# Hydrostatic Pressure Induced Anomalous Enhancement in the Thermoelectric
Performance of Monolayer MoS2

Saumen Chaudhuri  Department of Physics, Indian Institute of Technology
Kharagpur, Kharagpur 721302, India  Amrita Bhattacharya  Department of
Metallurgical Engineering and Materials Science, Indian Institute of
Technology Bombay, Mumbai 400076, India  A. K. Das  Department of Physics,
Indian Institute of Technology Kharagpur, Kharagpur 721302, India  G. P. Das
Research Institute for Sustainable Energy, TCG Centres for Research and
Education in Science and Technology, Sector V, Salt Lake, Kolkata 700091,
India  B. N. Dev<EMAIL_ADDRESS>Department of Physics and School of
Nano Science and Technology, Indian Institute of Technology Kharagpur,
Kharagpur, 721302, India  Centre for Quantum Engineering, Research and
Education, TCG Centres for Research and Education in Science and Technology,
Sector V, Salt Lake, Kolkata 700091, India

###### Abstract

The hydrostatic pressure induced changes in the transport properties of
monolayer (ML) MoS2 have been investigated using first-principles density
functional theory based calculations. The application of pressure induces
shift in the conduction band minimum (CBM) from K to $\Lambda$, while
retaining the band extrema at K in around the same energy at a pressure of 10
GPa. This increase in valley degeneracy is found to have a significant impact
on the electronic transport properties of ML-MoS2 via enhancement of the
thermopower (S) by up to 140% and power factor (S2$\sigma$/$\tau$) by up to
310% at 300 K. Besides, the very low deformation potential (E${}_{\text{DP}}$)
associated with the CB-$\Lambda$ valley results in a remarkably high
electronic mobility ($\mu$) and relaxation time ($\tau$). Additionally, the
application of pressure reduces the room temperature lattice thermal
conductivity ($\kappa_{\text{L}}$) by 20% of its unstrained value, owing to
the increased anharmonicity and resulting increase in the intrinsic phonon
scattering rates. The hydrostatic pressure induced increase in power factor
(S2$\sigma$) and the decrease in $\kappa_{\text{L}}$ act in unison to result
in a substantial improvement in the overall thermoelectric performance (zT) of
ML-MoS2. At 900 K with an external pressure of 25 GPa, zT values of 1.63 and
1.21 are obtained for electron and hole doping, respectively, which are
significantly higher compared to the zT values at zero pressure. For the
implementation in a thermoelectric module where both n-type and p-type legs
should be preferably made of the same material, the concomitant increase in zT
of ML-MoS2 for both types of doping with hydrostatic pressure can be highly
beneficial.

DFT, hydrostatic pressure, MoS2, thermoelectric properties

##  I Introduction

The search for renewable energy sources has become a quintessential challenge
for mankind [1]. Thermoelectric materials (TEMs), which can convert a
difference in temperature into useful voltage based on the principle of
Seebeck effect [2, 3], may aid to it via waste heat recovery [4]. The
efficiency of a thermoelectric material depends on a dimensionless parameter
called the thermoelectric figure of merit (zT), which is expressed as
$\text{zT}=\frac{\text{S}^{2}\sigma}{\kappa}\text{T}$. zT is directly
proportional to the product of square of Seebeck coefficient S and the
electrical conductivity $\sigma$, while is inversely proportional to the total
thermal conductivity, $\kappa=\kappa_{\text{e}}+\kappa_{\text{L}}$, which is
the sum of the electrical ($\kappa_{\text{e}}$) and the lattice thermal
conductivity ($\kappa_{\text{L}}$), at a given temperature T. Tuning the zT of
a given TEM is an extremely tedious task, since the transport parameters i.e.
the S, $\sigma$, and $\kappa_{\text{e}}$ are inter-dependent on each other, as
they all depend on the electronic carrier concentration (n). Therefore, the
two main routes that can be followed to maximize the zT are (a) engineering
the electronic band structure to optimize the power factor [5, 6] and (b)
phonon engineering for lowering the $\kappa_{\text{L}}$ of the material [7, 8,
9, 10, 11].

More recently, the two dimensional planar materials have garnered a lot of
interest in thermoelectric research owing to their unique properties arising
from quantum confinement effects. The excellent combination of electrical and
thermal transport in these 2D materials leads to a significantly large
thermoelectric zT [5, 7]. Over the years, various 2D materials, viz.
transition metal dichalcogenides (TMDCs) (e.g., MoS2, WS2, MoSe2, and HfS2),
group IVA-VA compounds (e.g., SnSe, SnS, GeSe) etc., have emerged as potential
TEM with reasonably good thermoelectric performance [12, 13, 14, 15]. In
recent times, especially the layered TMDCs have drawn significant interest
owing to their tunable band gap, high electrical mobility and low thermal
conductivity [16, 17, 18, 19, 20]. Some of the TMDCs, such as WS2 [21], WSe2
[22], HfS2 [23], SnSe2 [24] etc., have shown notably high thermoelectric
efficiency. Among the family of TMDCs, MoS2 has been explored extensively as a
potential thermoelectric material, both theoretically [25, 26, 27, 28, 29, 30,
19, 31] and experimentally [32, 33], and the monolayer (ML) counterpart
appears to be a reasonably good thermoelectric material [25, 28, 30, 32, 33].
Kedar et al [32] have measured the room temperature thermoelectric power
factor of ML-MoS2 to be 8.5 mWK-2m-1, which is comparable to the well known
thermoelectric materials, such as Bi2Te3. Buscema et al [34] have found a
large Seebeck coefficient in ML-MoS2, which is also tunable by an external
electric field. However, the zT value obtained with ML-MoS2 is very low (i.e.,
a zT value of only $\sim 0.11$ is achieved with ML-MoS2 at 500 K [13]), which
is not useful for practical purposes. The primary reason behind the low
thermoelectric efficiency of ML-MoS2 is derived from its rather high lattice
thermal conductivity ($\kappa_{\text{L}}$), which is considerably higher
compared to the other analogous TMDCs, such as WSe2, ZrS2, HfS2 etc. [12, 21,
35, 36, 37]. Therefore, finding out a way by which both the power factor
(S2$\sigma$) and the $\kappa_{\text{L}}$ can be optimized simultaneously in a
way to maximize the thermoelectric zT of ML-MoS2 has been an extensive quest.

Efforts have been made to improve the thermoelectric efficiency of ML-MoS2 by
applying an external electric field [34] or mechanical strain [11, 23, 25, 28,
30, 31]. Doping with impurity atoms and hybridization are the widely explored
avenues in improving the zT of a material [9, 38, 39]. In-plane strain
engineering has also been employed routinely to tune the electronic and
thermoelectric properties of various 2D materials, such as WS2, ZrS2, HfS2,
ZnO etc. [12, 23, 40, 41]. In one of our earlier studies, we have seen that
in-plane tensile strains, in general, and strains along the zig-zag direction,
in particular, can significantly enhance the thermoelectric efficiency of ML-
MoS2 [31]. However, the implementation of different in-plane strains in
experiments is a daunting task [42, 43, 44, 45, 46]. Application of
hydrostatic pressure, on the other hand, can be a potential alternative easier
route that can be achieved experimentally owing to the fairly simple procedure
and reversibility. Previous theoretical and experimental studies on the
pressure dependence of electronic properties, structural parameters and
elastic constants of bulk MoS2 suggest that hydrostatic pressure might be
effective in tuning its transport properties [29, 47, 48, 49]. Zhang et al
demonstrated that the thermoelectric performance of bulk MoS2 can be enhanced
by the application of hydrostatic pressure [29]. For ML-MoS2, the variation in
the thermoelectric parameters with various in-plane strain has been studied
thoroughly over the years [11, 23, 25, 28, 30, 31]. However, studies on the
effect of hydrostatic pressure on the transport properties of ML-MoS2 are
lacking.

Interestingly, in a recent study, a direct K-K to indirect K-$\Lambda$ band
gap transition has been optically detected in ML-MoS2 under hydrostatic
pressure [48]. Therefore, a pressure induced shift of the conduction band
minimum (CBM) from K to $\Lambda$ is inevitable. At the critical pressure,
exploiting the degeneracy of the conduction band edges at K and $\Lambda$, an
enhancement in the thermoelectric performance of ML-MoS2 can be achieved.
Also, the overall transport mechanism, both electronic and lattice, in ML-
MoS2, can be tuned continuously and controllably by the application of
pressure. The hydrostatic pressure induced modification in the transport
properties of ML-MoS2 has not been studied theoretically or experimentally.
Therefore, a thorough investigation of the underlying mechanism is necessary.
This may pave the way for the application of ML-MoS2 or any other
semiconducting TMDCs in future commercial thermoelectric devices. In the
present work, we have undertaken a theoretical investigation of hydrostatic
pressure-mediated modification in the thermoelectric properties of ML-MoS2.

##  II Computational details

First-principles calculations have been performed under the framework of ab-
initio density functional theory (DFT) as implemented in the Vienna Ab Initio
Simulation Package (VASP) [50, 51] with projector augmented wave (PAW)
potentials to account for the electron-ion interactions [52]. The electronic
exchange and correlation (XC) interactions are addressed within the
generalized gradient approximation (GGA) of Perdew-Burke-Ernzerhof (PBE) [53].
In all calculations, the Brillouin zone (BZ) is sampled using a well-converged
Monkhorst-Pack [54] k-point set ($21\times 21\times 1$), and a conjugate
gradient scheme is employed to optimize the geometries until the forces on
each atom are found to be less than 0.01 eV/Å. A vacuum thickness of
approximately 20 Å has been used to avoid the spurious interaction between the
periodic images of the layers. In order to generate the structure under
hydrostatic pressure, the bulk lattice of MoS2 is subjected to external stress
of equivalent amount. The electronically optimized strained structure of bulk
MoS2 is then used to cleave one single layer of the MoS2 from its bulk lattice
under different hydrostatic pressures i.e. for 5, 10 and 25 GPa. A similar
approach has also been adopted to generate hydrostatic strained structures in
earlier reports [48, 55].

The temperature- and carrier concentration-dependent changes in the
thermoelectric parameters such as the Seebeck coefficient (S), electrical
conductivity ($\sigma$), power factor (S${}^{2}\sigma$) etc. have been
calculated by using the energy eigen values to solve the semi-classical
Boltzmann transport equation as implemented within the BoltzTraP code [56].
All the transport properties have been calculated within the constant
relaxation time approximation (CRTA) which assumes that the charge carrier
relaxation time does not vary with energy or carrier concentration. To go
beyond the CRTA, the charge carrier mobility ($\mu$) and relaxation time
($\tau$) are explicitly determined using the acoustic deformation potential
(ADP) theory developed by Bardeen and Shockley [57]. Considering the
scattering between charge carriers and longitudinal acoustic phonons, the
$\mu$ and $\tau$ of the 2D materials are computed as
$\mu=\frac{2e{\hbar}^{3}\text{C}_{\text{2D}}}{3{\text{k}_{\text{B}}}\text{T}{\text{m}^{*}}^{2}(\text{E}_{\text{DP}})^{2}}$
and $\mu=\frac{e\tau}{\text{m}^{*}}$, where $\text{C}_{\text{2D}}$ is the
effective elastic constant, $\text{m}^{*}$ is the carrier effective mass, and
$\text{E}_{\text{DP}}$ is the deformation potential constant corresponding to
the conduction and the valence band edges. The material-specific inputs such
as the $\text{C}_{\text{2D}}$, $\text{m}^{*}$, and the $\text{E}_{\text{DP}}$
are determined from first-principles calculations using VASP.

The phonon dispersion curves are calculated based on the supercell approach
using the finite displacement method restricting the atomic vibrations to an
amplitude of 0.015 Å as implemented in the phonopy code [58]. To compute the
lattice transport properties, the Boltzmann Transport Equation (BTE) for
phonons is solved under the relaxation time approximation (RTA) as implemented
in the phono3py code [59]. The second- and third-order interatomic force
constants (IFC) are calculated using convergence-checked $4\times 4\times 1$
and $2\times 2\times 1$ supercells based on the relaxed unit cell,
respectively. Fourth- and higher-order IFCs are not taken into consideration
due to their presumably small contribution to lattice thermal transport.
Forces are optimized using a fine energy convergence criterion of 10-8 eV.
Well-converged k-meshes are used to optimize the structures. Supercell method
is used to ensure convergence, whereby convergence is reached for $4\times
4\times 1$ supercell. To accurately compute the lattice thermal conductivity,
a dense q-mesh of $51\times 51\times 1$ is used to sample the reciprocal space
of the primitive cells. The mode-resolved phonon transport parameters, such as
the phonon group velocity (vλ) and relaxation time ($\tau_{\lambda}$) are
extracted using python-based extensions.

Figure 1:  Schematic diagram illustrating the procedure of obtaining a
hydrostatic strained MoS2 monolayer by cleaving its equivalent strained bulk
MoS2 counterpart.

##  III Results and Discussion

###  III.1 Structure parameters and electronic properties

The single-layer counterpart of MoS2 forms a hexagonal honeycomb crystal
structure with the Mo and S atoms arranged in a trigonal prismatic
coordination, as can be seen from Fig. 1. A monolayer (ML) structure of MoS2
consists of one Mo and two S atomic planes arranged in a “sandwich” (S-Mo-S)
type structure. The primitive unit cell of bulk MoS2 comprises of two such
monolayers with an inter-layer separation (t) of 6.15 Å. The structural
parameters of both the bulk- and the ML-MoS2, calculated herein, are in good
agreement with earlier reports [28, 31]. To study the effect of hydrostatic
pressure on the electronic and transport properties of ML-MoS2, different
pressures of up to 25 GPa have been applied. A schematic of the pressure
application on ML-MoS2 is presented in Fig. 1. To confirm the dynamic
stability of the crystal structure, phonon dispersion has been calculated at
each pressure. From the electronic structure calculations, ML-MoS2 in its
pristine form is found to be a direct band gap semiconductor with both the
valence band maximum (VBM) and the conduction band minimum (CBM) located at
the same high-symmetry point i.e. at the K (0.33, 0.33, 0) point of the
Brillouin zone (BZ). The band gap value of pristine ML-MoS2 is found to be
1.68 eV, which is in agreement with previous theoretical works [19, 28, 60].

Figure 2:  (a) Band dispersion along the high symmetry path of ML-MoS2 at
different values viz. 0 GPa, 5 GPa, 10 GPa, and 25 GPa of hydrostatic
pressure. The pressure values are given at the top of the corresponding band
structure plots. (b) The variation in the band gap as a function of the
hydrostatic pressure (b: inset shows the energies corresponding to the valence
and conduction band edges at different k-points with reference to the Fermi
energy E${}_{\text{F}}$). (c) The various structural parameters i.e. the bong
length (shown in black) and bond angle (shown in red) of ML-MoS2 as a function
of hydrostatic pressure.

The hydrostatic pressure induced effects on the structural parameters and
electronic structure of ML-MoS2 are presented in Fig. 2. From the band
structure plots it can be seen that, with the application of pressure, the CBM
at K point (CB-K) shifts toward higher energy (i.e. away from the Fermi
level), while the energy shift in the valence band top at K (VB-K) is nominal.
Therefore, the band gap increases initially, until P = 10 GPa, as can be seen
from the variation in the band gap with pressure shown in Fig. 2 (b). At a
pressure around 10 GPa, the CBM shifts to the $\Lambda$-point i.e., midway
along the K-$\Gamma$ path, while the one extrema at the K point lying very
close in energy, thereby forming degenerate energy valleys in the conduction
band. Such valley or band degeneracies are found to have useful implications
in enhancing the thermoelectric performance of a material [5, 6]. Starting
from 15 GPa, when the CBM is at the $\Lambda$ point, the band gap starts to
decrease with increasing pressure. To quantify the energy shifts, the energies
corresponding to the VB and the CB edges are plotted as a function of pressure
and presented in Fig. 2 (b). It is clear that the energy shifts in the CB-K
and the VB-$\Gamma$ are significantly higher compared to the energy shifts in
other valence and conduction band extrema.

To understand these energy shifts in greater detail, we have calculated the
band decomposed charge densities corresponding to the relevant valence and
conduction band edges, such as the CB at K (CB-K), CB at $\Lambda$
(CB-$\Lambda$), VB at K (VB-K) and VB at $\Gamma$ (VB-$\Gamma$) (see Fig. S1
in supplementary material). The band edges at different k-points in the BZ are
found to have very different real space charge densities, however, mostly
comprising of the Mo-$d$ orbital. For example, the states at CB-K and VB-K are
predominantly comprising of Mo-d character arising from d${}_{z^{2}}$ and
d${}_{x^{2}-y^{2}}$ orbitals, respectively. The states at CB-$\Lambda$ and
VB-$\Gamma$, on the other hand, have a certain admixture of S-$p$ orbitals
along with the Mo-$d$ orbitals. With the application of pressure, both the Mo-
Mo and Mo-S bond length decreases with an increase in the S-Mo-S bond angle
and therefore, the S-S interplanar distance, as can be seen from Fig. 2 (c).
However, the magnitude of reduction in Mo-Mo bond length is much higher
compared to that of Mo-S bond length. It is, therefore, expected that the
states that are composed purely of Mo-$d$ orbitals (such as CB-K), and thus
depend entirely on the Mo-Mo bond length, will experience larger energy shifts
compared to the states that comprise of both Mo-$d$ and S-$p$ character (such
as CB-$\Lambda$). The large energy shifts of the CB-K, which is the most
relevant one in this case, can be understood by analysing the pressure induced
modifications in the structural parameters. With increasing bond angle and
decreasing bond length under pressure, the interaction strength between the
Mo-$d_{x^{2}-y^{2}}$ and S-$p$ orbitals weakens, while the coupling between
the $d_{z^{2}}$ orbitals will strengthen. Therefore, the energy splitting of
the pair of states with predominantly Mo-$d_{z^{2}}$ character, such as at
CB-K and VB-$\Gamma$, increases under pressure and thereby, shifting the CB-K
towards the higher energy.

###  III.2 Thermoelectric transport properties

To understand the implications of the pressure induced modifications in the
electronic band structure of ML-MoS2, various thermoelectric transport
parameters, such as the Seebeck coefficient or thermopower (S), electrical
conductivity ($\sigma$), power factor (PF: S${}^{2}\sigma$) and the charge
carrier mobility ($\mu$) have been investigated as a function of increasing
hydrostatic pressure. Specifically, the pressure induced degeneracy in the
conduction band is expected to have a significant impact on the thermoelectric
parameters of ML-MoS2. The pressure induced variation in the S and PF as a
function of carrier concentration at a fixed temperature T= 300 K, and as a
function of temperature at a fixed carrier concentration of N${}_{\text{2D}}$
= $1.3\times 10^{13}$ /cm2 are presented in Fig. 3 for both the n-type (shown
in top panel) and p-type (shown in the bottom panel) doping. With increasing
pressure, the S is found to increase significantly for n-type doping
concentration of $0.01-3\times 10^{13}$ /cm2. The highest value of S can be
achieved at a hydrostatic pressure of 10 GPa, which is due to the pressure
induced valley degeneracy at CB-K and CB-$\Lambda$. At this pressure owing to
the increased valley degeneracy at the CB edge, the density of states
effective mass (m${{}_{\text{d}}}^{*}$) increases, thereby resulting in an
enhancement of the S, without explicitly affecting the charge carrier mobility
($\mu$). Therefore, the power factor is significantly improved due to the
degenerate valleys. However, with pressure the valley degeneracy at the
valence band extrema is lost, since the band maximum at the $\Gamma$ point
shifts to lower energy and the band maximum is found to lie only at the
K-point. Due to this loss in valley degeneracy in the valence band, a
reduction in S is incurred upon hole doping. For both electron and hole
doping, the thermopower decreases with increasing carrier concentration, which
is a typical characteristic of any semiconducting material. Owing to the
increase in S, a significant enhancement in the relaxation time ($\tau$)
scaled power factor (S${}^{2}\sigma$/$\tau$) is observed as well under the
application of hydrostatic pressure for electron doping (see Fig. 3 (a) and
(b)). The power factor value reaches up to $8.5\times 10^{10}$ W/mK2s at 300 K
for electron doping at a pressure of 10 GPa, which is significantly higher
compared to the zero pressure power factor values. The power factor represents
the ability of a material to produce useful electrical power at a given
temperature gradient. Therefore, the large power factor is indicative of
better thermoelectric performance. Also, the Goldsmid-Sharp relation [61] is
found to hold well, as the variation in the maximum achievable thermopower
(S${}_{\text{max}}$) with applied pressure at a fixed temperature follows the
same trend as that of the band gap (see Fig. S2 in supplementary material).
The relation is given as, E${}_{\text{g}}$ = 2eS${}_{\text{max}}$T, where
E${}_{\text{g}}$ is the band gap, and S${}_{\text{max}}$ is the maximum
attainable thermopower at any temperature T. It is, therefore, clear that both
the electronic and the thermoelectric parameters are equally sensitive to the
applied pressure.

Figure 3: Variation in the thermoelectric parameters i.e., the thermopower (S)
and relaxation time ($\tau$) scaled power factor (S${}^{2}\sigma$/$\tau$) of
ML-MoS2 for different values of applied hydrostatic pressure as a function of
(a, c) carrier concentration (N${}_{\text{2D}}$) at a fixed temperature of 300
K and (b, d) as a function of temperature for a fixed electron (upper panel)
and hole (lower panel) doping of $1.3\times 10^{13}$ /cm2.

In order to understand the temperature dependency, the variation in the
thermoelectric parameters of ML-MoS2 is investigated for temperature in a
range of 100 K-900 K at a fixed carrier concentration of $1.3\times
10^{13}/\text{cm}^{2}$, which corresponds to $5\times 10^{19}/\text{cm}^{3}$
in bulk configuration. The upper limit of the temperature is well within the
thermal stability regime of ML-MoS2, which was analyzed using molecular
dynamics (MD) calculations in our earlier work [31]. The S and PF are plotted
as function of temperature for different hydrostatic pressure under electron
and hole doping (see Fig. 3 (b) and (d)). For the electron doping, both these
quantities are found to be highest at a pressure of 10 GPa for the entire
temperature range due to the increased degeneracy at the CBM. However, for the
hole doping the pristine one is found to show higher S and PF as compared to
their pressurized counterparts for the entire temperature range, which is due
to the loss of the valley degeneracy at the VBM under pressure (see Fig. 3
(d)). For both electron and hole doping, the PF is found to increase
dramatically with increasing temperature. At 900 K, the highest PF values
obtained for electron and hole doping are $16\times 10^{10}$ W/mK2s and
$8\times 10^{10}$ W/mK2s, respectively. This is indicative of the possible use
of ML-MoS2 as a high temperature thermoelectric material.

The transport parameters calculated herein, such as the $\sigma$,
$\kappa_{\text{e}}$ and PF (S${}^{2}\sigma$) are scaled by the charge carrier
relaxation time ($\tau$), since the calculations performed here are based on
the constant relaxation time approximation (CRTA). CRTA assumes that the
carrier relaxation time ($\tau$) and therefore, the mobility ($\mu$) does not
vary strongly with energy. Such an oversimplified assumption often lead to
misleading conclusions. In the present case, the explicit determination of
$\tau$ and $\mu$ is essential, since $\tau$ and $\mu$ itself are functions of
external applied pressure. In the calculations of $\mu$ and $\tau$, only the
scattering of charge carriers with longitudinal acoustic phonons within the
acoustic deformation potential (ADP) theory of Bardeen and Shockley [57] is
considered (see Table 1 in supplementary material). It is assumed that the
scattering with the polar optical phonons will have an inconsequential effect
owing to the intrinsically non-polar nature of MoS2. Other scattering events,
such as boundary scattering and impurity scattering are also ignored, since
these are strongly dependent on experimental conditions. Considering only a
part of the total scattering processes, the calculations performed herein are
expected to provide only an intrinsic limit of $\mu$ and $\tau$, which can be
largely overestimated compared to the experimental values.

Figure 4: Variation in the charge carrier relaxation time ($\tau$) of ML-MoS2 with temperature at different pressure values for (a) electron and (b) hole doping.  Table 1:  The calculated effective 2D elastic constant (C${}_{\text{2D}}$), the carrier effective mass (m∗), the absolute deformation potential constant (E${}_{\text{DP}}$) corresponding to the conduction and valence band edges and the total charge carrier mobility ($\mu$) of ML-MoS2 at 300 K for different hydrostatic pressures. The m∗ and E${}_{\text{DP}}$ are given for the different band edge positions (K and $\Lambda$ given in bracket). P (GPa) |  Carriers |  C${}_{\text{2D}}$ |  m∗ (m0) |  E${}_{\text{DP}}$ (eV) |  $\mu$ (cm2V-1s-1)
---|---|---|---|---|---
0 |  e |  212.5 |  $0.47$ (K) |  $6.61$ (K) |  193.64
|  h |  |  $0.58$ (K) |  $2.29$ (K) |  1056.25
5 |  e |  230.0 |  $0.48$ (K) |  $6.88$ (K) |  186.24
|  h |  |  $0.58$ (K) |  $1.94$ (K) |  1597.22
10 |  e |  246.2 |  $0.51$ (K), $0.52$ $(\Lambda)$ |  $6.97$ (K), $1.99$ $(\Lambda)$ |  1950.69
|  h |  |  $0.59$ (K) |  $1.63$ (K) |  2308.22
15 |  e |  295.0 |  $0.52$ $(\Lambda)$ |  $1.43$ $(\Lambda)$ |  2782.38
|  h |  |  $0.64$ (K) |  $1.22$ (K) |  4095.06

The charge carrier relaxation time ($\tau$) of ML-MoS2 as a function of
temperature at different pressure values is shown in Fig. 4, and the carrier
mobility ($\mu$) plots are provided in the supplementary material (see
supplementary material Fig. S3). The physical parameters that are required for
the analytical formulation of the deformation potential theory (DP), such as
the elastic constant (C${}_{\text{2D}}$), carrier effective mass (m∗) and the
deformation potential (E${}_{\text{DP}}$) are presented in Table 1, and the
computed mobility of ML-MoS2 in the zero pressure case is in good agreement
with previous reports [62, 63, 64]. Note that, due to the in-plane isotropic
nature of ML-MoS2, the values of the various physical parameters calculated
along different directions are found to be identical. Therefore, the simplest
form of the Bardeen and Shockley equation, which takes into account the
average values of the C${}_{\text{2D}}$, m∗, and E${}_{\text{DP}}$ is
implemented in this work. The mobility of electrons and holes at 300 K is
found to be 193.6 and 1056.2 cm2V-1s-1, respectively. The relaxation time
corresponding to electron and hole scattering decreases with temperature
following a parabolic function, as can be seen from Fig. 4 (a) and (b). For
both types of carriers, $\tau$ increases with increasing pressure. For
electrons, $\tau$ increases anomalously with pressure, whereas for holes it
increases in a monotonic fashion. A sharp increase in $\mu$ and $\tau$
starting from P = 10 GPa is seen for electron doping (see Fig. 4 (a)). As the
pressure increases to 10 GPa, the involvement of the conduction band valley at
$\Lambda$ (CB-$\Lambda$) in electronic conduction becomes apparent. It can be
seen from Table 1 that, the effective mass of electrons is nearly identical in
both the conduction band valleys at K and $\Lambda$. However, the deformation
potential (E${}_{\text{DP}}$) associated with the CB-$\Lambda$ is more than
three times smaller than that with the CB-K. Therefore, the electronic
mobility and relaxation time at the CB-$\Lambda$ valley is significantly
higher compared to that at the CB-K. Thus, the contribution from the
CB-$\Lambda$ results in the remarkable increase in total $\mu$ and $\tau$. For
holes, on the other hand, the effective mass increases slightly With
increasing pressure, however, due to the large reduction in E${}_{\text{DP}}$
corresponding to the VBM at K, the hole mobility and therefore, the relaxation
time increases. The enhancement in the mobility ($\mu$) and relaxation time
($\tau$) for both electrons and holes is highly beneficial for better
thermoelectric performance. The estimated $\tau$ values are incorporated in
the calculated electronic transport parameters, such as the $\sigma/\tau$,
$\kappa_{\text{e}}/\tau$ and PF (S${}^{2}\sigma/\tau$) to calculate the figure
of merit (zT) as discussed in the subsequent paragraph.

The pressure induced modifications in the structural parameters are expected
to have certain implications on the lattice transport properties. Therefore,
the effect of pressure on the lattice thermal conductivity
($\kappa_{\text{L}}$) of ML-MoS2 has been calculated and the variation in
$\kappa_{\text{L}}$ with temperature at some representative values of pressure
is shown in Fig. 5. For pristine ML-MoS2, the $\kappa_{\text{L}}$ at 300K is
found to be 24.41 Wm-1K-1, which agrees well with earlier theoretical [35, 65]
and experimental reports [36]. The $\kappa_{\text{L}}$ of ML-MoS2 decreases
with increasing pressure and reduces to 17.48 Wm-1K-1 at a pressure of 25 GPa.
Owing to the inverse relationship of $\kappa_{\text{L}}$ with the
thermoelectric figure of merit (zT), a reduction in $\kappa_{\text{L}}$ is
immensely advantageous in enhancing the thermoelectric performance. The
variation in $\kappa_{\text{L}}$ with temperature is found to follow the 1/T
law owing to the increased probability of Umklapp scattering. A reduced value
of 5.77 Wm-1K-1 is obtained at 900 K under an external pressure of 25 GPa. The
total thermal conductivity ($\kappa$), which is a sum of the electronic
($\kappa_{\text{e}}$) and the lattice contribution ($\kappa_{\text{L}}$), is
found to be largely dominated by the lattice counterpart, at all temperature
and pressure. This can be explained as the general behaviour of a
semiconductor that majority of the heat is carried by the phonons with
negligible contribution stemming from the electrons.

Figure 5: Variation in the lattice thermal conductivity ($\kappa_{\text{L}}$)
of ML-MoS2 with temperature at different values of hydrostatic pressure.
Figure 6: Phonon band structure (left) along the high-symmetry path
$\Gamma$-M-K-$\Gamma$, (middle) group velocity (vλ), and (right) scattering
time ($\tau_{\lambda}$) of ML-MoS2 at 300 K and at different hydrostatic
pressure of (a) 0 GPa, (b) 10 GPa, and (c) 25 GPa.

To investigate the pressure induced reduction in $\kappa_{\text{L}}$ in
greater detail, further analysis of the $\kappa_{\text{L}}$ is performed. The
mode-resolved values of the phonon group velocity (vλ) and scattering time
($\tau_{\lambda}$) for a particular phonon mode $\lambda$ are computed at 300
K as a function of phonon frequency at different pressures and presented with
the corresponding phonon band structures in Fig. 6. It can be seen that, both
vλ and $\tau_{\lambda}$ of ML-MoS2 is significantly higher within the
frequency range of 0 to 7 THz, which corresponds to the acoustic phonon modes.
It is, therefore, clear that the $\kappa_{\text{L}}$ stemming from the
acoustic modes is much higher than the optical modes. With increasing
pressure, due to the strengthening of the interatomic bonds, a stiffening of
the acoustic phonon modes and a blue shift in the frequency of the optical
phonon modes at the zone centre (at $\Gamma$) is observed. Thereby, the vλ
associated with the acoustic phonon modes increases slightly. However, the
phonon scattering time ($\tau_{\lambda}$) reduces significantly with
increasing pressure, as can be seen in Fig. 6. Owing to the large decrease in
$\tau_{\lambda}$, the $\kappa_{\text{L}}$ decreases with pressure.

Figure 7: Calculated mode Gr$\ddot{\text{u}}$neisen parameters ($\gamma$) of
ML-MoS2 as a function of phonon frequency at (a) 0 GPa and (b) 10 GPa. The
variation in the Gr$\ddot{\text{u}}$neisen parameter with temperature at 0 and
10 GPa is shown in the insets of (a) and (b). The phonon scattering rates at
300 K under 0 and 10 GPa hydrostatic pressure corresponding to the (c)
absorption and (d) emission processes.  Figure 8: Variation in the
thermoelectric figure of merit (zT) of ML-MoS2 under hole and electron doping
at various hydrostatic pressure as a function of (a) temperature at a fixed
carrier concentration of $1.3\times 10^{13}$ /cm2 and (b) carrier
concentration at a fixed temperature of 300 K.

In order to understand the pressure induced large reduction in
$\tau_{\lambda}$, the crystal anharmonicity and three-phonon scattering rate
of ML-MoS2 are computed at 300 K for two representative pressures, 0 GPa and
10 GPa, and shown in Fig. 7. The phonon scattering rate is strongly influenced
by the crystal anharmonicity. In fact, it is the anharmonic vibration that
limits the $\kappa_{\text{L}}$ to a finite value. The bond anharmonicity
measures the asymmetry in atomic vibration, which can be characterized by the
mode Gr$\ddot{\text{u}}$neisen parameters ($\gamma_{\text{i}}$), defined as
$\gamma_{\text{i}}$ = -$\frac{\text{V}_{0}}{\omega_{\text{i}}}$
${\frac{\Delta\omega_{\text{i}}}{\Delta\text{V}}}$, where $\omega_{\text{i}}$
is the frequency of the phonon mode i at the equilibrium volume V0. The
Gr$\ddot{\text{u}}$neisen parameter ($\gamma$) of ML-MoS2 has the highest
magnitude in the low frequency regime (0-2 THz), which belongs to the low
frequency acoustic modes, albeit mainly the out-of-plane acoustic mode or the
ZA mode. With increasing pressure, the mode Gr$\ddot{\text{u}}$neisen
parameter ($\gamma_{\text{i}}$) corresponding to the ZA mode increases, as can
be seen from Fig. 7 (a) and (b). Therefore, the anharmonic phonon scattering
rates are expected to increase with increasing pressure. Now, the
anharmonicity giving rise to the scattering events involving three phonons has
two possible scattering channels. One is the absorption process (+), where two
phonons combine to produce the third phonon. The other one is the emission
process (-), where one phonon splits into two different phonons. The
scattering rates corresponding to the two processes, absorption
(${\tau_{+}}^{-1}$) and emission (${\tau_{-}}^{-1}$), are plotted as a
function of phonon frequency (see Fig. 7), and the total scattering rate
(${\tau}^{-1}$= ${\tau_{+}}^{-1}$ \+ ${\tau_{-}}^{-1}$) is provided in the
supplementary material (see Fig. S4 in supplementary material). Due to the
energy conservation restriction, the absorption process (+) is dominant in the
low frequency regime, whereas the emission process (-) is dominant in the high
frequency regime. With increasing pressure, the scattering rates associated
with both processes increase, mainly the absorption one (${\tau_{+}}^{-1}$),
resulting in the decrease of the phonon relaxation time. The increase in the
probability of phonon scattering is attributed to the shortening of the
interatomic distances and the increased bond anharmonicity under the
application of pressure.

Finally, combining all the calculated transport parameters, such as the S,
S${}^{2}\sigma$, $\tau$, $\kappa$, the thermoelectric efficiency or the zT of
ML-MoS2 has been estimated and its variation with temperature (T) and carrier
concentration (N${}_{\text{2D}}$) has been explored under different pressure
conditions (see Fig. 8). The room temperature zT values obtained with pristine
ML-MoS2 with a carrier concentration of $1.3\times 10^{13}$ /cm2 are only 0.05
and 0.01 for hole and electron doping, respectively. Such a low value of zT is
not useful for practical application purposes. With the application of
pressure, the zT value under hole and electron doping is found to increase
significantly. For holes the zT value is found to increase monotonically with
pressure, whereas for electrons a dramatic enhancement is observed starting
from 10 GPa. At a pressure of 25 GPa, the room temperature zT values of ML-
MoS2 are found to be 0.17 and 0.35 for hole and electron doping, which are
significantly higher compared to the zero pressure values. Apart from the
reduction in lattice thermal conductivity, the gradual increase in hole
mobility and relaxation time with pressure act behind the enhancement of the
zT value under hole doping. However, for electrons, the increase in
thermopower (S) due to the valley convergence in the conduction band (CB) and
the involvement of the CB-$\Lambda$ in electronic transport with significantly
high carrier mobility acts in unison to result in the remarkable increase in
zT. In addition to the pressure induced enhancement, the zT values are found
to increase significantly with increasing temperature, as can be seen in Fig.
8 (a). The highest zT values at 900 K obtained with hole and electron doping
are 1.21 and 1.63, respectively. These zT values, obtained with ML-MoS2 are
comparable to any good commercial thermoelectric material. Also, it is worth
noting that the enhanced thermoelectric performance is not only achieved at a
specific doping concentration of $1.3\times 10^{13}$ /cm2; instead, the
enhancement can be achieved for a range of doping concentrations. To further
support this point, the variation in the zT of ML-MoS2 at 300 K as a function
of doping concentration under different pressures is shown in Fig. 8 (b). It
is evident that a much improved thermoelectric performance can be obtained at
the optimal doping concentration when a certain amount of pressure is applied.
Notably, the concurrent increase in zT for both electron and hole doping is
highly beneficial for its integration in thermoelectric devices, where both
the p- and n-legs are preferably made from a single material with high
thermoelectric efficiency.

##  IV Conclusions

In summary, first principles calculations have been performed to analyze the
hydrostatic strain induced changes in the electronic structure and transport
properties of ML-MoS2. The application of external pressure in ML-MoS2 induces
a direct to indirect band gap transition at around 10 GPa due to the shifting
of the CBM from K to $\Lambda$, which also increases the valley degeneracy
with the band edge at the K point still lying close in energy to the CBM at
$\Lambda$. Owing to this increase in valley degeneracy, the thermopower (S)
and thereby, the power factor (S${}^{2}\sigma$) show an increase for the
electron doping. Also, the application of hydrostatic pressure results in an
increase in electron mobility ($\mu$) and a dramatically enhanced relaxation
time ($\tau$), which follows from the low deformation potential associated
with CB-$\Lambda$. On the other hand, the lattice thermal conductivity
($\kappa_{\text{L}}$) is also found to decrease significantly with increasing
pressure due to the increase in anharmonicity and the resulting phonon
scattering. The pressure induced increase in S and $\mu$, and the decrease in
$\kappa_{\text{L}}$ act in unison to result in a large increase in zT. At 900
K with an external pressure of 25 GPa, zT values of 1.21 and 1.63 are achieved
for the hole and the electron doping, respectively, which are significantly
higher compared to that for the zero pressure case. This study, therefore,
highlights the importance of hydrostatic pressure in improving the
thermoelectric properties of ML-MoS2. Given that the application of
hydrostatic pressure increase the efficiency for both the n-type and p-type
doping, this fairly straight forward method can be beneficial for the
designing of thermoelectric module, where both legs should be preferably made
of the same material. Also, the proposed approach is expected to be equally
useful for all other semiconducting TMDCs with similar crystal structures.

###### Acknowledgements.

The first-principles calculations have been performed using the supercomputing
facility of IIT Kharagpur established under the National Supercomputing
Mission (NSM), Government of India and supported by the Centre for Development
of Advanced Computing (CDAC), Pune. AB acknowledges SERB POWER grant
(SPG/2021/003874) and BRNS regular grant (BRNS/37098) for the financial
assistance. SC acknowledges MHRD, India, for financial support.

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# Group invertibility of the sum in rings and its applications

Huanyin Chen  ,  Dayong Liu  and  Marjan Sheibani∗ School of Mathematics
Hangzhou Normal University
Hangzhou, China<EMAIL_ADDRESS>College of Science
Central South University of Forestry and Technology, China
<EMAIL_ADDRESS>Farzanegan Campus, Semnan University, Semnan, Iran
<EMAIL_ADDRESS>
###### Abstract.

We present new additive results for the group invertibility in a ring. Then we
apply our results to block operator matrices over Banach spaces and derive the
existence of group inverses of $2\times 2$ block operator matrices. These
generalize many known results, e.g., Benitez, Liu and Zhu(Linear Multilinear
Algebra, 59(2011), 279–289) and Zhou, Chen and Zhu(Comm. Algebra,
48(2020),676-690).

###### Key words and phrases:

group inverse; Drazin inverse; block operator matrix; Banach algebra; Banach
space.

###### 2010 Mathematics Subject Classification:

15A09, 47A08, 16U99.

Corresponding author: Marjan Sheibani

##  1\. Introduction

Let $R$ be a ring with an identity. An element $a\in R$ has Drazin inverse if
there exists $x\in R$ such that $x=xax,ax=xa,a^{n}=a^{n+1}x$ for some
$n\in{\mathbb{N}}.$ Such $x$ is unique, if it exists, and we denote it by
$a^{D}$. The smallest $n$ is called the Drazin index of $a$. If $a$ has Drazin
index $1$, $a$ is said to have group inverse $x$, and denote its group inverse
by $a^{\\#}$. As is well known, a square complex matrix $A$ has group inverse
if and only if $rank(A)=rank(A^{2})$. The group invertibility in a ring is
attractive. It has interesting applications of resistance distances to the
bipartiteness of graphs (see [15]). Many authors have studied group
invertibility from many different views, e.g., [2, 3, 4, 5, 6, 12, 13, 14, 17,
18]. It was also extensively investigated under the concept ”strongly
regularity” in ring theory (see [7]).

In [1, Theorem 2.1], Benitez et al. studied the group inverse of the sum of
two group invertible elements $a$ and $b$ in an algebra under the condition
$ab=0$. The group inverse of $P+Q$ of two group invertible complex matrices
$P$ and $Q$ was investigated in [10, Theorem 2.3] under the condition
$PQQ^{\\#}=QPP^{\\#}$. Zhou et al. extended the preceding result to a
Dedekind-finite ring in which $2$ is invertible (see [18, Theorem 3.1]). These
motivate us to explore the group inverse of the sum in a general setting.

Let $a\in R^{D}$. The element $a^{\pi}=1-aa^{D}$ is called the spectral
idempotent of $a$. In Section 2, we present new additive results for the group
invertibility by means of spectral idempotents in a ring. Let
$a,b,ba^{\pi},ab^{\pi}\in R^{\\#}$, $aba^{\pi}=0$ and $bab^{\pi}=0$. We prove
that $a+b\in R^{\\#}$ if and only if $aa^{\\#}b+bb^{\\#}a\in
R^{\\#},a^{\pi}b^{\pi}a=0$ and $b^{\pi}a^{\pi}b=0$. The preceding known
additive properties of the group invertibility are thereby extended to the
wider case.

In Section 3, we further investigate the additive properties of the group
invertibility under certain commutative-like conditions. Let $a,b\in R^{\\#}$.
If $a^{2}b=aba,b^{2}a=bab$, we prove that $a+b\in R^{\\#}$ if and only if
$1+a^{\\#}b\in R^{\\#}$ and $b^{\pi}a^{\pi}b=0$.

Let $X$ and $Y$ be Banach spaces, let $\mathcal{L}(X,Y)$ denote the set of all
bounded linear operators from $X$ to $Y$ and $\mathcal{L}(X)$ denote the set
of all bounded linear operators from $X$ to itself. The aim of the final
section is to explore the the group invertibility of a block operator matrix

$M=\left(\begin{array}[]{cc}A&B\\\
C&D\end{array}\right)~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(*)$

where $A\in\mathcal{L}(X),B\in\mathcal{L}(X,Y),C\in\mathcal{L}(Y,X)$ and
$D\in\mathcal{L}(Y)$. Here, $M$ is a bounded linear operator on $X\oplus Y$.
This problem is quite complicated and was studied by many authors (see [6, 13,
14, 17]). In [1], Benitez studied the group inverse of $2\times 2$ block
operator matrix $M$ under certain conditions. As applications of our results,
we provide many new conditions under which $M$ has group inverse. These extend
[1, Theorem 3.4-3.7] to the general setting.

Throughout the paper, all rings are associative with an identity. Let $p\in R$
be an idempotent, and let $x\in R$. Then we write
$x=pxp+px(1-p)+(1-p)xp+(1-p)x(1-p),$ and induce a Pierce representation given
by the matrix

$x=\left(\begin{array}[]{cc}pxp&px(1-p)\\\
(1-p)xp&(1-p)x(1-p)\end{array}\right)_{p}.$

##  2\. additive properties

The purpose of this section is to establish new additive results for group
inverses in a ring. We begin with

######  Lemma 2.1.

Let $p\in R$ be an idempotent and $x=\left(\begin{array}[]{cc}a&0\\\
c&d\end{array}\right)_{p}.$ If $a,d\in R^{\\#}$, then $x\in R^{\\#}$ if and
only if $d^{\pi}ca^{\pi}=0.$ In this case,
$x^{\\#}=\left(\begin{array}[]{cc}a^{\\#}&0\\\
u&d^{\\#}\end{array}\right)_{p},$ where

$u=d^{\pi}c(a^{\\#})^{2}+(d^{\\#})^{2}ca^{\pi}-d^{\\#}ca^{\\#}.$

###### Proof.

We obtain the result as in the proof of [11, Theorem 2.1].∎

######  Lemma 2.2.

Let $b\in R^{\\#}$ and $p\in R$ be an idempotent. If $pbp^{\pi}=0$ and
$bp^{\pi}\in R^{\\#}$, then $(bp^{\pi})^{\\#}=b^{\\#}p^{\pi}$ and
$(pb)^{\\#}=pb^{\\#}$.

###### Proof.

Since $(1-p)bp=0$, we have $b=\left(\begin{array}[]{cc}pb&0\\\
p^{\pi}bp&bp^{\pi}\end{array}\right)_{p}.$ Since $b,bp^{\pi}\in R^{\\#}$, it
follows by Lemma 2.1 that $pb\in R^{\\#}$. In this case, we have

$b^{\\#}=\left(\begin{array}[]{cc}(pb)^{\\#}&0\\\
&(bp^{\pi})^{\\#}\end{array}\right)_{p}.$

Therefore $(pb)^{\\#}=pb^{\\#}$ and $(bp^{\pi})^{\\#}=b^{\\#}p^{\pi}$, as
asserted.∎

######  Theorem 2.3.

Let $a,b,ba^{\pi}\in R^{\\#}$ and $aba^{\pi}=0$. Then $a+b\in R^{\\#}$ if and
only if $a(1+a^{\\#}b)\in R^{\\#}$ and $b^{\pi}a^{\pi}b=0$.

###### Proof.

Let $p=aa^{\\#}$. Since $aba^{\pi}=0$, we see that $aa^{\\#}ba^{\pi}=0$. Then

$a=\left(\begin{array}[]{cc}a_{1}&0\\\
0&0\end{array}\right)_{p},b=\left(\begin{array}[]{cc}b_{1}&0\\\
b_{3}&b_{4}\end{array}\right)_{p}.$

Here
$a_{1}=aa^{\\#}aaa^{\\#}=a,b_{3}=a^{\pi}baa^{\\#},b_{4}=a^{\pi}ba^{\pi}=ba^{\pi}$.
Then

$a+b=\left(\begin{array}[]{cc}a_{1}+b_{1}&0\\\
b_{3}&b_{4}\end{array}\right)_{p}.$

$\Longrightarrow$ By hypothesis, $b_{4}=ba^{\pi}\in R^{\\#}$. Since
$aba^{\pi}=0$, it follows by Lemma 2.2 that $b_{4}^{\\#}=b^{\\#}a^{\pi}$.
Clearly, $aa^{\\#}(a+b)a^{\pi}=0$ and $(a+b)a^{\pi}=ba^{\pi}\in R^{\\#}$.
Since $a+b\in R^{\\#}$, it follows by Lemma 2.2 that $aa^{\\#}(a+b)\in
R^{\\#}$ and $[aa^{\\#}(a+b)]^{\\#}=aa^{\\#}(a+b)^{\\#}$. Therefore
$aa^{\\#}(a+b)=a_{1}+b_{1}$ has group inverse. In view of Lemma 2.1, we have

$\begin{array}[]{lll}(a+b)^{\\#}&=&\left(\begin{array}[]{cc}w^{\\#}&0\\\
u&b^{\\#}a^{\pi}\end{array}\right)_{p},\end{array}$

where $w=aa^{\\#}(a+b)$ and

$\begin{array}[]{lll}u&=&(b_{4}^{\\#})^{2}b_{3}w^{\pi}+b_{4}^{\pi}b_{3}(w^{\\#})^{2}-b_{4}^{\\#}b_{3}w^{\\#}\\\
&=&b^{\\#}a^{\pi}b^{\\#}a^{\pi}baa^{\\#}w^{\pi}+b^{\pi}a^{\pi}b(w^{\\#})^{2}-b^{\\#}a^{\pi}bw^{\\#}.\end{array}$

We check that

$\begin{array}[]{ll}&(a+b)^{\pi}(a+b)\\\
=&\left(\begin{array}[]{cc}w^{\pi}&0\\\
-bw^{\\#}-ba^{\pi}u&b^{\pi}a^{\pi}\end{array}\right)_{p}\left(\begin{array}[]{cc}w&0\\\
a^{\pi}baa^{\\#}&ba^{\pi}\end{array}\right)_{p}\\\
=&\left(\begin{array}[]{cc}w^{\pi}w&0\\\
-(bw^{\\#}+ba^{\pi}u)w+b^{\pi}a^{\pi}baa^{\\#}&b^{\pi}a^{\pi}ba^{\pi}\\\
\end{array}\right)_{p}\\\ =&0.\end{array}$

Hence, $-(bw^{\\#}+ba^{\pi}u)w+b^{\pi}a^{\pi}baa^{\\#}=0$. Thus

$b^{\pi}a^{\pi}baa^{\\#}=b^{\pi}[-(bw^{\\#}+ba^{\pi}u)w+b^{\pi}a^{\pi}baa^{\\#}]=0.$

On the other hand,
$b^{\pi}a^{\pi}ba^{\pi}=b^{\pi}ba^{\pi}-b^{\pi}a^{\\#}aba^{\pi}=0$. Therefore
$b^{\pi}a^{\pi}b=b^{\pi}a^{\pi}b[aa^{\\#}+a^{\pi}]=0$, as required.

$\Longleftarrow$ We easily check that

$\begin{array}[]{lll}b_{4}^{\pi}b_{3}(a_{1}+b_{1})^{\pi}&=&(1-b^{\\#}a^{\pi}ba^{\pi})b_{3}(a_{1}+b_{1})^{\pi}\\\
&=&(a^{\pi}-b^{\\#}ba^{\pi})b_{3}(a_{1}+b_{1})^{\pi}\\\
&=&[b^{\pi}a^{\pi}b]aa^{\\#}(a_{1}+b_{1})^{\pi}\\\ &=&0.\end{array}$

According to Lemma 2.1, $a+b\in R^{\\#}$. Moreover, we have

$\begin{array}[]{lll}(a+b)^{\\#}&=&(a+b)^{D}\\\ &=&w^{\\#}+u+b^{\\#}a^{\pi}\\\
&=&w^{\\#}+b^{\\#}a^{\pi}b^{\\#}a^{\pi}baa^{\\#}w^{\pi}-b^{\\#}a^{\pi}bw^{\\#}+b^{\\#}a^{\pi}\\\
&=&[a(1+a^{\\#}b)]^{\\#}+b^{\\#}a^{\pi}b^{\\#}a^{\pi}baa^{\\#}[a(1+a^{\\#}b)]^{\pi}\\\
&-&b^{\\#}a^{\pi}b[a(1+a^{\\#}b)]^{\\#}+b^{\\#}a^{\pi},\end{array}$

as asserted.∎

######  Corollary 2.4.

Let $a,b,a^{\pi}b\in R^{\\#}$ and $a^{\pi}ba=0$. Then $a+b\in R^{\\#}$ if and
only if $(1+ba^{\\#})a\in R^{\\#}$ and $ba^{\pi}b^{\pi}=0$. In this case,

$\begin{array}[]{lll}(a+b)^{\\#}&=&[(1+ba^{\\#})a]^{\\#}+[(1+ba^{\\#})a]^{\pi}aa^{\\#}ba^{\pi}b^{\\#}a^{\pi}b^{\\#}\\\
&-&[(1+ba^{\\#})a]^{\\#}ba^{\pi}b^{\\#}+a^{\pi}b^{\\#}.\\\ \end{array}$

###### Proof.

Since $(R,\cdot)$ is a ring, $(R,*)$ is a ring with the multiplication
$a*b=b\cdot a$, i.e., it is the opposite ring. Then we complete the proof by
applying Theorem 2.3 to the ring $(R,*)$.∎

Now our first major result is demonstrated as follows.

######  Theorem 2.5.

Let $a,b,ba^{\pi},ab^{\pi}\in R^{\\#}$, $aba^{\pi}=0$ and $bab^{\pi}=0$. Then
$a+b\in R^{\\#}$ if and only if $aa^{\\#}b+bb^{\\#}a\in
R^{\\#},a^{\pi}b^{\pi}a=0$ and $b^{\pi}a^{\pi}b=0$.

###### Proof.

In view of Theorem 2.3, $a+b\in R^{\\#}$ if and only if
$a+aa^{\\#}b=a(1+a^{\\#}a)\in R^{\\#}$ and $b^{\pi}a^{\pi}b=0$. Since
$aa^{\\#}ba^{\pi}=0$ and $ba^{\pi}\in R^{D}$, it follows by Lemma 2.2 that
$(aa^{\\#}b)^{\\#}=aa^{\\#}b^{\\#}$. Let $q=[aa^{\\#}b][aa^{\\#}b]^{\\#}$.
Then $q=aa^{\\#}bb^{\\#}$. Obviously, we have
$qa(1-q)=aa^{\\#}bb^{\\#}a(1-aa^{\\#}bb^{\\#})=aa^{\\#}bb^{\\#}ab^{\pi}=0$.
Then

$a=\left(\begin{array}[]{cc}a_{1}&0\\\
a_{3}&a_{4}\end{array}\right)_{q},aa^{\\#}b=\left(\begin{array}[]{cc}b_{1}&0\\\
0&0\end{array}\right)_{q}.$

Here $a_{1}=qaq,a_{3}=q^{\pi}aq,a_{4}=q^{\pi}aq^{\pi}$. Then

$a+aa^{\\#}b=\left(\begin{array}[]{cc}a_{1}+b_{1}&0\\\
a_{3}&a_{4}\end{array}\right)_{q}.$

Here, we have

$\begin{array}[]{rll}a_{4}&=&(1-aa^{\\#}bb^{\\#})a(1-aa^{\\#}bb^{\\#})\\\
&=&(1-aa^{\\#}bb^{\\#})ab^{\pi}\\\ &=&ab^{\pi}\\\ &\in&R^{\\#}.\end{array}$

$\Longrightarrow$ Since $aba^{\pi}=0$, it follows by Theorem 2.3 that
$a^{\pi}b^{\pi}a=0$. Hence,

$\begin{array}[]{lll}a_{1}+b_{1}&=&qaq+qbq\\\
&=&aa^{\\#}bb^{\\#}abb^{\\#}+aa^{\\#}b^{2}b^{\\#}aa^{\\#}bb^{\\#}\\\
&=&aa^{\\#}bb^{\\#}a+aa^{\\#}b\\\ &=&bb^{\\#}a-a^{\pi}bb^{\\#}a+aa^{\\#}b\\\
&=&aa^{\\#}b+bb^{\\#}a.\end{array}$

Since $bab^{\pi}=0$, it follows by Lemma 2.2 that $a_{4}^{\\#}=a^{\\#}b^{\pi}$
and $a_{4}^{\pi}=q^{\pi}-aa^{\\#}b^{\pi}=a^{\pi}$. Moreover, we have
$a_{3}=q^{\pi}aq=(1-aa^{\\#}bb^{\\#})a^{2}a^{\\#}bb^{\\#}=abb^{\\#}-aa^{\\#}bb^{\\#}a$.
Hence $a^{\pi}a_{3}=0$.

By hypothesis, $ba^{\pi}\in R^{\\#}$. Since $a+b\in R^{\\#}$, it follows by
Theorem 2.3 that $aa^{\\#}(a+b)=a+aa^{\\#}b=a(1+a^{\\#}b)$ has group inverse.
Set $z=a_{1}+b_{1}$. Then we have

$\begin{array}[]{lll}(a+aa^{\\#}b)^{\\#}&=&\left(\begin{array}[]{cc}(a_{1}+b_{1})^{\\#}&\\\
v&a_{4}^{\\#}\end{array}\right)_{q}\\\
&=&\left(\begin{array}[]{cc}z^{\\#}&0\\\
v&a^{\\#}b^{\pi}\end{array}\right)_{q},\end{array},$

where

$\begin{array}[]{lll}v&=&(a_{4}^{\\#})^{2}a_{3}z^{\pi}+a_{4}^{\pi}a_{3}(z^{\\#})^{2}-a_{4}^{\\#}a_{3}z^{\\#}\\\
&=&[a^{\\#}b^{\pi}]^{2}a_{3}z^{\pi}+a^{\pi}a_{3}(z^{\\#})^{2}-a^{\\#}b^{\pi}a_{3}z^{\\#}\\\
&=&[a^{\\#}b^{\pi}]^{2}[abb^{\\#}-aa^{\\#}bb^{\\#}a]z^{\pi}-a^{\\#}b^{\pi}[abb^{\\#}-aa^{\\#}bb^{\\#}a]z^{\\#}.\end{array}$

We easily verify that $a_{1}+b_{1}\in R^{\\#}$, and so $aa^{\\#}b+bb^{\\#}a\in
R^{\\#}.$

$\Longleftarrow$ Clearly, $a_{4}\in R^{\\#}$ and

$\begin{array}[]{lll}a_{1}+b_{2}&=&q(a+aa^{\\#}b)q\\\
&=&aa^{\\#}bb^{\\#}(a+aa^{\\#}b)\\\
&=&bb^{\\#}a-a^{\pi}bb^{\\#}a+aa^{\\#}bb^{\\#}aa^{\\#}b\\\
&=&bb^{\\#}a+aa^{\\#}b-aa^{\\#}bb^{\\#}a^{\pi}b\\\
&=&bb^{\\#}a+aa^{\\#}b-aa^{\\#}(1-b^{\pi})a^{\pi}b\\\
&=&aa^{\\#}b+bb^{\\#}a\\\ &\in&R^{\\#}.\end{array}$

Moreover, we check that

$\begin{array}[]{lll}a_{4}^{\pi}a_{3}(a_{1}+b_{1})^{\pi}&=&a^{\pi}b^{\pi}abb^{\\#}(a_{1}+b_{1})^{\pi}\\\
&=&0.\end{array}$

According to Lemma 2.1,

$\begin{array}[]{lll}a(1+a^{\\#}b)&=&a+aa^{\\#}b\\\ &\in&R^{\\#}.\end{array}$

Moreover,

$\begin{array}[]{rll}[a(1+a^{\\#}b)]^{\\#}&=&[aa^{\\#}b+bb^{\\#}a]^{\\#}+a^{\\#}b^{\pi}\\\
&+&[a^{\\#}b^{\pi}]^{2}[abb^{\\#}-aa^{\\#}bb^{\\#}a][aa^{\\#}b+bb^{\\#}a]^{\pi}\\\
&-&a^{\\#}b^{\pi}[abb^{\\#}-aa^{\\#}bb^{\\#}a][aa^{\\#}b+bb^{\\#}a]^{\\#}.\end{array}$

This completes the proof.∎

In [18, Theorem 3.1], Zhou et al. investigated the group inverse of $a+b$
under the condition $aa^{\\#}b=bb^{\\#}a$ in a Dedekind ring with $2$ is
invertible. We now extend this result to a general setting.

######  Corollary 2.6.

Let $a,b,aa^{\\#}b\in R^{\\#}$ and $aa^{\\#}b=bb^{\\#}a$. Then $a+b\in
R^{\\#}$ if and only if $2aa^{\\#}b\in R^{\\#}$.

###### Proof.

Since $aa^{\\#}b=bb^{\\#}a$, we check that
$aba^{\pi}=a(aa^{\\#}b)a^{\pi}=a(bb^{\\#}a)a^{\pi}=0$. Similarly, we have
$bab^{\pi}=b^{\\#}(b^{2}a)b^{\pi}=b^{\\#}ba(bb^{\pi})=0$. Moreover, we have

$\begin{array}[]{c}a^{\pi}b^{\pi}a=a^{\pi}(1-bb^{\\#})a=a^{\pi}bb^{\\#}a=a^{\pi}aa^{\\#}b=0,\\\
b^{\pi}a^{\pi}b=b^{\pi}(1-aa^{\\#})b=b^{\pi}aa^{\\#}b=b^{\pi}bb^{\\#}a=0.\end{array}$

Since $aa^{\\#}b\in R^{\\#}$ and $aa^{\\#}ba^{\pi}=0$, similarly to Lemma 2.2,
$ba^{\pi}\in R^{\\#}$ and $[ba^{\pi}]^{\\#}=b^{\\#}a^{\pi}$. Likewise,
$ab^{\pi}\in R^{\\#}$. Therefore we complete the proof by Theorem 2.5.∎

Applying Corollary 2.6 to the opposite ring $(R,*)$, we dually derive

######  Corollary 2.7.

Let $a,b,abb^{\\#}\in R^{\\#}$ and $abb^{\\#}=baa^{\\#}$. Then $a+b\in
R^{\\#}$ if and only if $2abb^{\\#}\in R^{\\#}$.

##  3\. communicative-like conditions

The aim of this section is to investigate the group inverse in a ring under
some commutative-like conditions. We now prove:

######  Lemma 3.1.

Let $a,b\in R^{\\#}$. If $a^{2}b=aba$ and $b^{2}a=bab$, then $ab\in R^{\\#}$
and

$(ab)^{\\#}=a^{\\#}b^{\\#}.$

###### Proof.

Since $a,b\in R^{\\#}$, we have $a,b\in R^{D}$. In view of [19, Theorem 3.1],
$ab\in R^{D}$ and $(ab)^{D}=a^{D}b^{D}=a^{\\#}b^{\\#}$. One easily checks that

$(ab)^{2}(ab)^{D}=ababa^{\\#}b^{\\#}=(a^{2}a^{\\#})(b^{2}b^{\\#})=ab.$

Therefore $(ab)^{\\#}=a^{\\#}b^{\\#},$ as desired.∎

######  Theorem 3.2.

Let $a,b\in R^{\\#}$. If $a^{2}b=aba,b^{2}a=bab$, then $a+b\in R^{\\#}$ if and
only if $aa^{\\#}b+bb^{\\#}a\in R^{\\#},a^{\pi}b^{\pi}a=0,b^{\pi}a^{\pi}b=0$.

###### Proof.

Since $a^{2}b=aba$, we have
$aba^{\pi}=a^{\\#}(a^{2}b)a^{\pi}=a^{\\#}(aba)a^{\pi}=0$. It follows from
$b^{2}a=bab$ that $bab^{\pi}=0$. Since $a(ab)=a^{2}b=aba=(ab)a$, we have
$a^{\\#}(ab)=a^{D}(ab)=(ab)a^{D}=(ab)a^{\\#}$ by [8, Theorem 2.2]; hence,
$(aa^{\\#})^{2}b=a^{\\#}(ab)=(ab)a^{\\#}=aa^{\\#}baa^{\\#}$. Also we have
$b^{2}aa^{\\#}=(bab)a^{\\#}=baa^{\\#}b$. In view of Lemma 3.1,
$(aa^{\\#}b)^{\\#}=aa^{\\#}b^{\\#}.$ Since $aa^{\\#}ba^{\pi}=0$, analogously
to Lemma 2.2, $ba^{\pi}\in R^{\\#}$ and $(ba^{\pi})^{\\#}=b^{\\#}a^{\pi}$.
Similarly, we show that $ab^{\pi}\in R^{\\#}$. Therefore we complete the proof
by Theorem 2.5.∎

######  Corollary 3.3.

Let $a,b\in R^{\\#}$. If $ab=ba$, then $a+b\in R^{\\#}$ if and only if
$aa^{\\#}b+bb^{\\#}a\in R^{\\#}$.

###### Proof.

Since $ab=ba$, we have $a^{2}b=aba$ and $b^{2}a=ba$. The result follows by
Theorem 3.2.∎

We now illustrates Theorem 3.2 by the following example:

######  Example 3.4.

Let $a=\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&1&0&0\\\ 0&0&0&1\\\
0&0&0&0\end{array}\right),b=\left(\begin{array}[]{cccc}\frac{1}{3}&0&0&0\\\
0&0&0&0\\\ 0&0&0&0\\\
0&0&0&\frac{1}{3}\end{array}\right)\in{\mathbb{C}}^{4\times 4}$. Then
$a^{2}b=aba,b^{2}a=bab$. But $ab\neq ba$. We compute that

$a^{\\#}=\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&1&0&0\\\ 0&0&0&0\\\
0&0&0&0\end{array}\right),b^{\\#}=\left(\begin{array}[]{cccc}3&0&0&0\\\
0&0&0&0\\\ 0&0&0&0\\\ 0&0&0&3\end{array}\right).$

In this case,

$(aa^{\\#}b+bb^{\\#}a)^{\\#}=\left(\begin{array}[]{cccc}\frac{3}{4}&0&0&0\\\
0&0&0&0\\\ 0&0&0&0\\\
0&0&0&0\end{array}\right),(a+b)^{\\#}=\left(\begin{array}[]{cccc}\frac{3}{4}&0&0&0\\\
0&1&0&0\\\ 0&0&0&0\\\ 0&0&0&3\end{array}\right).$

We are now ready to prove:

######  Theorem 3.5.

Let $a,b\in R^{\\#}$. If $a^{2}b=aba,b^{2}a=bab$, then $a+b\in R^{\\#}$ if and
only if $1+a^{\\#}b\in R^{\\#}$ and $b^{\pi}a^{\pi}b=0$.

###### Proof.

$\Longrightarrow$ Since $a^{2}b=aba$, we see that
$aba^{\pi}=a^{\\#}(a^{2}b)a^{\pi}=a^{\\#}abaa^{\pi}=0$. As in the proof of
Theorem 3.2, we show that $ba^{\pi}$ has group inverse. By virtue of Theorem
2.3, $b^{\pi}a^{\pi}b=0$. Write $1+a^{\\#}b=a^{\pi}+a^{\\#}(a+b)$. One easily
checks that

$\begin{array}[]{rll}(a^{\\#})^{2}(a+b)&=&a^{\\#}+(a^{\\#})^{2}b\\\
&=&a^{\\#}+(a^{\\#})^{3}(ab)\\\ &=&a(a^{\\#})^{2}+(a^{\\#})^{2}(ab)a^{\\#}\\\
&=&a^{\\#}aa^{\\#}+a^{\\#}ba^{\\#}\\\ &=&a^{\\#}(a+b)a^{\\#}.\end{array}$

Likewise, we have $(a+b)^{2}a^{\\#}=(a+b)a^{\\#}(a+b).$ In light of Lemma 3.1,
$[a^{\\#}(a+b)]^{\\#}=a(a+b)^{\\#}$. Clearly, $a^{\pi}[a^{\\#}(a+b)]=0$. Also
we have
$[a^{\\#}(a+b)]a^{\pi}=a^{\\#}ba^{\pi}=(a^{\\#})^{3}(a^{2}b)a^{\pi}=(a^{\\#})^{3}(aba)a^{\pi}=0$.
According to Corollary 2.6, $a^{\pi}+a^{\\#}(a+b)\in R^{\\#}$. Then
$1+a^{\\#}b\in R^{\\#}$, as desired.

$\Longleftarrow$ Clearly, $aa^{\\#}(a+b)=a(1+a^{\\#}b)$. By hypothesis, we
have

$a(1+a^{\\#}b)=a+aa^{\\#}b=a+aa^{\\#}b=a+(a^{\\#})^{2}aba=(1+a^{\\#}b)a.$

By virtue of Lemma 3.1, $[aa^{\\#}(a+b)]^{\\#}=a^{\\#}(1+a^{\\#}b)^{\\#}$. As
in the proof of Theorem 3.2, $ba^{\pi}\in R^{\\#}$. Since $b^{\pi}a^{\pi}b=0$,
we complete the proof by Theorem 2.3.∎

######  Corollary 3.6.

Let $a,b\in R^{\\#}$. If $ab^{2}=bab,ba^{2}=aba$, then $a+b\in R^{\\#}$ if and
only if $1+ba^{\\#}\in R^{\\#}$ and $ba^{\pi}b^{\pi}=0$.

###### Proof.

Since $(R,\cdot)$ is a ring, $(R,*)$ is a ring with the multiplication
$a*b=b\cdot a$. Then we complete the proof by applying Theorem 3.5 to the
opposite ring $(R,*)$.∎

##  4\. applications

We will investigate the group invertibility of a block operator matrix $M$ as
in $(*)$. The main idea is to split $M$ as the sum of two special block
operator matrices. Then we apply our additive results and find new conditions
on bounded linear operators $A,B,C$ and $D$ under which $M$ is group
invertible.

######  Theorem 4.1.

Let $A,D$ and $BC$ have group inverses. If
$B(CB)^{\pi}=0,C(BC)^{\pi}=0,ABD^{\pi}=0$ and $DCA^{\pi}=0$, then $M$ has
group inverse if and only if

  1. (1)

$A^{\pi}(BC)^{\pi}A=0,D^{\pi}(CB)^{\pi}D=0$;

  2. (2)

$\left(\begin{array}[]{cc}A&AA^{\\#}B\\\ DD^{\\#}C&D\end{array}\right)$ has
group inverse.

###### Proof.

Write $M=P+Q$, where

$P=\left(\begin{array}[]{cc}A&0\\\
0&D\end{array}\right),Q=\left(\begin{array}[]{cc}0&B\\\
C&0\end{array}\right).$

Then

$P^{\\#}=\left(\begin{array}[]{cc}A^{\\#}&0\\\
0&D^{\\#}\end{array}\right),P^{\pi}=\left(\begin{array}[]{cc}A^{\pi}&0\\\
0&D^{\pi}\end{array}\right).$

Since $BC$ has group inverse, we have

$(Q^{2})^{D}=\left(\begin{array}[]{cc}BC&0\\\
0&CB\end{array}\right)^{D}=\left(\begin{array}[]{cc}(BC)^{\\#}&0\\\
0&(CB)^{\\#}\end{array}\right).$

Since $\mathcal{L}(X\oplus Y)$ is a Banach algebra, it follows by [9, Theorem
2.1] that $Q$ has Drazin inverse and

$Q^{D}=Q(Q^{2})^{D}=\left(\begin{array}[]{cc}0&B(CB)^{\\#}\\\
C(BC)^{\\#}&0\end{array}\right).$

Moreover, we have

$Q^{\pi}=\left(\begin{array}[]{cc}(BC)^{\pi}&0\\\
0&(CB)^{\pi}\end{array}\right).$

Hence,

$QQ^{D}=Q^{D}Q,Q^{D}=Q^{D}QQ^{D}.$

It is easy to verify that

$\begin{array}[]{lll}QQ^{\pi}&=&\left(\begin{array}[]{cc}0&B\\\
C&0\end{array}\right)\left(\begin{array}[]{cc}(BC)^{\pi}&0\\\
0&(CB)^{\pi}\end{array}\right)\\\ &=&\left(\begin{array}[]{cc}0&B(CB)^{\pi}\\\
C(BC)^{\pi}&0\end{array}\right)\\\ &=&0;\end{array}$

whence, $Q=Q^{2}Q^{D}$. Then $Q$ has group inverse and

$Q^{\\#}=\left(\begin{array}[]{cc}0&B(CB)^{\\#}\\\
C(BC)^{\\#}&0\end{array}\right).$

Therefore we compute that

$\begin{array}[]{lll}P^{\pi}Q^{\pi}P&=&\left(\begin{array}[]{cc}A^{\pi}(BC)^{\pi}&0\\\
0&D^{\pi}(CB)^{\pi}\end{array}\right)\left(\begin{array}[]{cc}A&0\\\
0&D\end{array}\right)\\\ &=&\left(\begin{array}[]{cc}A^{\pi}(BC)^{\pi}A&0\\\
0&D^{\pi}(CB)^{\pi}D\end{array}\right)\\\ &=&0.\end{array}$

We easily check that

$\begin{array}[]{rll}PQP^{\pi}&=&\left(\begin{array}[]{cc}0&ABD^{\pi}\\\
DCA^{\pi}&0\end{array}\right),\\\
PP^{\\#}(P+Q)&=&\left(\begin{array}[]{cc}A&AA^{\\#}B\\\
DD^{\\#}C&D\end{array}\right).\end{array}$

Therefore we complete the proof by Theorem 2.3.∎

######  Corollary 4.2.

Let $A,D$ and $BC$ have group inverses. If $AB=0,B(CB)^{\pi}=0,C(BC)^{\pi}=0$
and $DCA^{\pi}=0$, then $M$ has group inverse if and only if $BCA=0$ and
$D^{\pi}(CB)^{\pi}D=0$.

###### Proof.

Clearly, $D^{\pi}(DD^{\\#}C)A^{\pi}=0$. As in the proof of [1, Theorem 3.2],
$\left(\begin{array}[]{cc}A&AA^{\\#}B\\\ DD^{\\#}C&D\end{array}\right)$ has
group inverse. This completes the proof by Theorem 4.1.∎

######  Corollary 4.3.

Let $A,D$ and $BC$ have group inverses. If $DC=0,B(CB)^{\pi}=0,C(BC)^{\pi}=0$
and $ABD^{\pi}=0$, then $M$ has group inverse if and only if
$A^{\pi}(BC)^{\pi}A=0$ and $CBD=0$.

###### Proof.

Obviously, $A^{\pi}(AA^{\\#}B)D^{\pi}=0$. Similarly to [1, Theorem 3.1],
$\left(\begin{array}[]{cc}A&AA^{\\#}B\\\ DD^{\\#}C&D\end{array}\right)$ has
group inverse. There we obtain the result by Theorem 4.1.∎

######  Corollary 4.4.

Let $A\in{\mathbb{C}}^{m\times m}$ and $D\in{\mathbb{C}}^{n\times n}$ have
group inverses. If $r(B)=r(C)=r(BC)=r(CB)$, then

  1. (1)

If $AB=0$ and $DCA^{\pi}=0$, then $M$ has group inverse if and only if $BCA=0$
and $D^{\pi}(CB)^{\pi}D=0$.

  2. (2)

If $DC=0$ and $ABD^{\pi}=0$, then $M$ has group inverse if and only if
$A^{\pi}(BC)^{\pi}A=0$ and $CBD=0$.

###### Proof.

Since $r(B)=r(C)=r(BC)=r(CB)$, by virtue of [3, Lemma 2.3], $BC$ and $CB$ have
group inverses. Let $N=\left(\begin{array}[]{cc}0&B\\\ C&0\end{array}\right)$.
Then $N^{2}=\left(\begin{array}[]{cc}BC&0\\\ 0&CB\end{array}\right).$ Hence,
$rank(N^{2})=rank(BC)+rank(CB)=rank(B)+rank(C)=rank(N)$. Therefore $N$ has
group inverse. This implies that

$NN^{\pi}=\left(\begin{array}[]{cc}0&B(CB)^{\pi}\\\
C(BC)^{\pi}&0\end{array}\right)=0,$

and so $B(CB)^{\pi}=0,C(BC)^{\pi}=0.$ Therefore we complete the proof by
Corollary 4.2 and Corollary 4.3.∎

We now ready to prove:

######  Theorem 4.5.

Let $A$ and $D$ have group inverses. If
$A^{\pi}B=0,D^{\pi}C=0,BCA^{\pi}=0,ABC=0$ and $BD=BCA^{\\#}B$, then $M$ has
group inverse.

###### Proof.

Write $M=P+Q$, where

$P=\left(\begin{array}[]{cc}A&B\\\
0&0\end{array}\right),Q=\left(\begin{array}[]{cc}0&0\\\
C&D\end{array}\right).$

Since $A^{\pi}B=0,D^{\pi}C=0$, we easily see that $P$ and $Q$ have group
inverses. Moreover, we have

$\begin{array}[]{c}P^{\\#}=\left(\begin{array}[]{cc}A^{\\#}&(A^{\\#})^{2}B\\\
0&0\end{array}\right),P^{\pi}=\left(\begin{array}[]{cc}A^{\pi}&-A^{\\#}B\\\
0&I_{n}\end{array}\right);\\\ Q^{\\#}=\left(\begin{array}[]{cc}0&0\\\
(D^{\\#})^{2}C&D^{\\#}\end{array}\right),Q^{\pi}=\left(\begin{array}[]{cc}I_{m}&0\\\
-D^{\\#}C&D^{\pi}\end{array}\right).\end{array}$

We compute that

$\begin{array}[]{lll}PP^{\\#}Q&=&\left(\begin{array}[]{cc}AA^{\\#}&A^{\\#}B\\\
0&0\end{array}\right)\left(\begin{array}[]{cc}0&0\\\ C&D\end{array}\right)\\\
&=&\left(\begin{array}[]{cc}A^{\\#}BC&A^{\\#}BD\\\ 0&0\end{array}\right)\\\
&=&0.\end{array}$

Then

$[QP^{\pi}]^{D}=Q[(P^{\pi}Q)^{D}]^{2}P^{\pi}=Q(Q^{D})^{2}P^{\pi}=Q^{D}P^{\pi}.$

We check that $[QP^{\pi}]^{2}[QP^{\pi}]^{D}=Q^{2}Q^{D}P^{\pi}=QP^{\pi}.$
Therefore $QP^{\pi}$ has group inverse. Moreover, we check that

$\begin{array}[]{rll}PQP^{\pi}&=&\left(\begin{array}[]{cc}A&B\\\
0&0\end{array}\right)\left(\begin{array}[]{cc}0&0\\\
C&D\end{array}\right)\left(\begin{array}[]{cc}A^{\pi}&-A^{\\#}B\\\
0&I_{n}\end{array}\right)\\\ &=&\left(\begin{array}[]{cc}BCA^{\pi}&BD-
BCA^{\\#}B\\\ 0&0\end{array}\right)\\\ &=&0,\\\
Q^{\pi}P^{\pi}Q&=&\left(\begin{array}[]{cc}I&0\\\
-D^{\\#}C&D^{\pi}\end{array}\right)\left(\begin{array}[]{cc}A^{\pi}&-A^{\\#}B\\\
0&I\end{array}\right)\left(\begin{array}[]{cc}0&0\\\ C&D\end{array}\right)\\\
&=&\left(\begin{array}[]{cc}I&0\\\
-D^{\\#}C&D^{\pi}\end{array}\right)\left(\begin{array}[]{cc}-A^{\\#}BC&-A^{\\#}BD\\\
C&D\end{array}\right)\\\ &=&\left(\begin{array}[]{cc}-A^{\\#}BC&-A^{\\#}BD\\\
D^{\\#}CA^{\\#}BC&D^{\\#}CA^{\\#}BD\end{array}\right)\\\ &=&0.\end{array}$

Further, we have

$\begin{array}[]{lll}PP^{\\#}(P+Q)&=&\left(\begin{array}[]{cc}AA^{\\#}&A^{\\#}B\\\
0&0\end{array}\right)\left(\begin{array}[]{cc}A&B\\\ C&D\end{array}\right)\\\
&=&\left(\begin{array}[]{cc}A+A^{\\#}BC&AA^{\\#}B+A^{\\#}BD\\\
0&0\end{array}\right)\\\ &=&\left(\begin{array}[]{cc}A&AA^{\\#}B\\\
0&0\end{array}\right).\end{array}$

Since $A^{\pi}(AA^{\\#}B)=0$ and $A$ has group inverse, $PP^{\\#}(P+Q)$ has
group inverse. Therefore we complete the proof by Theorem 2.3.∎

######  Corollary 4.6.

( [1, Theorem 3.4]) Let $A$ and $D$ have group inverses. If
$A^{\pi}B=0,D^{\pi}C=0,BC=0$ and $BD=0$, then $M$ has group inverse.

###### Proof.

This is obvious by Theorem 4.5.∎

######  Theorem 4.7.

Let $A,D$ and have group inverses. If $CA^{\pi}=0,BD^{\pi}$
$=0,A^{\pi}BC=0,BCA=0$ and $DC=CA^{\\#}BC$, then $M$ has group inverse.

###### Proof.

Write $M=P+Q$, where

$P=\left(\begin{array}[]{cc}A&0\\\
C&0\end{array}\right),Q=\left(\begin{array}[]{cc}0&B\\\
0&D\end{array}\right).$

Since $CA^{\pi}=0,BD^{\pi}=0$, $P$ and $Q$ have group inverses. Moreover, we
have

$\begin{array}[]{c}P^{\\#}=\left(\begin{array}[]{cc}A^{\\#}&0\\\
C(A^{\\#})^{2}&0\end{array}\right),P^{\pi}=\left(\begin{array}[]{cc}A^{\pi}&0\\\
-CA^{\\#}&I\end{array}\right);\\\
Q^{\\#}=\left(\begin{array}[]{cc}0&B(D^{\\#})^{2}\\\
0&D^{\\#}\end{array}\right),Q^{\pi}=\left(\begin{array}[]{cc}I&-BD^{\\#}\\\
0&D^{\pi}\end{array}\right).\end{array}$

We check that

$\begin{array}[]{lll}QPP^{\\#}&=&\left(\begin{array}[]{cc}0&B\\\
0&D\end{array}\right)\left(\begin{array}[]{cc}AA^{\\#}&0\\\
CA^{\\#}&0\end{array}\right)\\\ &=&\left(\begin{array}[]{cc}BCA^{\\#}&0\\\
DCA^{\\#}&0\end{array}\right)\\\ &=&0.\end{array}$

By using Cline’s formula, we have

$\begin{array}[]{lll}[P^{\pi}Q]^{D}&=&P^{\pi}[(QP^{\pi})^{D}]^{2}Q\\\
&=&P^{\pi}[Q^{D}]^{2}Q\\\ &=&P^{\pi}Q^{D}.\end{array}$

Since $[P^{\pi}Q]^{2}[P^{\pi}Q]^{D}=P^{\pi}Q$, it follows that $P^{\pi}Q$ has
group inverse. Moreover, we verify that

$\begin{array}[]{lll}P^{\pi}QP&=&\left(\begin{array}[]{cc}0&A^{\pi}B\\\
0&D-CA^{\\#}B\end{array}\right)\left(\begin{array}[]{cc}A&0\\\
C&0\end{array}\right)\\\ &=&\left(\begin{array}[]{cc}A^{\pi}BC&0\\\
(D-CA^{\\#}B)C&0\end{array}\right).\end{array}$

Moreover, we compute that

$\begin{array}[]{lll}QP^{\pi}Q^{\pi}&=&\left(\begin{array}[]{cc}0&B\\\
0&D\end{array}\right)\left(\begin{array}[]{cc}A^{\pi}&0\\\
-CA^{\\#}&I\end{array}\right)\left(\begin{array}[]{cc}I&-BD^{\\#}\\\
0&D^{\pi}\end{array}\right)\\\
&=&\left(\begin{array}[]{cc}-BCA^{\\#}&BCA^{\\#}BD^{\\#}\\\
-DCA^{\\#}&DCA^{\\#}BD^{\\#}\end{array}\right)\\\ &=&0.\end{array}$

We also see that

$\begin{array}[]{lll}(I+QP^{\\#})P&=&(P+Q)PP^{\\#}\\\
&=&\left(\begin{array}[]{cc}A&B\\\
C&D\end{array}\right)\left(\begin{array}[]{cc}AA^{\\#}&0\\\
CA^{\\#}&0\end{array}\right)\\\ &=&\left(\begin{array}[]{cc}A&0\\\
CAA^{\\#}&0\end{array}\right).\end{array}$

Since $[CAA^{\\#}]A^{\pi}=0$ and $A$ has group inverse, $(I+QP^{\\#})P$ has
group inverse. Therefore $M$ has group inverse by Corollary 2.4.∎

######  Corollary 4.8.

( [1, Theorem 3.6]) Let $A$ and $D$ have group inverses. If
$CA^{\pi}=0,BD^{\pi}=0,BC=0$ and $DC=0$, then $M$ has group inverse.

###### Proof.

This is obvious by Theorem 4.7.∎

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