’s $S$-resolvent
equation in quaternionic setting.

###### Key words and phrases:

Quaternion, Riesz projection, Essential 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 Operators acting on right quaternionic Hilbert space
    3. 2.3 The quaternionic functional calculus
  3. 3 Eigenvalue of finite type
  4. 4 Browder $S$-resolvent equation in quaternionic setting

##  1\. Introduction

In the theory of complex Banach spaces, the search for the eigenvalues of
finite type of linear operator aroused the interest and attracted the
attention of many researchers, see for instance [7, 9, 12, 13, 26, 28, 29] and
references therein. Sometimes this type of eigenvalue is known as a Riesz
point. In doing so, one can develop the spectral theory of operators. An
attractive characterization of eigenvalues of finite type by using Riesz
projection is discussed and determined in [26]. In particular, they show that
this part of the spectrum is only the set of isolated point of the spectrum
such that the corresponding Riesz projections are finite dimensional. Thus an
extension of the usual resolvent is studied in [30]. We refer to [16, 28] for
the applications to the Frobenius Schur factorizationfor $2\times 2$ Matrices
and to the transport operators.

In the quaternionic setting, there has long been an apparent problem in
defining the concept of spectrum of a quaternionic operator. In fact, the
quaternionic multiplication is not commutative. This makes it possible to
observe three types of Banach spaces: right, left and bilateral, according to
the operation of the multiplication on the vectors. It was only in $2006$ that
F. Colombo and I. Sabbadini succeed in giving a new attractive and useful
concept for the study of quaternionic operations, namely the $S$-spectrum. We
refer to [19, Section 1.2.1], see also [20] for the precise history and
motivation of this new concept. Some years later, D. Alpay, F. Colombo and
D.P. Kimsey in [5] gave the spectral theorem for the bounded and unbounded
quaternionic operator related to the concept of $S$-spectrum. In the book
[18], the authors have studied and discussed the spectral theory for the
Clifford operators. We refer to [15] for some results on operators
perturbation, to [21] for a version of functional calculus for bounded and
unbounded normal operators on a Clifford module, and to [22] for the study and
discussion of slice monogenic function of a Clifford variable.

The first aim of this article is to study the $S$-eigenvalue of finite type of
a bounded right quaternionic linear operator acting in a right quaternionic
Hilbert space. In fact, if $T\in\mathcal{B}(V^{R}_{\mathbb{H}})$ (the set of
all right bounded operator) and $q\in\sigma_{S}(T)\setminus\mathbb{R}$ (where
$\sigma_{S}(T)$ denote the $S$-spectrum of $T$), then $[q]:=\\{hqh^{-1}:\
q\in\mathbb{H}^{*}\\}\subset\sigma_{S}(T)$ since the $S$-spectrum of $T$ is
axially symmetric. In particular, $q$ is never isolated in $\sigma_{S}(T)$.
However, we can speak of an isolated $2$-sphere in $\sigma_{S}(T)$. In this
case, we can associate to $T$ and $[q]$ with a Riesz projection. Under there
circumstances, the $S$-eigenvalue of finite type will be considered as an
isolated $2$-sphere with an associated Riesz projection $P_{[q]}$ of finite
rank. Especially, if $T\in\mathcal{B}(V_{\mathbb{C}})$ (i.e. $T$ is a linear
operator acting on complex Banach space) and $\lambda$ is in the complex
spectrum of $T$, then $[\lambda]=\\{\lambda\\}$ and this gives the usual
version of the Riesz point in the complex case. We turn to the understanding
of the $S$-eigenvalue of finite type. To begin with, let
$T\in\mathcal{B}(V^{R}_{\mathbb{H}})$ and $q\in\sigma_{d}^{S}(T)$ (the set of
$S$-eigenvalues of finite type), we refer to Section 3 for precise definition.
The first result of this paper characterizes the range of the Riesz projection
$P_{[q]}$ associated with the $2$-sphere $[q]$ and the operator $T$. Next,
thank to the $S$-spectral mapping theorem [19, Theorem 4.2.1], we show that if
we perturb the pseudo-resolvent $\mathcal{Q}_{q}(T):=T^{2}-2{\rm
Re}(q)T+|q|^{2}\mathbb{I}_{V^{R}_{\mathbb{H}}}$ by the Riesz projection
$P_{[q]}$ we obtain an invertible operator. We end the first part of the paper
with a discussion on the localization of the $S$-eigenvalue of finite type of
sequence of quaternionic operators.

The second aim of this article is to determine the quaternionic version of
Browder’s resolvent equation. Let $T$ be a linear operator acting on a complex
Banach space $V_{\mathbb{C}}$. The spectrum of $T$ will be denoted by
$\sigma(T)$ and the Riesz point will be denoted by $\sigma_{d}(T)$ (the set of
isolated point $\lambda\in\mathbb{C}$ in the spectrum such that the
corresponding Riesz projection $P_{\\{\lambda\\}}$ are finite dimensional).
For $\lambda,\ \mu\in(\mathbb{C}\setminus\sigma(T))\cup\sigma_{d}(T)$, the
Browder’s resolvent equation is given by

$R_{B}^{-1}(\lambda,T)-R_{B}^{-1}(\mu,T)=(\lambda-\mu)R_{B}(\lambda,T)R_{B}(\mu,T)+M_{T}(\lambda,\mu),$
(1)

where

$\displaystyle
R_{B}^{-1}(\lambda,T)=(T-\lambda\mid_{P^{-1}_{\\{\lambda\\}}(\\{0\\})})^{-1}(I-P_{\\{\lambda\\}})+P_{\\{\lambda\\}}$

and

$\displaystyle
M_{T}(\lambda,\mu)=R_{B}^{-1}(\lambda,T)([T-(\lambda+1)]P_{\\{\lambda\\}}-[T-(\mu+1)]P_{\\{\mu\\}})R_{B}^{-1}(\mu,T).$

We refer to [30] for a brief discussion and for a full proof. We turn to the
case quaternionic. Set $T\in\mathcal{B}(V^{R}_{\mathbb{H}})$ and
$q\in\sigma_{d}^{S}(T)$. Let $P_{[q]}$ denote the corresponding Riesz
projector with rang and kernel denoted by $R(P_{[q]})$ and $N(P_{[q]})$,
respectively. Thanks to the Riesz decomposition theorem [34, Theorem 6] in
quaternionic setting, we have

$\displaystyle\sigma_{S}(T\mid_{R(P_{[q]})})=[q]\mbox{ and
}\sigma_{S}(T\mid_{N(P_{[q]})})=\sigma_{S}(T)\setminus[q].$

In this way, we can define the left Browder $S-$resolvent operator

$\displaystyle
S_{L,B}^{-1}(q,T):=-[\mathcal{Q}_{q}(T)\mid_{N(P_{[q]})}]^{-1}(T-\overline{q}\mathbb{I}_{V^{R}_{\mathbb{H}}})(\mathbb{I}_{V^{R}_{\mathbb{H}}}-P_{[q]})-P_{[q]}$

and the right Browder $S-$resolvent operator

$\displaystyle
S_{R,B}^{-1}(q,T):=-(T-\overline{q}\mathbb{I}_{V^{R}_{\mathbb{H}}})[\mathcal{Q}_{q}(T)\mid_{N(P_{[q]})}]^{-1}(\mathbb{I}_{V^{R}_{\mathbb{H}}}-P_{[q]})-P_{[q]}.$

Motivated by this, we obtain a generalization of the classical Browder’s
resolvent equation (1). Precisely, we have

$S^{-1}_{R,B}(s,T)S^{-1}_{L,B}(p,T)\mathcal{Q}_{s}(p)$

$\displaystyle=[S^{-1}_{R,B}(s,T)-S^{-1}_{L,B}(p,T)]p+\overline{s}[S^{-1}_{L,B}(p,T)-S^{-1}_{R,B}(s,T)]$
$\displaystyle+[S^{-1}_{R,B}(s,T)(T-(p+1)\mathbb{I}_{V^{R}_{\mathbb{H}}})P_{[p]}-(T-(s+1)\mathbb{I}_{V^{R}_{\mathbb{H}}})P_{[s]}S^{-1}_{L,B}(p,T)]p$
$\displaystyle+\overline{s}[(T-(s+1)\mathbb{I}_{V^{R}_{\mathbb{H}}})P_{[s]}S^{-1}_{L,B}(p,T)-S^{-1}_{R,B}(s,T)(T-(p+1)\mathbb{I}_{V^{R}_{\mathbb{H}}})P_{[p]}],$

where $p,s\in(\mathbb{H}\setminus\sigma_{S}(T))\cup\sigma_{d}^{S}(T)$ and
$\mathcal{Q}_{s}(p)=p^{2}-2{\rm Re}(s)p+|s|^{2}$. The technique of the proof
is inspired from the proof of [19, Theorem 3.1.15]. It is remarkable that the
Browder’s resolvent equation extend [19, Theorem 3.1.15] to
$(\mathbb{H}\setminus\sigma_{S}(T))\cup\sigma_{d}^{S}(T)$. Indeed, if
$q\in\mathbb{H}\setminus\sigma_{S}(T)$ with the convention $P_{[q]}=0$, then

$\displaystyle S_{L,B}^{-1}(q,T)=S_{L}^{-1}(q,T)=-(T^{2}-2{\rm
Re}(q)T+|q|^{2}\mathbb{I}_{V^{R}_{\mathbb{H}}})^{-1}(T-\overline{q}\mathbb{I}_{V^{R}_{\mathbb{H}}})$

and

$\displaystyle
S_{R,B}^{-1}(q,T)=S_{R}^{-1}(q,T)=-(T-\overline{q}\mathbb{I}_{V^{R}_{\mathbb{H}}})(T^{2}-2{\rm
Re}(q)T+|q|^{2}\mathbb{I}_{V^{R}_{\mathbb{H}}})^{-1}.$

As for the rest of this paper, it is structured as follows. The next Section
is devoted to some basic notions of operator theory and slice functional
calculus. In Section 3, we discuss some properties of the $S$-eigenvalue of
finite type. Finally, in Section 4, we give and provide the Browder’s
$S$-resolvent equation in quaternionic setting.

##  2\. Mathematical preliminaries

In order to make the paper detailed, we collect some definitions and recall
some results needed in the rest of the paper. We refer to [1, 5, 19, 20] for
surveys on the matter.

###  2.1. Quaternions

We denote by $\mathbb{H}$ the Hamiltonian skew field of quaternions with the
standard basis $\\{1,i,j,k\\}$. Formally, we have

$\displaystyle\mathbb{H}=\Big{\\{}q=x_{0}+x_{1}i+x_{2}j+x_{3}k:\
x_{i}\in\mathbb{R},\ i=0,1,2,3\Big{\\}}.$

The three imaginary units $i,j,k$ satisfy the relations

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

Let $q=x_{0}+x_{1}i+x_{2}j+x_{3}k\in\mathbb{H}$. The real part of $q$ is given
by ${\rm Re}(q)=x_{0}$ and its imaginary part is defined as ${\rm
Im}(q)=x_{1}i+x_{2}j+x_{3}k$, then the conjugate and the usual norm of $q$ are
defined, respectively, by

$\displaystyle\overline{q}={\rm Re}(q)-{\rm Im}(q)\mbox{ and
}|q|=\sqrt{x_{0}^{2}+x_{1}^{2}+x_{2}^{2}+x_{3}^{2}}.$

The unit sphere of purely imaginary quaternions is given by

$\displaystyle\mathbb{S}=\Big{\\{}q\in\mathbb{H}:\ {\rm Re}(q)=0\mbox{ and
}\overline{q}q=1\Big{\\}}.$

It is remarkable that $\mathbb{S}$ is a two-dimensional sphere in
$\mathbb{R}^{4}$. If $q\in\mathbb{H}\setminus\mathbb{R}$, then

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

where $I_{q}=\frac{{\rm Im}(q)}{|{\rm Im}(q)|}\in\mathbb{S}$. In this way we
can associated to $q$ a two-dimensional sphere defined by

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

Note that $[q]$ has center at the real point ${\rm Re}(q)$ and has radius
$|{\rm Im}(q)|$. This sphere $[q]$ coincides with the set $\\{hqh^{-1}:\
h\in\mathbb{H}^{*}\\}$. We refer the reader to [6] for the full proof. For
$I\in\mathbb{S}$, we set

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

In this case, we have

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

###  2.2. Operators acting on right quaternionic Hilbert space

Let $V^{R}_{\mathbb{H}}$ be a right quaternionic Hilbert space and
$\mathcal{O}=\\{\phi_{k}:\ k\in\mathbb{N}\\}$ be an orthonormal subset of
$V^{R}_{\mathbb{H}}$. $\mathcal{O}$ is said to be Hilbert basis of
$V^{R}_{\mathbb{H}}$ if for every $\phi,\psi\in V_{\mathbb{H}}^{R}$, the
series
$\sum_{k\in\mathbb{N}}\langle\phi,\phi_{k}\rangle\langle\phi_{k},\psi\rangle$
converges absolutely and

$\displaystyle\langle\phi,\psi\rangle=\sum_{k\in\mathbb{N}}\langle\phi,\phi_{k}\rangle\langle\phi_{k},\psi\rangle.$

A more detailed discussion about right Hilbert space can be found in [1, 19].
We start by recalling the following classical result whose proof is analogous
to that in the complex version, see [36].

######  Proposition 2.1.

Let $V_{\mathbb{H}}^{R}$ be a separable right quaternionic Hilbert space. We
have the following three assertions

  1. (1)

$V_{\mathbb{H}}^{R}$ admits at least one Hilbert basis.

  2. (2)

If $X$ and $Y$ are two Hilbert basis of $V_{\mathbb{H}}^{R}$, then $\sharp
X=\sharp Y,$ where $\sharp X$ denote the cardinal of $X$.

  3. (3)

If $\mathcal{O}=\\{\phi_{k}:\ k\in\mathbb{N}\\}$ is a Hilbert basis of
$V_{\mathbb{H}}^{R}$, then every $\phi\in V_{\mathbb{H}}^{R}$ can be uniquely
decomposed as follows

$\displaystyle\phi=\sum_{k\in\mathbb{N}}\phi_{k}\langle\phi_{k},\phi\rangle,$

where the series $\sum_{k\in\mathbb{N}}\phi_{k}\langle\phi_{k},\phi\rangle$
converges absolutely in $V_{\mathbb{H}}^{R}$.

Let $\mathcal{O}=\\{\phi_{k}:\ ,k\in\mathbb{N}\\}$ be a Hilbert basis of
$V^{R}_{\mathbb{H}}.$ The left scalar multiplication on $V^{R}_{\mathbb{H}}$
induced by $\mathcal{O}$ is defined as the map

$\displaystyle\mathbb{H}\times V^{R}_{\mathbb{H}}$
$\displaystyle\longrightarrow V^{R}_{\mathbb{H}}$ $\displaystyle(q,\phi)$
$\displaystyle\longrightarrow
q\phi=\sum_{k\in\mathbb{N}}\phi_{k}q\langle\phi_{k},\phi\rangle.$

A function $T:\ V^{R}_{\mathbb{H}}\longrightarrow V^{R}_{\mathbb{H}}$ is said
to be quaternionic right linear if

$\displaystyle T(\phi+\psi q)=T(\phi)+T(\psi)q,$

for all $\phi$, $\psi$ $\in V^{R}_{\mathbb{H}}$ and $q\in\mathbb{H}.$ We call
a quaternionic right operator $T$ is bounded if

$\displaystyle\|T\|\displaystyle=\sup_{\phi\in
V^{R}_{\mathbb{H}}\backslash\\{0\\}}\frac{\|T\phi\|}{\|\phi\|}<+\infty.$

The set of all bounded right operators on $V^{R}_{\mathbb{H}}$ is denoted by
$\mathcal{B}(V^{R}_{\mathbb{H}})$ and the identity operator on
$V^{R}_{\mathbb{H}}$ will be denoted by $\mathbb{I}_{V^{R}_{\mathbb{H}}}.$ If
$T$ $\in\mathcal{B}(V^{R}_{\mathbb{H}})$, then we write $N(T)$ and $R(T)$,
respectively, for the null space and range of $T$. We set

$\alpha(T)=\dim N(T)\mbox{ and }\beta(T)={\rm codim\ }R(T).$

######  Definition 2.2.

[31, 33] Let $T\in\mathcal{B}(V^{R}_{\mathbb{H}})$, then

  1. (1)

$T$ is a Fredholm operator if both $\alpha(T)$ and $\beta(T)$ are finite.

  2. (2)

If $T$ is a Fredholm operator, then the index of $T$ is the number

$\displaystyle i(T)=\alpha(T)-\beta(T).$

  3. (3)

$T$ is a Weyl operator if $T$ is a Fredholm operator and $i(T)=0$.

Let $\Phi(V^{R}_{\mathbb{H}})$ be denote the set of Fredholm operators and
$\mathcal{W}(V^{R}_{\mathbb{H}})$ be denote the set of Weyl operators.

######  Definition 2.3.

[31, 33] Let $T\in\mathcal{B}(V^{R}_{\mathbb{H}})$.

  1. (1)

$T$ is said a finite rank if $\dim R(T)<\infty$.

  2. (2)

$T$ is said compact if $T$ maps bounded set into precompact sets.

We denote by $\mathcal{K}(V^{R}_{\mathbb{H}})$ the set of all compact
operators on $V^{R}_{\mathbb{H}}$. In the sequel of the paper, we equip
$V^{R}_{\mathbb{H}}$ with a Hilbert basis $\mathcal{O}$. In this way,
$\mathcal{B}(V^{R}_{\mathbb{H}})$ is a two-sided ideal quaternionic Banach
algebras with respect to the two multiplications:

$\displaystyle(qT)\phi=\displaystyle\sum_{\psi\in\mathcal{O}}\psi
q\langle\psi,T\phi\rangle\mbox{ and
}(Tq)\phi=\displaystyle\sum_{\psi\in\mathcal{O}}T(\psi)q\langle\psi,\phi\rangle.$

for all $\phi\in V^{R}_{\mathbb{H}}$. In the next proposition we will recall
some well-known properties of the compact and Fredholm-set, see [31, 33].

######  Proposition 2.4.

  1. (1)

$\mathcal{K}(V^{R}_{\mathbb{H}})$ is a closed two-sided ideal of
$\mathcal{B}(V^{R}_{\mathbb{H}}).$

  2. (2)

If $A\in\Phi(V^{R}_{\mathbb{H}})$ and $K\in\mathcal{K}(V^{R}_{\mathbb{H}})$,
then $A+K\in\Phi(V^{R}_{\mathbb{H}})$ and $i(A+K)=i(A)$.

###  2.3. The quaternionic functional calculus

In this subsection, we recall some definitions and basis properties for the
Sabadini spectrum ($S$-spectrum), slice regular functions and Riesz projectors
necessary for development of this manuscript. For more details see [3, 5, 18,
19, 20].

For $T\in\mathcal{B}(V^{R}_{\mathbb{H}})$ and $q\in\mathbb{H}$, we define the
associated operator $\mathcal{Q}_{q}(T):\ V^{R}_{\mathbb{H}}\longrightarrow
V^{R}_{\mathbb{H}}$ by setting

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

######  Definition 2.5.

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

  1. (1)

The $S$-spectrum of $T$ is defined as

$\displaystyle\sigma_{S}(T)=\Big{\\{}q\in\mathbb{H}:\ \mathcal{Q}_{q}(T)\mbox{
is not invertible in }\mathcal{B}(V^{R}_{\mathbb{H}})\Big{\\}}.$

  2. (2)

We define the $S$-resolvent set of $T$ as

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

  3. (3)

The point $S$-spectrum of $T$ is given by

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

The concept of $S$-spectrum is motivated by both the left Cauchy kernel series

$\displaystyle\displaystyle\sum_{n=0}^{+\infty}T^{n}q^{-n-1}=-(T^{2}-2{\rm
Re}(q)T+|q|^{2}\mathbb{I}_{V^{R}_{\mathbb{H}}})^{-1}(T-\overline{q}\mathbb{I}_{V^{R}_{\mathbb{H}}}),\
|q|>\parallel T\parallel$

and the right Cauchy kernel series

$\displaystyle\displaystyle\sum_{n=0}^{+\infty}q^{-n-1}T^{n}=-(T-\overline{q}\mathbb{I}_{V^{R}_{\mathbb{H}}})(T^{2}-2{\rm
Re}(q)T+|q|^{2}\mathbb{I}_{V^{R}_{\mathbb{H}}})^{-1},\ |q|>\parallel
T\parallel.$

We refer to [19] for a full explanation. Note that $\sigma_{S}(T)$ is a non-
empty compact set, see [20]. If $Tu=uq$ for some $u\in
V^{R}_{\mathbb{H}}\backslash\\{0\\}$ and $q\in\mathbb{H}$, then $u$ is called
eigenvector of $T$ with right eigenvalue $q$. We recall that the set of right
eigenvalue coincides with point $S$-spectrum $\sigma_{S}(T)$, see [23, Theorem
2.5].

######  Definition 2.6.

A set $\Omega\subset\mathbb{H}$ is called
$(i)$ axially symmetric if $\\{hqh^{-1}:\ h\in\mathbb{H}\\}\subset\Omega$ for
any $q\in\Omega$ and
$(ii)$ a slice domain (or $s$-domain for short) if $\Omega$ is open,
$\Omega\cap\mathbb{R}\neq\emptyset$ and $\Omega\cap\mathbb{C}_{I}$ is a domain
in $\mathbb{C}_{I}$, for any $I\in\mathbb{S}$.

Note that The $S$-spectrum $\sigma_{S}(T)$ and the $S$-resolvent $\rho_{S}(T)$
are axially symmetric, see [19].

######  Definition 2.7.

[19, Definition 2.1.2] (Slice hyperholomorphic functions) Let
$\Omega\subset\mathbb{H}$ be an axially symmetric open set and $f:\
\Omega\longrightarrow\mathbb{H}$ be a function. Set

$\displaystyle\Omega_{\mathbb{R}^{2}}:=\Big{\\{}(u,v)\in\mathbb{R}^{2}:\
u+Iv\in\Omega\mbox{, for all }I\in\mathbb{S}\Big{\\}}.$

We say that $f$ is a left slice hyperholomorphic function if it is of the form

$\displaystyle f(q)=P(u,v)+I_{q}Q(u,v)\mbox{, for }q=u+I_{q}v\in\Omega$

with $P$,$Q$ take value in $\mathbb{H}$ such that

$P(u,-v)=P(u,v),\ Q(u,-v)=-Q(u,v)$ (2)

and satisfy the Cauchy Riemann equation

$\displaystyle\frac{\partial P}{\partial u}-\frac{\partial Q}{\partial v}=0,\
\frac{\partial P}{\partial v}+\frac{\partial Q}{\partial u}=0.$ (3)

We say that $f$ is a right hyperholomorphic function if it is of the form

$\displaystyle f(q)=P(u,v)+Q(u,v)I_{q}\mbox{ for }q=u+I_{q}v\in\Omega$

with $P$,$Q$ $:\ \Omega_{\mathbb{R}^{2}}\longrightarrow\mathbb{H}$ satisfy
$(\ref{r1})$ and $(\ref{r2})$.
If $f$ is left or right with $P(u,v),\ Q(u,v)\in\mathbb{R}$ for all
$(u,v)\in\Omega_{\mathbb{R}^{2}}$, then $f$ is said intrinsic function.

Let $\mathcal{SH}_{L}(\Omega)$ $($resp. $\mathcal{SH}_{R}(\Omega))$ be denote
the set of left $($resp. right) slice hyperholomorphic functions on $\Omega$
and $\mathcal{N}(\Omega)$ be denotes the set of intrinsic functions. This
class of functions is a generalization of the set of holomorphic functions in
the complex setting.

######  Definition 2.8.

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

$\displaystyle S_{L}^{-1}(q,T):=-(T^{2}-2{\rm
Re}(q)T+|q|^{2}\mathbb{I}_{V^{R}_{\mathbb{H}}})^{-1}(T-\overline{q}\mathbb{I}_{V_{\mathbb{H}}^{R}}),$

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

$\displaystyle
S_{R}^{-1}(q,T):=-(T-\overline{q}\mathbb{I}_{V_{\mathbb{H}}^{R}})(T^{2}-2{\rm
Re}(q)T+|q|^{2}\mathbb{I}_{V^{R}_{\mathbb{H}}})^{-1}.$

######  Proposition 2.9.

[19, Lemma 3.1.11] The left $S$-resolvent operator $q\longmapsto
S_{L}^{-1}(q,T)$ is right slice hyperholomorphic and the right $S$-resolvent
operator $q\longrightarrow S_{R}^{-1}(q,T)$ is left slice hyperholomorphic.

Let $\mathcal{SH}_{L}(\sigma_{S}(T))$, $\mathcal{SH}_{R}(\sigma_{S}(T)))$ and
$\mathcal{N}(\sigma_{S}(T))$ be denote, respectively, the set of all $f$ left,
right and intrinsic slice hyperholomorphic functions $f$ such that
$\sigma_{S}(T)\subset\mathcal{D}(f)$, where $\mathcal{D}(f)$ denote the domain
of $f$.

######  Remark 2.10.

[19, Remark 3.2.4] Let
$f\in\mathcal{SH}_{L}(\sigma_{S}(T))\cup\mathcal{SH}_{R}(\sigma_{S}(T)))\cup\mathcal{N}(\sigma_{S}(T))$,
then there exists a bounded slice Cauchy domain $\Omega$ such that

$\displaystyle\sigma_{S}(T)\subset\Omega\mbox{ and
}\overline{\Omega}\subset\mathcal{D}(f).$

Now, we can give the version of the quaternionic functional calculus.

######  Definition 2.11.

[19, Definition 3.2.5] Let $T\in\mathcal{B}(V_{\mathbb{H}}^{R})$. We define

$f(T)=\displaystyle\frac{1}{2\pi}\int_{\partial(\Omega\cap\mathbb{C}_{I})}S_{L}^{-1}(q,T)dq_{I}f(q),\
\forall f\in\mathcal{SH}_{L}(\sigma_{S}(T))$ (4)

and

$f(T)=\displaystyle\frac{1}{2\pi}\int_{\partial(\Omega\cap\mathbb{C}_{I})}f(q)dq_{I}S_{R}^{-1}(q,T),\
\forall f\in\mathcal{SH}_{R}(\sigma_{S}(T))$ (5)

where $dq_{I}=-dqI$ and $\Omega$ is a slice Cauchy domain as in the remark
2.10.

######  Theorem 2.12.

$($Riesz’s projectors$)$[19, Theorem 4.1.5] Let
$T\in\mathcal{B}(V^{R}_{\mathbb{H}})$ and assume that
$\sigma_{S}(T)=\sigma_{1}\cup\sigma_{2}$ with

$\displaystyle dist(\sigma_{1},\sigma_{2})>0.$

Let $\mathcal{O}$ be an open axially symmetric set with
$\sigma_{1}\subset\mathcal{O}$ and
$\overline{\mathcal{O}}\cap\sigma_{2}=\emptyset.$ We define
$\chi_{\sigma_{1}}(s)=1$ for $s\in\mathcal{O}$ and $\chi_{\sigma_{2}}(s)=0$
for $s\notin\mathcal{O},$ Then,
$\chi_{\sigma_{1}}\in\mathcal{N}(\sigma_{S}(T)),$ and

$\displaystyle
P_{\sigma_{1}}:=\chi_{\sigma_{1}}(T)=\frac{1}{2\pi}\int_{\partial(\mathcal{O}\cap\mathbb{C}_{I})}S^{-1}_{L}(q,T)dq_{I}.$

Further, $P_{\sigma_{1}}$ is a continuous projection operator that commute
with $T$ and $P_{\sigma_{1}}V^{R}_{\mathbb{H}}$ is a right linear subspace of
$V^{R}_{\mathbb{H}}$ that is invariant under $T$.

######  Theorem 2.13.

[19, Lemma 4.1.1] Let T $\in\mathcal{B}(V^{R}_{\mathbb{H}}),$ then

  1. (1)

If $f$, $g$ $\in\mathcal{SH}_{L}(\sigma_{S}(T))$ and $q\in\mathbb{H},$ then

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

  2. (2)

If $f$, $g$ $\in\mathcal{SH}_{R}(\sigma_{S}(T))$ and $q\in\mathbb{H},$ then

$\displaystyle(f+g)(T)=f(T)+g(T)\ and\ (qf)(T)=qf(T).$

######  Theorem 2.14.

(The spectral mapping theorem) [19, Theorem 4.2.1]
Let $T\in\mathcal{B}(V^{R}_{\mathbb{H}})$ and
$f\in\mathcal{N}(\sigma_{S}(T)),$ then

$\displaystyle\sigma_{S}(f(T))=f(\sigma_{S}(T)).$

######  Remark 2.15.

Let $P_{\sigma}$ be a Riesz projector associated to the spectral set $\sigma,$
then

$\displaystyle qP_{\sigma}=P_{\sigma}q,\mbox{ for all }q\in\mathbb{H}.$

In particular $R(P_{\sigma})$ is a left linear subspace of
$V^{R}_{\mathbb{H}}.$ Indeed,

$\displaystyle(qP_{\sigma})(T)=qP_{\sigma}(T)=q\chi_{\sigma}(T)=(\chi_{\sigma}q)(T)=\chi_{\sigma}(T)q=P_{\sigma}q.$

##  3\. Eigenvalue of finite type

Let

$\displaystyle\mathbb{R}_{+}^{2}:=\\{(x,y)\in\mathbb{R}^{2}:\
y\in\mathbb{R}_{+}\\}.$

We consider the following equation

$\displaystyle\Psi:\mathbb{H}\longrightarrow\mathbb{R}_{+}^{2}$
$\displaystyle\quad\quad q\longmapsto\ \ ({\rm Re}(q),|{\rm Im(q)}|).$

We refer to [24] for more properties of $\Psi$. In particular, the author
prove that $\Psi$ is continuous, open, closed and

$\displaystyle[\Omega]=\Psi^{-1}(\Psi(\Omega))\mbox{ for all
}\Omega\subset\mathbb{H},$

see [24, Corollary 3.16 and Lemma 3.18].
Let $T\in\mathcal{B}(V_{\mathbb{H}}^{R})$. A subset
$\sigma\subset\sigma_{S}(T)$ is called an isolated part of $\sigma_{S}(T)$ if
both $\sigma$ and $\sigma_{S}(T)\backslash\sigma$ are closed subsets of
$\sigma_{S}(T)$. We start with the following result:

######  Proposition 3.1.

Let $T\in\mathcal{B}(V_{\mathbb{H}}^{R})$ and $q\in\sigma_{S}(T)$, then $[q]$
is an isolated part of $\sigma_{S}(T)$ if, and only if, there exist
$\varepsilon>0$ such that

$[B(q,\varepsilon)]\cap\sigma_{S}(T)=[q],$ (6)

where $B(q,\varepsilon)$denote the open boule of center $q$ and radius
$\varepsilon.$

###### Proof.

If $[q]$ is an isolated $2$-sphere of $\sigma_{S}(T)$, then $[q]$ is an open
set of $\sigma_{S}(T)$. Let $U_{q}$ be an open set of $\mathbb{H}$ such that

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

Since $q\in U_{q}$, then there exists $\varepsilon>0$ such that
$B(q,\varepsilon)\subset U_{q}$. This implies that

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

Indeed, assume that there exists
$p\in[B(q,\varepsilon)]\cap\sigma_{S}(T)\backslash[q]$. So, $p\in[q^{\prime}]$
for some $q^{\prime}\in U_{q}\backslash[q]$. Since $\sigma_{S}(T)$ is axially
symmetric, then $q^{\prime}\in\sigma_{S}(T)$, contradiction.
Conversely, if $(\ref{e})$ is satisfied, then $[q]$ is open in
$\sigma_{S}(T)$. Since $[q]$, is closed, we deduce that $[q]$ is an isolated
part of $\sigma_{S}(T)$. ∎

We recall that in a complex setting, the eigenvalue of finite type is
introduced and studied in [26]. In particular, the authors gave a
characterization of this type of spectrum by using the Riesz projection. A
version in the quaternionic case is introduced in [11] in the following
definition.

######  Definition 3.2.

Let $T\in\mathcal{B}(V_{\mathbb{H}}^{R})$. A point $q\in\sigma_{S}(T)$ is
called a $S$-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

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

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

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

We start by recalling the following decomposition theorem:

######  Theorem 3.3.

[2, Theorem 4.4] Let $T\in\mathcal{B}(V_{\mathbb{H}}^{R})$. Suppose that
$P_{1}$ is a projector in $\mathcal{B}(V_{\mathbb{H}}^{R})$ commuting with $T$
and set $P_{2}=\mathbb{I}_{V_{\mathbb{H}}^{R}}-P_{1}$. Let
$V_{j}=P_{j}(V_{\mathbb{H}}^{R})$, $j=1,2$, and define the operators
$T_{j}=TP_{j}=P_{j}T$. Denote by $\widetilde{T}_{j}:=T_{j}\mid_{V_{j}}$,
$j=1,2$, then

$\displaystyle\sigma_{S}(T)=\sigma(\widetilde{T}_{1})\cup\sigma(\widetilde{T}_{2}).$

######  Remark 3.4.

Let $T\in\mathcal{B}(V_{\mathbb{H}}^{R})$ and assume that $[q]$ is an isolated
part of $\sigma_{S}(T)$. By using Theorem 3.3 and [20, Theorem 3.7.8], we have

$\displaystyle\sigma_{S}(T)=[q]\cup\sigma_{S}(T(\mathbb{I}_{V_{\mathbb{H}}^{R}}-P_{[q]}))\mbox{
and }[q]\cap\sigma_{S}(T(\mathbb{I}_{V_{\mathbb{H}}^{R}}-P_{[q]}))=\emptyset,$

where $P_{[q]}$ is the Riesz projection related to $[q]$ and $T$.

We recall:

######  Theorem 3.5.

[11, Theorem 3.10] Let $T\in\mathcal{B}(V_{\mathbb{H}}^{R})$ and $[q]$ be an
isolated part of $\sigma_{S}(T)$, then $q$ is a right eigenvalue of finite
type if and only if $\dim R(P_{[q]})<\infty$.

We turn to the pseudo $S$-resolvent operator

$\displaystyle\mathcal{Q}_{q}(T)^{-1}:=(T^{2}-2{\rm
Re}(q)T+|q|^{2}\mathbb{I}_{V_{\mathbb{H}}^{R}})^{-1},\ q\in\rho_{S}(T).$

As in complex case, one generalize this concept by using the Riesz projection.
Let $\sigma_{d}^{S}(T)$ be denote the set of all $S$-eigenvalues of
$T\in\mathcal{B}(V_{\mathbb{H}}^{R})$ of finite type. By using [20, Theorem
3.7.8], we have

$\displaystyle\sigma_{S}(T\mid_{R(P_{[q]})})=[q]\mbox{ and
}\sigma_{S}(T\mid_{N(P_{[q]})})=\sigma_{S}(T)\backslash[q]$

for all $q\in\sigma_{d}^{S}(T)$. Thus, $q\in\rho_{S}(T\mid_{N(P_{[q]})})$ for
every $q\in\sigma_{d}^{S}(T)$. This allows us to extend the pseudo
$S$-resolvent operator. More precisely, set

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

If $q\in\rho_{B,S}(T)$, then the operator

$\displaystyle P\mathcal{Q}_{q}(T):=(T^{2}-2{\rm
Re}(q)T+|q|^{2}\mathbb{I}_{V_{\mathbb{H}}^{R}})(\mathbb{I}_{V_{\mathbb{H}}^{R}}-P_{[q]})+P_{[q]}$

is invertible and its inverse is given by

$\displaystyle PR_{B,S}(q,T)=((T^{2}-2{\rm
Re}(q)T+|q|^{2}\mathbb{I}_{V_{\mathbb{H}}^{R}})\mid_{N(P_{[q]})})^{-1}(\mathbb{I}_{V_{\mathbb{H}}^{R}}-P_{[q]})^{-1}+P_{[q]}.$

Using this new concept, we prove the following result.

######  Proposition 3.6.

Let $T\in\mathcal{B}(V^{R}_{\mathbb{H}})$. If $q\in\sigma^{S}_{d}(T)$ and
$x\in V^{R}_{\mathbb{H}}$, then

$\displaystyle x\in R(P_{[q]})\mbox{ if and only if
}\lim_{n\longrightarrow+\infty}\|\mathcal{Q}_{q}^{n}(T)x\|^{\frac{1}{n}}=0,$

where $P_{[q]}$ is the Riesz projection related to $[q]$ and $T$.

###### Proof.

Using the Riesz decomposition [34, Theorem 6], we have

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

Set

$\displaystyle\mathcal{Q}_{q}(X)=X^{2}-2Re(q)X+|q|^{2}.$

Observe that
$X\longmapsto\mathcal{Q}_{q}(X)\in\mathcal{N}(\sigma_{S}(TP_{[q]}\mid_{R(P_{[q]})}))$.
On the other hand, the polynomial $\mathcal{Q}_{q}$ vanishes exactly at $[q]$,
see [6, Lemma 4.2.3]. Now, by the $S$-spectral mapping theorem 2.14, we have

$\displaystyle\sigma_{S}(\mathcal{Q}_{q}(T)P_{[q]}\mid_{R(P_{[q]})})=\mathcal{Q}_{q}(\sigma_{S}(TP_{[q]}\mid_{R(P_{[q]})}))=\mathcal{Q}_{q}([q])=\\{0\\}.$

In particular, for $0\neq x\in R(P_{[q]})$, we have

$\displaystyle\lim_{n\longrightarrow+\infty}\|\mathcal{Q}_{q}^{n}(T)x\|^{\frac{1}{n}}$
$\displaystyle=\lim_{n\longrightarrow+\infty}\|\mathcal{Q}_{q}^{n}(T)P_{[q]}\mid_{R(P_{[q]})}x\|^{\frac{1}{n}}$
$\displaystyle\leq\lim_{n\longrightarrow+\infty}\|Q_{q}^{n}P_{[q]}\mid_{R(P_{[q]})}\|^{\frac{1}{n}}\|x\|^{\frac{1}{n}}$
$\displaystyle=r_{S}(\mathcal{Q}_{q}(T)P_{[q]}\mid_{R(P_{[q]})})=0.$

Conversely, set $0\neq x\in V^{R}_{\mathbb{H}}$ such that
$\lim_{n\longrightarrow+\infty}\|\mathcal{Q}_{q}^{n}(T)x\|^{\frac{1}{n}}=0$.
Take

$\displaystyle
x_{n}:=P\mathcal{Q}_{q}^{n}(T)x=(\mathcal{Q}_{q}(T))^{n}(\mathbb{I}_{V_{\mathbb{H}}^{R}}-P_{[q]})x+P_{[q]}x{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}.}$

It is clear that

$\displaystyle\|(\mathbb{I}_{V_{\mathbb{H}}^{R}}-P_{[q]})x_{n}\|^{\frac{1}{n}}\leq\|(\mathbb{I}_{V_{\mathbb{H}}^{R}}-P_{[q]})\|\|(\mathcal{Q}_{q}(T))^{n}x\|^{\frac{1}{n}}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}.}$

In this way, we say that

$\displaystyle\lim_{n\longrightarrow+\infty}\|(I-P_{[q]})x_{n}\|^{\frac{1}{n}}=0.$

Since $q\in\rho^{S}_{B}(T)$, then

$\displaystyle
x=PR_{B,S}^{n}(q,T)x_{n}=(\mathcal{Q}_{q}(T)\mid_{N(P_{[q]})})^{-n}(\mathbb{I}_{V_{\mathbb{H}}^{R}}-P_{[q]})x_{n}+P_{[q]}x_{n}.$

Finally, we obtain

$\displaystyle\|(\mathbb{I}_{V_{\mathbb{H}}^{R}}-P_{[q]})x\|^{\frac{1}{n}}\leq\|(\mathbb{I}_{V_{\mathbb{H}}^{R}}-P_{[q]})x_{n}\|^{\frac{1}{n}}\times\|(\mathcal{Q}_{q}(T)P_{[q]}|_{N(P_{[q]})})^{-1}\|.$

This implies that
$\lim_{n\longrightarrow+\infty}\|(I-P_{[q]})x\|^{\frac{1}{n}}=0$ and so $x\in
R(P_{[q]})$.∎

We recall that $T$ is quasi-nilpotent if $\sigma_{S}(T)=\\{0\\}.$

######  Proposition 3.7.

Let $T\in\mathcal{B}(V^{R}_{\mathbb{H}})$ and $[q]$ be an isolated $2$-sphere
of $\sigma_{S}(T).$ Then,

  1. (1)

$\mathcal{Q}_{q}(T)+P_{[q]}$ and
$\mathcal{Q}_{q}(T)+[2T+(1-2Re(q))\mathbb{I}_{V^{R}_{\mathbb{H}}}]P_{[q]}$ are
invertible,

  2. (2)

$\mathcal{Q}_{q}(T)P_{[q]}$ and
$\mathcal{Q}_{q}(T)[2T+(1-2Re(q))\mathbb{I}_{V^{R}_{\mathbb{H}}}]P_{[q]}$ are
quasi-nilpotent.

###### Proof.

$(1)$ Since $[q]$ is an isolated $2$-sphere of $\sigma_{S}(T)$, then there
exist $\varepsilon>0$ such that
$[\overline{B}(q,\varepsilon)]\cap\sigma_{S}(T)=[q]$. Set

$\displaystyle U:=[B(q,\varepsilon)]\mbox{ and
}V:=\mathbb{H}\backslash[\overline{B}(q,\varepsilon)].$

Observe that $U$ and $V$ are two axially symmetric open sets,

$\displaystyle U\cap V=\emptyset,\ [q]\subset U\mbox{ and
}\sigma_{S}(T)\setminus[q]\subset V.$

Let us define the functions

$\displaystyle g(p):=\left\\{\begin{array}[]{rl}\noindent 1&\mbox{for }p\in
U,\\\ \nobreak\leavevmode\nobreak\leavevmode\hfil\\\ \displaystyle 0&\mbox{for
}p\in V.\end{array}\right.$

and

$\displaystyle h(p):=p^{2}-2Re(q)p+|q|^{2},\ p\in\mathbb{H}.$

Then

$\displaystyle g(T)=P_{[q]}\mbox{ and }h(T)=\mathcal{Q}_{q}(T).$

Recall that $h(p)=0$ if, and only, if $p\in[q]$, see [6, Lemma 4.2.3]. So,
$(g+h)(p)\neq 0$ for all $p\in\sigma_{S}(T)$. Indeed, if $p\in[q]\subseteq U$,
then $g(p)=1$ and $h(p)=0$. If $p\in\sigma_{S}(T)\backslash[q]$, then $g(p)=0$
and $h(p)\neq 0.$ Now, by using the algebraic properties of the quaternionic
functional calculus [19, Lemma 4.1.1], we have

$\displaystyle\mathcal{Q}_{q}(T)+P_{[q]}=(g+h)(T).$

Finally, since $g+h\in\mathcal{N}(\sigma_{S}(T)),$ then thanks to the
$S$-spectral mapping theorem 2.14, we conclude that
$\mathcal{Q}_{q}(T)+P_{[q]}$ is an invertible operator.
We turn to the operator
$\mathcal{Q}_{q}(T)+[2T+(1-2Re(q))\mathbb{I}_{V^{R}_{\mathbb{H}}}]P_{[q]}$. We
consider the function

$\displaystyle k(p):=\left\\{\begin{array}[]{rl}\noindent 2p+1-2Re(q)&\mbox{if
}p\in U,\\\ \nobreak\leavevmode\nobreak\leavevmode\hfil\\\ 0&\mbox{if }p\in
V.\end{array}\right.$

A similar argument as before, we have

$\displaystyle k(T)=[2T+(1-2Re(q))\mathbb{I}_{V^{R}_{\mathbb{H}}}]P_{[q]}$

and

$\displaystyle(h+k)(T)=\mathcal{Q}_{q}(T)+[2T+(1-2Re(q))\mathbb{I}_{V^{R}_{\mathbb{H}}}]P_{[q]}.$

On the other hand, we have $(k+h)(p)\neq 0$ for all $p\in\sigma_{S}(T)$.
Indeed, if $p={\rm Re}(q)+I_{p}|{\rm Im}(q)|$ for some $I_{p}\in\mathbb{S}$
(i.e, $p\in[q]$), then $h(p)=0$ (by using [6, Lemma 4.2.3]) and

$\displaystyle k(p):$ $\displaystyle=1-2Re(q)+2p$ $\displaystyle=1+2I_{q}|{\rm
Im}(q)|\neq 0.$

Now, if $p\in\sigma(T)\backslash[q]$, then we have easily $h(p)\neq\\{0\\}$
and $k(p)=0$. Finally, we can conclude that
$\mathcal{Q}_{q}(T)+[1-2Re(T)+2T]P_{[q]}$ is an invertible operator.
$(2)$ Since $(hg)(p)=h(p)g(p)=0$ and $(hk)(p)=0$ for all $p\in\sigma_{S}(T)$,
then by using [19, Lemma 3.2.8] and the $S$-spectral mapping theorem 2.14, we
have

$\displaystyle\sigma_{S}(Q(T)P_{[q]})=\sigma_{S}(h(T)g(T))=\\{0\\}$

and

$\displaystyle\sigma_{S}(\mathcal{Q}_{q}(T)[2T+(1-2Re(q))\mathbb{I}_{V^{R}_{\mathbb{H}}}]P_{[q]})=\sigma_{S}(h(T)k(T))=\\{0\\}.$

This completes the proof.∎

######  Proposition 3.8.

Let $T\in\mathcal{B}(V^{R}_{\mathbb{H}})$ and $[q]$ be an isolated $2$-sphere
of $\sigma_{S}(T).$ Then,

$\displaystyle q\in\sigma_{d}^{S}(T)\mbox{ if and only if
}\mathcal{Q}_{q}(T)\in\mathcal{W}(V^{R}_{\mathbb{H}}).$

###### Proof.

Suppose that $q\in\sigma_{d}^{S}(T)$. By using the previous proposition 3.7,
we have $\mathcal{Q}_{q}(T)+P_{[q]}$ is an invertible operator. In this way,
we see that $\mathcal{Q}_{q}(T)+P_{[q]}\in\mathcal{W}(V^{R}_{\mathbb{H}})$.
Since $P_{[q]}\in\mathcal{K}(V^{R}_{\mathbb{H}})$, then thanks to proposition
2.4 we deduce that $\mathcal{Q}_{q}(T)\in\mathcal{W}(V^{R}_{\mathbb{H}})$.

Conversely, Let $\varepsilon>0$ such that
$\sigma_{S}(T)\cap[B(q,\varepsilon)]=[q]$, then the Riesz projection $P_{[q]}$
associated with $T$ and $q$ is given by

$\displaystyle P_{[q]}$
$\displaystyle=\displaystyle\frac{-1}{2\pi}\int_{\partial([B(q,\varepsilon)]\cap\mathbb{C}_{I})}\mathcal{Q}_{p}^{-1}(T)(T-\overline{p}\mathbb{I}_{V^{R}_{\mathbb{H}}})dp_{I}.$

Now, denote by $\pi$ the natural quotient map into the Calkin algebra
$\mathcal{C}(V_{\mathbb{H}^{R}})=\mathcal{B}(V_{\mathbb{H}^{R}})/\mathcal{K}(V_{\mathbb{H}^{R}})$,
then $q\in\rho_{S}(\pi(T))$ and

$\displaystyle\pi(P_{[q]})=\displaystyle\frac{-1}{2\pi}\int_{\partial([B(q,\varepsilon)]\cap\mathbb{C}_{I})}\mathcal{Q}_{p}^{-1}(\pi(T))(\pi(T)-\overline{p}\mathbb{I}_{\mathcal{C}(V_{\mathbb{H}^{R}})})dp_{I}=0.$

So, $P_{[q]}\in\mathcal{K}(V^{R}_{\mathbb{H}})$. This implies that
$\mathbb{I}_{R(P_{[q]})}:\ R(P_{[q]})\longrightarrow R(P_{[q]})$ is compact,
we deduce that $\dim(R(P_{[q]}))<\infty$. ∎

######  Remark 3.9.

As in the complex setting, we have if $T\in\mathcal{B}(V^{R}_{\mathbb{H}})$ is
invertible and $N$ is a

nilpotent operator that commute with $T$, then $T+N$ is also invertible.
Indeed, let $m\in\mathbb{N}^{*}$ such that $N^{m}=0$. Then,
$\mathbb{I}_{V^{R}_{\mathbb{H}}}+N$ is invertible and its inverse is given by

$\displaystyle(I+N)^{-1}=\displaystyle\sum_{k=0}^{m-1}(-1)^{k}N^{k}.$

Since $TN=NT$, then $T^{-1}N$ is nilpotent. In this way, we see that
$T+N=T(\mathbb{I}_{V^{R}_{\mathbb{H}}}+T^{-1}N)$ is invertible.

######  Theorem 3.10.

Let $T_{n}$ and $T$ be belong to $\mathcal{B}(V_{\mathbb{H}}^{R})$ with
$n\in\mathbb{N}$ and $\|T_{n}-T\|\longrightarrow 0.$ We suppose that
$0\in\sigma_{d}^{S}(T).$ For an axially symmetric $V\subset\mathbb{H},$ we set

$\displaystyle E_{T}^{V\bigcap\sigma_{S}(T)}=(V\cap\sigma_{S}(T))/\cong,$

where $p\cong q$ if, and only, if $p\in[q]$, then there exist $N\in\mathbb{N}$
and an open axially symmetric $V_{0}\subset\mathbb{H}$ such that $\sharp
E_{T_{n}}^{V_{0}\cap\sigma_{S}(T_{n})}<\infty$ and
$V_{0}\bigcap\sigma_{S}(T_{n})\subset\sigma_{d}^{S}(T_{n})$ for all $n\geq N.$

To prove this theorem, we first need to show the following results.

######  Lemma 3.11.

Let $T$ and $S$ $\in Inv(\mathcal{B}(V^{R}_{\mathbb{H}}))\ (i.e.\
0\in\rho_{S}(T)\cap\rho_{S}(S)).$ We assume that
$\|T-S\|\leq\frac{1}{2}\|S^{-1}\|^{-1},$ then

$\displaystyle\|T^{-1}-S^{-1}\|\leq 2\|S^{-1}\|^{2}\|T-S\|.$

###### Proof.

The proof is exactly similar to the proof of [14, Lemma 5, p.11] in the
complex setting.

######  Definition 3.12.

[14, Definition 15, p.25] Let $X$, $Y$ be two topological spaces and let
$\phi$ be a function defined on the space X and whose values are subsets of
the space Y. The mapping $\phi$ is upper semi-continuous on $x_{0}$ if for
each neighborhood $V_{\phi(x_{0})}$ of $\phi(x_{0})$, there exist a
neighborhood $U_{x_{0}}$ of $x_{0}$ such that

$\phi(x)\subset V_{\phi(x_{0})},\ x\in U_{x_{0}}.$

$\phi$ is said to be upper semi-continuous if x is a point of upper semi-
continuity for $\phi$ for each $x\in X.$

######  Lemma 3.13.

[14, Lemma 16, p.25] Let $X$, $Y$ be metric spaces, let $Y$ be compact and let
$\phi$ be a mapping of $X$ into the closed subsets of $Y$, then $\phi$ is
upper semi-continuous if and only if the following conditions holds

$\displaystyle x_{n}\in X,\ y_{n}\in\phi(x_{n}),\
x=\lim_{n\longrightarrow+\infty}x_{n},\ y=\lim_{n\longrightarrow+\infty}y_{n}\
\Longrightarrow y\in\phi(x).$

######  Proposition 3.14.

Let $\psi_{\mathcal{B}(V^{R}_{\mathbb{H}})}$: $T\longrightarrow\sigma_{S}(T)$
be the function defined on the space $\mathcal{B}(V^{R}_{\mathbb{H}})$ and
whose values are in the compact subset of $\mathbb{H}.$ Then,
$\psi_{\mathcal{B}(V^{R}_{\mathbb{H}})}$ is upper semi-continuous.

###### Proof.

Let $A\in\mathcal{B}(V^{R}_{\mathbb{H}})$ and $R>\|A\|.$ We set

$\displaystyle X_{A}:=\Big{\\{}B\in\mathcal{B}(V^{R}_{\mathbb{H}}):\|B\|\leq
R\Big{\\}}.$

and

$\displaystyle Y_{A}:=\Big{\\{}q\in\mathbb{H}:|q|\leq R\Big{\\}}.$

First, if
$\psi_{\mathcal{B}(V^{R}_{\mathbb{H}})}\mid_{X_{A}}:X_{A}\longrightarrow\\{\mbox{
the compact subset of }Y_{A}\\}$ is upper semi-continuous, then
$\psi_{B(V^{R}_{\mathbb{H}})}$ is upper semi-continuous on $A$. Secondly, let
$(T_{n})_{n}$ be a sequence of operators in $X_{A}$,
$q_{n}\in\sigma_{S}(T_{n})$,

$\lim_{n\longrightarrow+\infty}\|T_{n}-T\|=0\mbox{ and
}\lim_{n\longrightarrow+\infty}|q_{n}-q|=0$

We have to show that $q\in\sigma_{S}(T).$ Indeed, we assume that
$\mathcal{Q}_{q}(T)\in Inv(\mathcal{B}(V^{R}_{\mathbb{H}})),$ then

$\mathcal{Q}_{q}(T)=\lim_{n\longrightarrow+\infty}\mathcal{Q}_{q_{n}}(T_{n}).$

In fact,

$\displaystyle\|\mathcal{Q}_{q}(T)-\mathcal{Q}_{q_{n}}(T_{n})\|\leq\|T_{n}^{2}-T^{2}\|+\|2Re(q_{n})T_{n}-2Re(q)T\|+||q_{n}|^{2}-|q|^{2}|.$

In this way, we see that

$\displaystyle\|\mathcal{Q}_{q}(T)-\mathcal{Q}_{q_{n}}(T_{n})\|\longrightarrow
0\mbox{ since }T\longrightarrow T^{2}\mbox{ is continuous }.$

Let $\varepsilon>0$ be such that $B(\mathcal{Q}_{q}(T),\varepsilon)\subset
Inv(\mathcal{B}(V^{R}_{\mathbb{H}})).$ Then, there exist $N_{q}\in\mathbb{N}$
such that $\mathcal{Q}_{q_{n}}(T_{n})\in B(\mathcal{Q}_{q}(T),\varepsilon)$
for all $n\geq N_{q}.$ This is a contradiction since
$q_{n}\in\sigma_{S}(T_{n})$ for all $n$ $\in\mathbb{N}.$ ∎

######  Lemma 3.15.

Let $P$ and $Q$ be two projections in $\mathcal{B}(V^{R}_{\mathbb{H}}).$ We
assume that $\|P-Q\|<1$, then

  1. (1)

$R(P)\cong R(Q).$

  2. (2)

The operator
$T=QP+(\mathbb{I}_{V^{R}_{\mathbb{H}}}-Q)(\mathbb{I}_{V^{R}_{\mathbb{H}}}-P)$
is bijective.

  3. (3)

$T(R(P))\subset R(Q)$ and $T(N(P))\subset N(Q).$

###### Proof.

The proof is exactly similar to the proof of [35, Theorem 12.4] in the complex
setting. ∎

######  Lemma 3.16.

[20, Lemma 3.1.3] Let $T\in\mathcal{B}(V^{R}_{\mathbb{H}}).$ The functions
$q\longrightarrow\mathcal{Q}_{q}(T)^{-1}$ and $q\longrightarrow
T\mathcal{Q}_{q}(T)^{-1}$ which are defined on $\rho_{S}(T)$ and take values
in $\mathcal{B}(V^{R}_{\mathbb{H}})$ are continuous.

Proof of Theorem 3.10. Let $\varepsilon$ $\in]0,1[$ and
$\mathcal{O}\subset\mathbb{H}$ be an open axially symmetric subset with
$\mathcal{O}\cap\overline{B}(0,\varepsilon)=\emptyset$ and
$\sigma_{S}(T)\backslash\\{0\\}\subset\mathcal{O}.$ By using Lemma 3.16 for
all I $\in\mathbb{S},$ there is $M_{I}\geq 1$ such

$\displaystyle\sup_{q\in\partial(\mathbb{C}_{I}\bigcap
B(0,\varepsilon))}\|\mathcal{Q}_{q}(T)^{-1}\|\leq M_{I}.$

On the other hand, $\mathcal{O}\cup B(0,\varepsilon)$ is a neighborhood of
$\sigma_{S}(T).$ By using Proposition 3.14, there exist $N_{\varepsilon}>0$
such that

$\displaystyle\sigma_{S}(T_{k})\subset B(0,\varepsilon)\cup\mathcal{O}$

for all $k\geq N_{\varepsilon}$. We choose $N_{\varepsilon}$ large enough such
that

$\displaystyle\|T^{2}_{k}-T^{2}+2Re(q)(T_{k}-T)\|\|T-\overline{q}\mathbb{I}_{V^{R}_{\mathbb{H}}}\|+\|T-T_{k}\|\leq\frac{1}{4M^{2}_{I}},$

for all $q\in\partial(\mathbb{C}_{I}\bigcap B(0,\varepsilon))$.

In view of Lemma 3.11, we have

$\displaystyle\|\mathcal{Q}_{q}(T)^{-1}-\mathcal{Q}_{q}(T_{k})^{-1}\|\leq
2M^{2}_{I}\|T^{2}_{k}-T^{2}+2Re(q)(T_{k}-T)\|,$

$q\in\partial(\mathbb{C}_{I}\bigcap B(0,\varepsilon))$. In this way, we see
that

$\displaystyle\|S_{L}^{-1}(q,T)-S_{L}^{-1}(q,T_{k})\|$ $\displaystyle\leq
M_{I}\|T-T_{k}\|+\|\mathcal{Q}_{q}^{-1}(T)-\mathcal{Q}_{q}^{-1}(T_{k})\|\|T-\overline{q}\mathbb{I}_{V^{R}_{\mathbb{H}}}\|\|$
$\displaystyle\leq 2M_{I}^{2}\times\frac{1}{4M_{I}^{2}}=\frac{1}{2}<1.$

Let $P_{0}$ be the Riesz projection associated to $0$ and $T$. Set

$P_{\sigma_{N_{\varepsilon}}^{k}}:=\frac{1}{2\pi}\int_{\partial(B(0,\varepsilon)\bigcap\mathbb{C_{I}})}S_{L}^{-1}(s,T_{k})ds_{I},$

where
$\sigma_{N_{\varepsilon}}^{k}:=D(0,\varepsilon)\bigcap\sigma_{S}(T_{k}).$

By using Lemma 3.15, we have

$R(P_{0})\cong R(P_{\sigma^{k}_{N_{\varepsilon}}}).$

In particular, $\dim R(P_{\sigma^{k}_{N_{\varepsilon}}})<\infty$ for all
$k\geq N_{\varepsilon}.$ Applying [11, theorem 3.17], we have

$\displaystyle\sharp E_{T_{k}}^{\sigma^{k}_{N_{\varepsilon}}}<\infty,$

$q\in\sigma_{d}^{S}(T_{k})$ for all $k\geq N_{\varepsilon}$ and
$q\in\sigma^{k}_{N_{\varepsilon}}.$ ∎

##  4\. Browder $S$-resolvent equation in quaternionic setting

Let $[q]$ be an isolate $2$-sphere of $\sigma_{S}(T).$In view of spectral
decomposition theorem, we have

$\displaystyle\sigma_{S}(T|_{R(P_{[q]})})=[q]\ \mbox{ and }\
\sigma_{S}(T|_{N(P_{[q]})})=\sigma_{S}(T)\backslash[q].$

Let $T\in\mathcal{B}(V^{R}_{\mathbb{H}}).$ For
$S\in\rho^{S}_{B}(T):=\rho_{S}(T)\cup\sigma^{S}_{d}(T).$ We define the left
Browder $S$-resolvent operator as

$\displaystyle
S^{-1}_{L,B}(s,T)=-[Q_{s}(T)|_{N(P_{[s]})}]^{-1}(T-\overline{s}\mathbb{I_{V^{R}_{\mathbb{H}}}})(\mathbb{I_{V^{R}_{\mathbb{H}}}}-P_{[s]})-P_{[s]}.$

and the right Browder $S$-resolvent operator as

$\displaystyle
S^{-1}_{R,B}(s,T)=-(T-\overline{s}\mathbb{I_{V^{R}_{\mathbb{H}}}})[Q_{s}(T)|_{N(P_{[s]})}]^{-1}(\mathbb{I_{V^{R}_{\mathbb{H}}}}-P_{[s]})-P_{[s]}.$

######  Remark 4.1.

The Browder $S$-resolvent operator extend the $S$-resolvent operator to
$\rho_{S}(T)\cup\sigma_{d}^{S}(T)$. Indeed, if $q\in\rho_{S}(T)$ with the
convention $P_{[q]}=0$, we have

$\displaystyle S^{-1}_{R,B}(q,T)=S^{-1}_{R}(q,T)\ \mbox{ and }\
S^{-1}_{L,B}(q,T)=S^{-1}_{L}(q,T).$

######  Theorem 4.2.

Let $T\in\mathcal{B}(V^{R}_{\mathbb{H}})$ and
$q\in\rho^{S}_{B}(T):=\sigma^{S}_{d}(T)\cup\rho_{S}(T).$ Then, the left
Browder $S$-resolvent operator satisfy the left Browder $S$-resolvent equation

$\displaystyle
S^{-1}_{L,B}(q,T)(\mathbb{I_{V^{R}_{\mathbb{H}}}}-P_{[q]})q-T(\mathbb{I_{V^{R}_{\mathbb{H}}}}-P_{[q]})S^{-1}_{L,B}(q,T)+P_{[q]}=\mathbb{I_{V^{R}_{\mathbb{H}}}}.$

and the right Browder $S$-resolvent operator satisfy the right Browder
$S$-resolvent equation

$\displaystyle
q(\mathbb{I_{V^{R}_{\mathbb{H}}}}-P_{[q]})S^{-1}_{R,B}(q,T)-S^{-1}_{R,B}(q,T)(\mathbb{I_{V^{R}_{\mathbb{H}}}}-P_{[q]})T+P_{[q]}=\mathbb{I_{V^{R}_{\mathbb{H}}}}.$

###### Proof.

Let $q\in\rho^{S}_{B}(T).$ It is clear that,

$\displaystyle(\mathbb{I_{V^{R}_{\mathbb{H}}}}-P_{[q]})q=q(\mathbb{I_{V^{R}_{\mathbb{H}}}}-P_{[q]}),\
P(\mathbb{I_{V^{R}_{\mathbb{H}}}}-P_{[q]})=(\mathbb{I_{V^{R}_{\mathbb{H}}}}-P_{[q]})P_{[q]}$

and

$\displaystyle
T(\mathcal{Q}_{q}(T)|_{N(P_{[q]})})^{-1}=(\mathcal{Q}_{q}(T)|_{N(P_{[q]})})^{-1}T\mid_{N(P_{[q]})}.$

We obtain

$\displaystyle
S^{-1}_{L,B}(q,T)(\mathbb{I_{V^{R}_{\mathbb{H}}}}-P_{[q]})q-T(\mathbb{I_{V^{R}_{\mathbb{H}}}}-P_{[q]})S^{-1}_{L,B}(q,T)$
$\displaystyle=-[\mathcal{Q}_{q}(T)|_{N(P_{[q]})}]^{-1}(Tq-|q|^{2}\mathbb{I_{V^{R}_{\mathbb{H}}}})(\mathbb{I_{V^{R}_{\mathbb{H}}}}-P_{[q]})+[\mathcal{Q}_{q}(T)|_{N(P_{[q]})}]^{-1}(T^{2}-T\overline{q})(\mathbb{I_{V^{R}_{\mathbb{H}}}}-P_{[q]})$
$\displaystyle=[\mathcal{Q}_{q}(T)|_{N(P_{[q]})}]^{-1}\mathcal{Q}_{q}(T)|_{N(P_{[q]})}(\mathbb{I_{V^{R}_{\mathbb{H}}}}-P_{[q]})$
$\displaystyle=\mathbb{I_{V^{R}_{\mathbb{H}}}}-P_{[q]}.$

The right $S$-resolvent equation follows by similar computations.∎

######  Remark 4.3.

  1. (1)

The left and the right $S$-resolvent equation implies,

$\displaystyle
S_{L,B}^{-1}(q,T)q-TS_{L,B}^{-1}(q,T)-(T-(q+1)\mathbb{I_{V^{R}_{\mathbb{H}}}})P_{[q]}=\mathbb{I_{V^{R}_{\mathbb{H}}}}$

and

$\displaystyle
qS_{R,B}^{-1}(q,T)-S_{R,B}^{-1}(q,T)T-(T-(q+1)\mathbb{I_{V^{R}_{\mathbb{H}}}})P_{[q]}=\mathbb{I_{V^{R}_{\mathbb{H}}}}.$

  2. (2)

If $q\in\rho_{S}(T)$, then $P_{[q]}=0$. In this case, we obtain the two
equations in [19, Theorem 3.1.14]:

$\displaystyle
S_{L}^{-1}(q,T)q-TS_{L}^{-1}(q,T)=\mathbb{I_{V^{R}_{\mathbb{H}}}}$

and

$\displaystyle
qS_{R}^{-1}(q,T)-S_{R}^{-1}(q,T)T=\mathbb{I_{V^{R}_{\mathbb{H}}}}.$

Let $X$ be a complex Banach space and $A$ be a bounded operator on $X$. For
$\lambda\in\mathbb{C}\backslash\sigma(A)\cup\sigma_{d}(T),$ where
$\sigma_{d}(A)$ is the set of the Riesz points of $A$. We consider the
operator

$\displaystyle
R_{B}(\lambda,A)=(A-\lambda\mathbb{I}_{X})(\mathbb{I}_{X}-P_{\\{\lambda\\}})+P_{\\{\lambda\\}}.$

Thus, $R_{B}(\lambda,A)$ is invertible and

$\displaystyle
R_{B}^{-1}(\lambda,A)=(A-\lambda\mathbb{I}_{X}\mid_{N(P_{\\{\lambda\\}})})^{-1}(I-P_{\\{\lambda\\}})+P_{\\{\lambda\\}},$

where $P_{\\{\lambda\\}}$ is the Riesz projection complex associated to
$\lambda.$ The Browder resolvent operator satisfies the Browder resolvent
equation:

$\displaystyle R_{B}^{-1}(\lambda,A)-R_{B}^{-1}(\mu,A)$
$\displaystyle=(\lambda-\mu)R_{B}^{-1}(\lambda,A)R_{B}^{-1}(\mu,A)$
$\displaystyle+R^{-1}(\lambda,A)[(A-(\lambda+1)\mathbb{I}_{X})P_{\\{\lambda\\}}-(A-(\mu+1)\mathbb{I}_{X})P_{\\{\mu\\}}]R^{-1}_{B}(\mu,A),$

for $\lambda$, $\mu$ $\in\rho(T)\cup\sigma_{d}(T)$.

Now, we give the Browder $S$-resolvent equation in quaternionic setting.

######  Theorem 4.4.

Let $T\in\mathcal{B}(V^{R}_{\mathbb{H}})$ and let $s$, $p$
$\in\sigma_{d}^{S}(T)\cup\rho_{S}(T)$ with $p\not\in[s]$, then the left and
right Browder $S-$resolvent operators satisfies the following equation

$\displaystyle S^{-1}_{R,B}(s,T)S^{-1}_{L,B}(p,T)\mathcal{Q}_{s}(p)$
$\displaystyle=[S^{-1}_{R,B}(s,T)-S^{-1}_{L,B}(p,T)]p+\overline{s}[S^{-1}_{L,B}(p,T)-S^{-1}_{R,B}(s,T)]$
$\displaystyle+[S^{-1}_{R,B}(s,T)(T-(p+1)\mathbb{I_{V^{R}_{\mathbb{H}}}})P_{[p]}-(T-(s+1)\mathbb{I_{V^{R}_{\mathbb{H}}}})P_{[s]}S^{-1}_{L,B}(p,T)]p$
$\displaystyle+\overline{s}[(T-(s+1)\mathbb{I_{V^{R}_{\mathbb{H}}}})P_{[s]}S^{-1}_{L,B}(p,T)-S^{-1}_{R,B}(s,T)(T-(p+1)\mathbb{I_{V^{R}_{\mathbb{H}}}})P_{[p]}].$

###### Proof.

Set

$\displaystyle\sigma_{B}(s,p,T):=S^{-1}_{R,B}(s,T)S^{-1}_{L,B}(p,T)(p^{2}-2Re(s)p+|s|^{2})\
$

The left Browder $S$-resolvent equation implies

$\displaystyle
S^{-1}_{L,B}(p,T)p=\mathbb{I_{V^{R}_{\mathbb{H}}}}+TS^{-1}_{L,B}(p,T)+(T-(p+1)\mathbb{I_{V^{R}_{\mathbb{H}}}})P_{[p]}.$

In this way, we have

$\displaystyle\sigma_{B}(s,p,T)$
$\displaystyle=S^{-1}_{R,B}(s,T)[\mathbb{I_{V^{R}_{\mathbb{H}}}}+TS^{-1}_{L,B}(p,T)+(T-(p+1)\mathbb{I_{V^{R}_{\mathbb{H}}}})P_{[p]}]p$
$\displaystyle-2Re(s)S^{-1}_{R,B}(s,T)[\mathbb{I_{V^{R}_{\mathbb{H}}}}+TS^{-1}_{L,B}(p,T)+(T-(p+1)\mathbb{I_{V^{R}_{\mathbb{H}}}})P_{[p]}]$
$\displaystyle+|s|^{2}S^{-1}_{R,B}(s,T)S^{-1}_{L,B}(p,T)$
$\displaystyle=S^{-1}_{R,B}(s,T)p+S^{-1}_{R,B}(s,T)T[\mathbb{I_{V^{R}_{\mathbb{H}}}}+TS^{-1}_{L,B}(p,T)+(T-(p+1)\mathbb{I_{V^{R}_{\mathbb{H}}}})P_{[p]}]$
$\displaystyle+S^{-1}_{R,B}(s,T)(T-(p+1)\mathbb{I_{V^{R}_{\mathbb{H}}}})P_{[p]}p-2Re(s)S^{-1}_{R,B}(s,T)$
$\displaystyle-2Re(s)S^{-1}_{R,B}(s,T)TS^{-1}_{L,B}(p,T)$
$\displaystyle-2Re(s)S^{-1}_{R,B}(s,T)[(T-(p+1)\mathbb{I_{V^{R}_{\mathbb{H}}}})P_{[p]}]+|s|^{2}S^{-1}_{R,B}(s,T)S^{-1}_{L,B}(p,T).$

The right Browder $S$-resolvent equation implies

$\displaystyle
S^{-1}_{R,B}(s,T)T=sS^{-1}_{R,B}(s,T)-(T-(s+1)\mathbb{I_{V^{R}_{\mathbb{H}}}})P_{[s]}-\mathbb{I_{V^{R}_{\mathbb{H}}}}.$

So, we get

$\displaystyle\sigma_{B}(s,p,T)$
$\displaystyle=S^{-1}_{R,B}(s,T)p+sS^{-1}_{R,B}(s,T)-(T-(s+1)\mathbb{I_{V^{R}_{\mathbb{H}}}})P_{[s]}-\mathbb{I_{V^{R}_{\mathbb{H}}}}$
$\displaystyle+[sS^{-1}_{R,B}(s,T)-(T-(s+1)\mathbb{I_{V^{R}_{\mathbb{H}}}})P_{[s]}-\mathbb{I_{V^{R}_{\mathbb{H}}}}]TS^{-1}_{L,B}(s,T)$
$\displaystyle+[sS^{-1}_{R,B}(s,T)-(T-(s+1)\mathbb{I_{V^{R}_{\mathbb{H}}}})P_{[s]}-\mathbb{I_{V^{R}_{\mathbb{H}}}}](T-(p+1)\mathbb{I_{V^{R}_{\mathbb{H}}}})P_{[p]}$
$\displaystyle+S^{-1}_{R,B}(s,T)(T-(p+1)\mathbb{I_{V^{R}_{\mathbb{H}}}})P_{[p]}p$
$\displaystyle-2Re(s)S^{-1}_{R,B}(s,T)-2Re(s)[sS^{-1}_{R,B}(s,T)-(T-(s+1)\mathbb{I_{V^{R}_{\mathbb{H}}}})P_{[s]}-\mathbb{I_{V^{R}_{\mathbb{H}}}}]S^{-1}_{L,B}(p,T)$
$\displaystyle-2Re(s)S^{-1}_{R,B}(s,T)[T-(p+1)\mathbb{I_{V^{R}_{\mathbb{H}}}})P_{[p]}]+|s|^{2}S^{-1}_{R,B}(s,T)S^{-1}_{L,B}(p,T).$

If $s\not\in[p],$ then $P_{[s]}P_{[p]}=0.$ In particular, we have

$\displaystyle(T-(s+1)\mathbb{I_{V^{R}_{\mathbb{H}}}})P_{[s]}(T-(p+1)\mathbb{I_{V^{R}_{\mathbb{H}}}})P_{[p]}=0.$

The left Browder $S$-resolvent equation implies

$\displaystyle-\mathbb{I_{V^{R}_{\mathbb{H}}}}-TS^{-1}_{L,B}(p,T)=(T-(p+1)\mathbb{I_{V^{R}_{\mathbb{H}}}})P_{{}_{[s]}}-S^{-1}_{L,B}(p,T)p.$

Then, we obtain

$\displaystyle\sigma_{B}(s,p,T)$
$\displaystyle=S^{-1}_{R,B}(s,T)p+sS^{-1}_{R,B}(s,T)-(T-(s+1)\mathbb{I_{V^{R}_{\mathbb{H}}}})P_{[s]}-S^{-1}_{L,B}(p,T)p$
$\displaystyle+s[sS^{-1}_{R,B}(s,T)-(T-(s+1)\mathbb{I_{V^{R}_{\mathbb{H}}}})P_{[s]}-\mathbb{I_{V^{R}_{\mathbb{H}}}}]S^{-1}_{L,B}(p,T)$
$\displaystyle-(T-(s+1)\mathbb{I_{V^{R}_{\mathbb{H}}}})P_{[s]}TS^{-1}_{L,B}(p,T)+sS^{-1}_{R,B}(s,T)(T-(p+1))P_{[p]}$
$\displaystyle+S^{-1}_{R,B}(s,T)(T-(p+1))P_{[p]}p$
$\displaystyle-2Re(s)S^{-1}_{R,B}(s,T)-2Re(s)sS^{-1}_{R,B}(s,T)S^{-1}_{L,B}(p,T)$
$\displaystyle-2Re(s)[-(T-(s+1)\mathbb{I_{V^{R}_{\mathbb{H}}}})P_{[s]}S^{-1}_{L,B}(p,T)+S^{-1}_{R,B}(s,T)(T-(p+1)\mathbb{I_{V^{R}_{\mathbb{H}}}})P_{p}]$
$\displaystyle+2Re(s)S^{-1}_{L,B}(p,T)+|s|^{2}S^{-1}_{R,B}(s,T)S^{-1}_{L,B}(p,T).$
$\displaystyle=S^{-1}_{R,B}(s,T)p+sS^{-1}_{R,B}(s,T)-(T-(s+1)\mathbb{I_{V^{R}_{\mathbb{H}}}})P_{[s]}-S^{-1}_{L,B}(p,T)p$
$\displaystyle+s^{2}S^{-1}_{R,B}(s,T)S_{L,B}^{-1}(p,T)-s(T-(s+1)\mathbb{I_{V^{R}_{\mathbb{H}}}})P_{[s]}S^{-1}_{L,B}(p,T)-sS^{-1}_{L,B}(p,T)$
$\displaystyle-(T-(s+1)\mathbb{I_{V^{R}_{\mathbb{H}}}})P_{[s]}S^{-1}_{L,B}(p,T)p+(T-(s+1)\mathbb{I_{V^{R}_{\mathbb{H}}}})P_{[s]}$
$\displaystyle+sS^{-1}_{R,B}(s,T)(T-(p+1)\mathbb{I_{V^{R}_{\mathbb{H}}}})P_{[p]}+S^{-1}_{R,B}(s,T)(T-(p+1)\mathbb{I_{V^{R}_{\mathbb{H}}}})P_{[p]}p$
$\displaystyle-2Re(s)S^{-1}_{R,B}(s,T)-2Re(s)sS^{-1}_{R,B}(s,T)S^{-1}_{L,B}(p,T)$
$\displaystyle-2Re(s)[-(T-(s+1)\mathbb{I_{V^{R}_{\mathbb{H}}}})P_{[s]}S^{-1}_{L,B}(p,T)+S^{-1}_{R,B}(s,T)(T-(p+1)\mathbb{I_{V^{R}_{\mathbb{H}}}})P_{[p]}]$
$\displaystyle+2Re(s)S^{-1}_{L,B}(p,T)+|s|^{2}S^{-1}_{R,B}(s,T)S^{-1}_{L,B}(p,T).$
$\displaystyle=[S^{-1}_{R,B}(s,T)-S^{-1}_{L,B}(p,T)]p+(s^{2}-2Re(s)s+|s|^{2})S^{-1}_{R,B}(s,T)S^{-1}_{L,B}(p,T)$
$\displaystyle+[S^{-1}_{R,B}(s,T)(T-(p+1)\mathbb{I_{V^{R}_{\mathbb{H}}}})P_{[p]}-(T-(s+1)\mathbb{I_{V^{R}_{\mathbb{H}}}})P_{[s]}S^{-1}_{L,B}(p,T)]p-\overline{s}S^{-1}_{R,B}(s,T)$
$\displaystyle+\overline{s}[(T-(s+1)\mathbb{I_{V^{R}_{\mathbb{H}}}})P_{[s]}S^{-1}_{L,B}(p,T)-S^{-1}_{R,B}(s,T)(T-(p+1)\mathbb{I_{V^{R}_{\mathbb{H}}}})P_{[p]}]+\overline{s}S^{-1}_{L,B}(p,T).$

On the other hand

$\displaystyle s^{2}+2{\rm Re}(s)s+|s|^{2}=0.$

We conclude that

$\displaystyle\sigma_{B}(s,p,T)$
$\displaystyle=[S^{-1}_{R,B}(s,T)-S^{-1}_{L,B}(p,T)]p+\overline{s}[S^{-1}_{L,B}(p,T)-S^{-1}_{R,B}(s,T)]$
$\displaystyle+[S^{-1}_{R,B}(s,T)(T-(p+1)\mathbb{I_{V^{R}_{\mathbb{H}}}})P_{[p]}-(T-(s+1)\mathbb{I_{V^{R}_{\mathbb{H}}}})P_{[s]}S^{-1}_{L,B}(p,T)]p$
$\displaystyle+\overline{s}[(T-(s+1)\mathbb{I_{V^{R}_{\mathbb{H}}}})P_{[s]}S^{-1}_{L,B}(p,T)-S^{-1}_{R,B}(s,T)(T-(p+1)\mathbb{I_{V^{R}_{\mathbb{H}}}})P_{[p]}].$

∎

######  Remark 4.5.

  1. (1)

If $s$, $p$ $\in\rho_{S}(T),$ then $P_{[s]}=P_{[p]}=0.$ Hence, we find the
$S$-resolvent equation, see [19, theorem 3.1.15]:
$S^{-1}_{R}(s,T)S^{-1}_{L}(p,T)$

$\displaystyle=[(S^{-1}_{R}(s,T)-S^{-1}_{L}(p,T))p-\overline{s}[S^{-1}_{R}(s,T)-S^{-1}_{L}(p,T)](p^{2}-2Re(s)p+|s|^{2})^{-1}.$

  2. (2)

Let’s test the Browder $S$-resolvent equation in the commutative case, if
Tq=qT for all $q\in\mathbb{H},$ then for $s$, $q$ $\in\rho_{S}(T),$ we have

$\displaystyle M(s,p)$
$\displaystyle:=[S^{-1}_{R,B}(s,T)(T-(p+1)\mathbb{I_{V^{R}_{\mathbb{H}}}})P_{[p]}-(T-(s+1)\mathbb{I_{V^{R}_{\mathbb{H}}}})P_{[s]}S_{R,B}^{-1}(p,T)]p$
$\displaystyle+\overline{s}[(T-(s+1)\mathbb{I_{V^{R}_{\mathbb{H}}}})P_{[s]}S^{-1}_{L,B}(p,T)-S^{-1}_{R,B}(s,T)(T-(p+1)\mathbb{I_{V^{R}_{\mathbb{H}}}})P_{[p]}]$
$\displaystyle=p[S^{-1}_{R,B}(s,T)(T-(p+1)\mathbb{I_{V^{R}_{\mathbb{H}}}})P_{[p]}-S^{-1}_{L,B}(p,T)(T-(s+1)\mathbb{I_{V^{R}_{\mathbb{H}}}})P_{[s]}]$
$\displaystyle+\overline{s}[S^{-1}_{L,B}(p,T)(T-(s+1)\mathbb{I_{V^{R}_{\mathbb{H}}}})P_{[s]}-S^{-1}_{R,B}(s,T)(T-(p+1)\mathbb{I_{V^{R}_{\mathbb{H}}}})P_{[p]}]$
$\displaystyle=(p-\overline{s})[S^{-1}_{R,B}(s,T)(T-(p+1)\mathbb{I_{V^{R}_{\mathbb{H}}}})P_{[p]}-S^{-1}_{L,B}(p,T)(T-(s+1)\mathbb{I_{V^{R}_{\mathbb{H}}}})P_{[s]}].$

In particular in the complex case, if $T\in\mathcal{B}(V_{\mathbb{C}})$ then,

$\displaystyle S_{R,B}^{-1}(p,T)=R_{B}^{-1}(p,T)\mbox{ and
}S_{L,B}^{-1}(s,T)=R_{B}^{-1}(s,T).$

Therefore, we obtain
$M(s,p)$

$\displaystyle=(p-\overline{s})R_{B}^{-1}(s,T)[(T-(p+1)\mathbb{I_{V^{R}_{\mathbb{H}}}})P_{[p]}-R_{B}(s,T)(T-(s+1)\mathbb{I_{V^{R}_{\mathbb{H}}}})P_{[s]}R^{-1}_{B}(p,T)]$
$\displaystyle=(p-\overline{s})R^{-1}_{B}(s,T)[(T-(p+1)\mathbb{I_{V^{R}_{\mathbb{H}}}}P_{[s]}R_{B}(p,T)-R_{B}(s,T)(T-(s+1)\mathbb{I_{V^{R}_{\mathbb{H}}}})P_{[s]}]R^{-1}_{B}(p,T).$

Hence,

$P_{[p]}R_{B}(p,T)=P_{[p]}\ \mbox{and}\ R_{B(s,T)}P_{[s]}=P_{[s]}.$

So, we get

$\displaystyle
M(s,p):=(p-\overline{s})R^{-1}_{B}[(s,T)[(T-(p+1)\mathbb{I_{V^{R}_{\mathbb{H}}}})P_{[p]}-(T-(s+1)\mathbb{I_{V^{R}_{\mathbb{H}}}})P_{[s]}]R^{-1}_{B}(p,T).$

Thus, we obtain the classic Browder resolvent equation in complex case.

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