# At the Edge of a Seamless Cloud Experience ††thanks: This work has been made
with support in partnership with Proximus Luxembourg S.A.

Samuel Rac1 and Mats Brorsson2  Interdisciplinary Centre for Security,
Reliability, and Trust
University of Luxembourg, Luxembourg
Email<EMAIL_ADDRESS>2<EMAIL_ADDRESS>
###### Abstract

There is a growing need for low latency for many devices and users. The
traditional cloud computing paradigm can not meet this requirement,
legitimizing the need for a new paradigm. Edge computing proposes to move
computing capacities to the edge of the network, closer to where data is
produced and consumed. However, edge computing raises new challenges. At the
edge, devices are more heterogeneous than in the data centre, where everything
is optimized to achieve economies of scale. Edge devices can be mobile, like a
car, which complicates architecture with dynamic topologies. IoT devices
produce a considerable amount of data that can be processed at the Edge.

In this paper, we discuss the main challenges to be met in edge computing and
solutions to achieve a seamless cloud experience. We propose to use
technologies like containers and WebAssembly to manage applications’ execution
on heterogeneous devices.

###### Index Terms:

Cloud computing, Edge computing, Container orchestration, WebAssembly

##  I Introduction

Cloud computing is evolving and so are the requirements that modern
applications put on the infrastructure and services provided. On the
application side, we particularly see the need for Edge computing growing
rapidly. The definition of the _edge_ varies in the literature. From a cloud
provider perspective, the point-of-presence that a user connects to is the
edge. In a cyber-physical system (CPS), the edge is likely to be the system to
which IoT devices are connected. For the purpose of this paper, we define Edge
computing to be application processing that takes place outside a data centre.
Some people also call this Fog computing [15].

An infrastructure able to support different kinds of edge applications (given
the above definition of the Edge), might be quite complex. IoT, and other,
devices might be connected to the data centres directly or through gateways
acting as aggregators of device data. Some connections are directly through
the internet, others through other means like 5G/LTE or edge radio protocols.
In any and all of these locations, devices, gateways, telecommunication base
station and cloud nodes, there might be compute and storage capabilities, and
we then have an infrastructure from device to network edge, to cloud edge and
to central cloud data centre with many steps in between, each possibly having
compute capabilities. See Figure 1, which illustrates one possibility for this
infrastructure architecture.

Figure 1: A hierarchical cloud infrastructure.

Irrespective of what applications are run on such an infrastructure, the
current software deployment model is very fragmented and highly manual.
Assuming the situation in Figure 1, software may be deployed at several
levels:

  * •

At a data centre in the cloud (where there is a choice of multiple points-of-
presence).

  * •

At a data centre of another cloud provider.

  * •

At a gateway node.

  * •

At a network edge node.

  * •

At an edge device.

The state-of-the-art method to deploy an application consisting of multiple
components, a distributed system, is to use a container orchestration method
such as Kubernetes. This allows for a descriptive model of specifying the
desired state of the application and Kubernetes will work at making sure that
the current state matches the desired state even during disturbances such as
node failure and changes of requirements such as sudden surge in traffic.

This model works well for an application in one, or spanning a few, data
centres in one cloud. However, if you want to deploy an application spanning
multiple or all levels of a hierarchical cloud according to Figure 1, all
levels outside the traditional cloud are mostly managed manually without any
adaptation on the changing needs of an application.

Furthermore, while data centre nodes tend to be quite homogeneous, gateways
and edge devices come in many shapes and forms and may use different processor
architectures (currently mainly x86 or ARM). Currently, you need to explicitly
target the architecture of the node where the code should run. Finally, there
is another level of heterogeneity: accelerators. In data centres there are
GPU-accelerators, often used for training machine learning models.
Increasingly we see other types of accelerators as well, such as FPGAs and
Tensor Processor Units (TPUs). At the edge, you often see many kinds of
accelerators for machine learning inference, graphics processing, etc.

All-in-all, developing for this heterogeneous infrastructure landscape is
difficult and may lead to cloud vendor lock-in and infrastructure over-
provisioning as well as inferior performance and/or power consumption
characteristics. We believe deploying applications to an Edge application
should be as easy as deploying an application to services like Google GKE
Autopilot [2].

In this paper, we explore the state of the art in seamless cloud computing
across such an infrastructure along three dimensions: i) architecture, ii)
execution, and iii) orchestration. The Architecture dimension is about
technologies and concepts on how to connect components to form a seamless
cloud experience. The Execution dimension is about various solutions to deal
with heterogeneous platforms, and, finally, the Orchestration dimension is
about solutions to manage an application spreading across the infrastructure.

##  II Architecture

Finding an architecture for a seamless cloud computing experience is not an
easy task. The challenge here is to identify the different layers and
aggregate them to form a seamless cloud experience. We review here the main
architectural principles that has been proposed to extend traditional cloud
architectures to the edge and in between.

###  II-A Topology

####  II-A1 Cloudlet topology

Cloudlet is an essential paradigm to achieve a seamless cloud. Cloudlet can be
a server, a computer, or a little cluster located at the edge of the network,
typically at one hop of the devices. With mobile devices and Cloud, they
achieve a 3-tier continuum architecture [15]. Cloudlets are designed to
provide services to mobile devices [10]. The key idea of Cloudlet is to enable
mobile devices offloading in their vicinity. On these devices, energy
consumption is one of the major parameters for offloading. Cloudlets must be
close to mobile devices to satisfy the Quality of Service (QoS). Cloudlets
should also be able to work without the Internet. Working in a local area
would also avoid congestion in the core network. To have a seamless cloud
experience, Cloudlet, traditional Cloud, and Mobile devices have to work in
synergy. Offloading on Cloudlet or traditional Cloud should be transparent for
the end-user. Choosing one or the other must consider the following
parameters: latency, task size, energy, resource availability.

####  II-A2 Ad-hoc topology

Ad-hoc edge computing aims to create a dynamic, distributed and decentralized
infrastructure to take into account edge constraints [5]. Ad-hoc edge is a
very resilient architecture that relies on a decentralized network, this
should help avoid having a single point of failure. Ad-hoc network should have
a lightweight implementation not to disturb the normal functioning of edge
devices[4].

Mobility is a game-changer for the practices of resources management. When
devices change their location, this causes problems: unreliable connection,
resource availability and intermittent connection to the internet. Mobility
affects the availability of edge nodes. Dynamicity present in resource
availability, also named node churn, reflects the behaviour of resources that
appear and disappear from the system[4],[5]. Also, the availability of devices
can be determined by their energy management and battery discharge.

###  II-B Networking

Networking is a keystone in the construction of a seamless cloud. A reliable
networking architecture is required for running distributed applications. The
development of paradigms like micro-services architectures also shows
dependences on the network. In addition, networking devices can have an active
role in edge computing. In-network computing is a paradigm where applications
are distributed through the network infrastructure [3]. E. g., a switch or a
router should become a small edge node and propose computing power to run
small applications.

Software-Defined Networking (SDN) is a technology that can provide seamless
networking configuration. Configuring networking takes time, there are devices
to configure manually (e. g., routers, firewalls). SDN permits to automate
networking configuration and management. The idea of SDN is similar to
Operating System, adding abstraction on top of hardware to simplify uses. SDN
has a separation between the control plane and the user plane to increase
network management flexibility. Edge devices can be mobile, having an
intermittent connection or limited bandwidth. Therefore, it is important to
easily manage networking to be able to adapt to this dynamic networking
topology.

SDN flexibility allows having a service-centric architecture [1]. Networking
can be dynamically adapted to the task’s needs. The SDN controller has a high-
level view of the network that helps deploy an application and its networking.
If the application needs to scale up or down, networking will dynamically
follow its needs.

A goal of edge computing is to provide low latency and location-aware
services. The idea of a seamless cloud experience is to execute tasks at the
edge of the network as if they were processed in the traditional Cloud, with a
low latency benefit. When tasks or data goes to the Cloud through the edge
node, routing should not affect performances. SDN and Networking
virtualization provides the flexibility needed for a seamless cloud
experience.

Besides, edge computing will take advantage of the fifth generation of mobile
networks (5G). It seems difficult to achieve mobility and use cases like V2X
without using those networks. The 5G network will bring three major
innovations: enhanced Mobile Broad Band (eMBB), Ultra-Reliable Low-Latency
Communications (URLLC), and massive Machine-Type Communications (mMTC). 5G
technologies rely in part on the virtualization of networking.

SDN will make networking easier to deploy for telco operators. There will be
less need for specific and expensive hardware thanks to virtualization. Also,
resources will be mutualized for better performance.

Network slicing is a 5G feature that proposes a new way of virtualization for
network bandwidth usage. It allows a more dynamic and flexible infrastructure.
Network slices can be design for a particular purpose, e. g. reserve resources
to achieve ultra-low latency on one slice and manage many users on another.
Slices can guarantee some levels of Quality of Service (QoS). SDN has a role
in network slicing deployment.

Virtualization is more and more important in networking. With Network Function
Virtualisation (NFV), network function (NF) can be managed like a VM.
According to [9], NF can be containerized, and VNF should become Containerized
Network Functions (CNFs). Containerization allows even more flexibility and
enables Kubernetes-based networking.

##  III Execution

The purpose of Edge computing is to transfer computation abilities at the edge
of the network. Systems can be composed of many kinds of devices. It then
becomes necessary to find a way to manage all of them. Managing a cluster with
a lot of similar nodes is a problem with a known solution. However, it becomes
problematic when we need to manage heterogeneous nodes. In a heterogeneous
cluster, there are multiple computing platforms; the challenge is to address
all of them.

There are many ways to address most of those computing platforms. However, we
will not discuss virtual-machine (VM). VM is a significant paradigm in cloud
computing, but it often requires too much computation power for small devices
usually found at the edge. Then, we focus on lighter virtualization technology
like containers.

WebAssembly is also a technology to consider in an edge computing environment.
It brings lightweight and fast execution, like containers, but in a more
secure way by providing sandboxing and with architectural independence.
Security is fundamental in a multi-tenant context where providers execute
untrusted codes on their platforms.

Figure 2 compares WebAssembly’s architecture to other models found in Cloud
Computing.

Figure 2: Comparison of architecture

###  III-A Containers

Containers is rapidly becoming the standard deployment technology in cloud
computing. They allow better management of resources and make the
development/deployment of applications easier. Also for Edge computing,
container technology can be considered.

According to Hong and Varghese, [8], containers propose good performance for
edge computing. However, they are not ready to deal with heterogeneity. They
do not support specific hardware like graphic processing units (GPU), data
processing units (DPU), tensor processing units (TPU), or field-programmable
gate array (FPGA). Those hardware accelerators are essential to achieve ultra-
low latency on specific applications at the edge. However, building container
for multiple targets requires additional steps. Containers strongly depend on
their host; e.g., a Linux image cannot be directly executed on a Windows host.

Kata containers [13] are really lightweight VM that can be managed like a
container (e.g., in Kubernetes). Using them allows having workload isolation
and better security. Combining VM and containers could be a solution to manage
hardware accelerators and security.

####  III-A1 OCI

OCI111https://opencontainers.org/ propose standards for container formats and
runtimes. It allows simpler management of container images (stored in
registries). Docker containers, Kata containers, and even WebAssembly modules
are compatible with this structure. In order to manage heterogeneous nodes at
the edge, following OCI standards will help to deploy applications.

####  III-A2 Live migration

Live migration between edge devices is an important mechanism. Some devices at
the edge can be mobile, and their applications have to follow them to always
be as close as possible. Xu et al [14] present a tool for Docker container
live migration. They explain that container migration is more complicated than
VM migration. Docker container migration needs to migrate image, runtime, and
context.

###  III-B WebAssembly

WebAssembly (also called WASM) is a new technology design for web applications
that now rapidly gains interes outside the web. The idea behind WASM is to
create a new assembly standard that proposes a unique binary that can run on
every processor with good performance. WASM also comes with a memory-safe
structure, lightweight sandboxed execution that allows a high-security level
and an efficient execution. It is compatible with most high-level programming
languages (e. g. Rust, C/C++, …).

WebAssembly should have a significant contribution to edge computing. This
improvement should allow developers to maintain a single code-base and address
most devices with a single binary. Considering that edge devices will have
different architecture, WebAssembly should become a game-changer. WebAssembly
applications should be deployed on every edge device seamlessly without
worrying about the platform or architecture.

The major web browsers (Chrome, Edge, Firefox, and Safari) now support
WebAssembly. Apart from browsers, WASM can run with many different runtimes.

WebAssembly Standard Interface (WASI)222https://wasi.dev/ aims to propose
standards to facilitate communication between WebAssembly conceptual machine
and its host system. WASI has a goal quite similar to that of POSIX.

Wasmtime is a runtime that implements the WASI standards. It is still under
development as WASI is not yet completed. Wasmtime is, therefore, still
limited in its uses because of missing networking functionalities. However,
implementations are already tested. It is only a matter of time before the
standardization is complete.

The flexibility and performances of WebAssembly are strongly linked to its
runtime. WebAssembly can be interpreted; it supports more platforms but has
lesser performance. Some runtimes support Just In Time Compilation (JIT),
performances are better for heavier tasks, but binaries are larger. Then,
Ahead-Of-Time (AOT) compilation provides the best performances. Execution
performance really depends on the runtime because they do not have the same
level of optimization. Runtimes evolve quickly and often, therefore, it is not
so easy to evaluate WASM performance.

####  III-B1 Running WebAssembly everywhere

WASM binary can be executed on almost every architecture (x86, x86_64, ARM,
RISC-V, PowerPC, MIPS, Xtensa, ARC32, …) and platform (laptop, smartphones,
SBC like Raspberry Pi, MCUs like Arduino, …) thanks to runtimes like
wasm3333https://github.com/wasm3/wasm3. This wide support means that the same
software can run on almost every device at the edge, even with limited
resources. The task of managing resources becomes easier with WASM. In a world
where most data will be collected and processed at the edge, managing a
heterogeneous swarm of devices will be simpler when executing the same binary.

Most high-level programmation languages can target WebAssembly. However, it is
still challenging to uses some. It is tricky to implement Garbage collectors
when the available memory is set before the execution. Languages like Java
cannot be used with WebAssembly yet. However, those languages with a high-
level virtual machine do not seem to be good candidates for edge computing.
They are not optimized enough for edge device constraints like memory or CPU.

Figure 3 shows the different platforms and architectures that can be targeted.

Figure 3: WASM

####  III-B2 Krustlet

Krustlet is a tool that allows WebAssembly modules to run on Kubernetes nodes.
The edge environment is fundamentally heterogeneous; it is essential to have
this in mind to deploy a cluster at the Edge. Krustlet permits using the state
of the art of orchestrator at the Edge. WebAssembly modules can be run on
every node, which enables to have a cluster with heterogeneous nodes. Krustlet
implements the Kubelet API, the pod manager present on every node in a
Kubernetes cluster (the pod is the atomic unit manageable on Kubernetes).
WebAssembly modules can be pushed in a registry following the OCI standards.
Then WASM modules can be scheduled on krustlet nodes. Nodes that run Krustlet
can execute natively WebAssembly modules, using WASI runtime (wasmtime) by
default. Currently, this project is still limited by the progression of WASI,
especially for networking missing features.

####  III-B3 Wasmcloud

Wasmcloud444https://wasmcloud.dev/ is an open-source platform for developing
applications in WebAssembly and deploy them to the Cloud. The ability to
deploy a WebAssembly module on a cluster is important for edge computing,
where nodes are heterogeneous. The project wants to improve application
development in a faster and safer way with less boilerplate code. Wasmcloud
wants to facilitate micro-service deployment with the actor model. In
wasmcloud, an actor is the smallest deployable unit. Every actor is a small
WebAssembly module that can handle messages from a capability provider.
Capability providers are dynamic libraries that provide interfaces like an
HTTP server, access to databases, or storage to an actor. Actors are reactive,
and they react to messages they receive. They are secure because of
privileged-based security and the inherent advantages of WebAssembly. With the
help of Krustlet, Wasmcloud actors’ can be deployed into a cluster at the
edge.

###  III-C Accelerators

Edge computing has to manage a growing demand in data processing while
reducing latency. CPU alone can not meet this increasing demand. Hardware
accelerators respond to this challenge, they are circuits designed for a
specific use case. E.g., accelerators can be designed to handle data for the
V2X use case in real-time. In their field of applications, hardware
accelerators are really better than CPU and can consume less energy. It is
also possible to virtualize accelerators to make the most of their
capabilities [3], virtualization brings flexibility in utilization. GPU, FPGA,
and ASIC are the main accelerators for edge computing. They can be used
together with CPU to form heterogeneous platforms [16].

####  III-C1 GPU

Graphics processing unit (GPU) are circuits design to execute parallel
applications faster. They were initially designed to process images and are
used in a multitude of applications like AI. They can be found everywhere: in
data centers, personal computers, or embedded devices. GPUs are the more
common accelerators, they also are the more energy-consumer. Frameworks like
OpenCL or CUDA can be used to program GPU.

####  III-C2 ASIC

Application-specific integrated circuit (ASIC) are circuits designed for a
specific purpose. They propose high performances and security for small size
and lower power consumption. [12]. ASICs are the best alternative for a very
specific task, but the flip side of the coin is their lack of flexibility.
They are totally incapable of doing anything different than what they were
designed to do.

####  III-C3 FPGA

Field-programmable gate array (FPGA) are circuits designed to be reconfigured
after manufacturing. Reconfiguration can be static, i.e., before program
execution, or dynamic, i.e., during runtime. Reconfiguration can be partial or
global depending on the circuit [16]. They are designed to achieve specific
tasks like ASICs, but they are more versatile and can be reconfigured for many
other usages. FPGAs provide low latency and low power consumption that make
them suitable for edge or embedded devices. Public cloud providers propose
access to FPGA, analogous access could be available for edge computing. FPGA
are secure devices from an external perspective, but giving access to
reconfigurable hardware in a multi-tenant environment could raise security
issues. FPGAs’ reconfigurability can be very useful for a scheduler at the
edge. If a computing unit is missing or not available, an FPGA can be
dynamically reconfigured to execute a new task. Hardware description language
(HDL) like VHDL can be used to configure FPGAs.

##  IV Orchestration and management

Edge computing will extend the cloud with many different devices. The
challenge will be to make every device work together. For this, orchestration
methods from cloud computing need to be extended for edge computing. Edge
computing orchestration needs to take into consideration the volatility and
the heterogeneity of resources. Tools like Kubernetes should be improved to
support the edge computing challenges. In order to manage edge complexity,
automation methods should be used.

###  IV-A Scheduling

Scheduling at the edge is a fundamental challenge. There is a huge number of
parameters to take into account. In addition, this heterogeneous distributed
architecture will be multi-tenant.

Kubernetes has become a standard in scheduling. However, some requirements are
needed in a heterogeneous cluster. Kubernetes has to manage the different
node’s architecture. In addition, hardware accelerators have to be taken into
account. In order to have a seamless cloud, a heterogenous cluster has to
identify its different capabilities, and developers have to add constraints to
their applications. A good scheduler has to ensure that application
constraints meet the cluster and node capabilities. Other parameters like
energy consumption should also be taken into account by the scheduler. Auto-
scaling and auto-discovery mechanisms should also be addressed.

There is, therefore, a need for standardization to achieve scheduling at the
edge. The container standard defined by the OCI could be a good candidate. As
explained in the previous section, most of the containers support it, like
Docker or WebAssembly. Having a standardized container will allow having an
atomic unit to manageable by the scheduler. It is important to provide
resource abstraction (i.e., CPU architecture, hardware accelerator) to permit
more efficient scheduling by removing a significant constraint. It is
necessary insofar as devices are more and more heterogeneous.

This atomic unit should include information to consider the following
parameters: available resources, need of bandwidth, low latency requirement,
real-time execution, the energy available, device location or context,
hardware accelerator.

Another issue of such a scheduler is being integrated into a context of
cooperation between edge and Cloud. A distributed application may need to
execute some tasks at the edge, while others may need the powerful computing
capabilities of the Cloud.

###  IV-B Multi-access Edge Computing

Multi-access Edge Computing (MEC) is a new paradigm that aims to provide cloud
abilities at the network’s edge. MEC also permits application offloading and
data aggregation [7]. ETSI555https://www.etsi.org/technologies/multi-access-
edge-computing defines many aspects of MEC and presents some use cases. MEC
can be used to achieve ultra-low latency, high bandwidth, and Real-Time
performance at the edge. These technologies enable many applications like
gaming, Virtual Reality (VR), Augmented Reality (AR), Internet of Things
(IoT), or Vehicle-to-everything (V2X). With MEC, tasks can be offloaded from
small devices to more powerful ones. Energy is another essential criterion for
offloading. A low battery device can offload a task to save battery.

MEC is a key enabler for 5G technologies because of its position at the edge
of a network. MEC can be deployed in a Radio access network (RAN) and provides
computation and storage capabilities. Data generated by edge devices can be
collected and processed directly in MEC. This can provide better privacy and
security of personal data. Data are not systematically sent to a central
authority but processed locally. Processing data at the edge will help to
unclog the network and data centers. RAN and MEC can enable context-aware
services. Taking into account devices and users’ mobility is essential, e. g.
a connected smart car needs to process its data locally to have ultra-low
latency. While a vehicle is a mobile device, data processing tasks should
migrate from one node to the devices’ closest. MEC has to be able to realize
the live migration of tasks to go with devices.

Larger computing abilities at the edge can be used to train Artificial
Intelligence (AI) and Deep Learning (DL). All data required for this task can
be found directly at the source. Those data should help to have better models
while saving bandwidth to the data centers. Using the models during the
inference phase should also be done at the edge. Managing and scheduling tasks
while considering all of those criteria in this heterogeneous environment is a
significant research field. Scheduling algorithms should find the better node
at the better location to improve MEC performance.

MEC has a key role in edge computing. It is a link between Cloud and telco.

Kubernetes should be extended to be usable at the edge. In this case, MEC
would become an interesting use case. However, it is necessary to package the
application in a container-friendly format in order to manage it. Projects
like KubeEdge666https://kubeedge.io/ proses this extension. KubeEdge has a
control plane on the cloud side and worker nodes on the edge side. This tool
also considers network constraints like unreliable connections to allow edge
nodes to continue to work even in the case of networking failure.

###  IV-C Serverless

Serverless is a paradigm where a cloud provider offers an abstraction of the
resources allocated to a client. According to [11], serverless is a technology
that is increasingly being used. Developers only have to pay attention to the
code they wrote; resources and infrastructure will be dynamically provided
according to the need. It is easy to use with the support of most high-level
programming languages. In contrast to a dedicated server, serverless is cost-
efficient; only used resources are billed. However, serverless is less
suitable for long-running tasks. When execution times become too long,
serverless become more expensive. Dedicated servers are better candidates for
long-run resource-intensive execution. Serverless is an event-driven paradigm;
an external event triggers the application execution. Apart from this case, no
resources are consumed.

Serverless can be seen as the addition of Function as a Service (FaaS) and
Backend as a Service (BaaS). FaaS is a service that permits to trigger code
execution by events. Execution is stateless and needs external storage to save
data. BaaS are services needed for FaaS, e.g., external storage, database,
networking functions…

A common practice in serverless is Extract, Transform and Load (ETL)
applications. The serverless application will get some data, transform with a
FaaS and store it in a database with BaaS. This pattern should be found at the
edge in a distributed way. Small devices could collect and aggregate data
acquired. Then data can be transformed and processed on the same device or an
edge node and finally transferred to storage in the cloud. Data aggregation
would reduce bandwidth costs to the cloud. Data are processed locally, and
less data is transferred to the cloud to be stored or to be aggregate in a
larger process involving many data from many places.

However, there are still challenges to be addressed. Security is essential but
difficult to achieve. All interaction with BaaS increases the attack surface.
It is also currently difficult to manage the hardware heterogeneity.
Serverless at the edge should involve many different devices at every stage of
fog computing, from the sensor to the cloud with cloudlet and edge nodes.

Sledge [6] is an open-source for ServerLess at the EDGE. It is designed to run
on a single-host server; with Wasm support, it can run on most servers from
the smallest like Raspberry Pi to the more powerful ones. To achieve ultra-low
latency, Sledge bypasses the kernel scheduler and takes advantage of Wasm.
This helps to have a rapid startup time. Sledge uses a custom aWams compiler
and runtimes. Wasm also provides benefits in terms of security with process
isolation and sandboxing. With Wasm, it is possible to execute the same binary
on all platforms running Sledge.

Nuclio, OpenFaaS are other frameworks for serverless at the edge. Unlike
Sledge, they are not based on Wasm but on container technology. They can run
on multiple platforms and be scheduled by Kubernetes.

The Content Delivery Network (CDN) notion can be extended to Access Delivery
Network (ADN). The idea is not only to provide content but also computation
capabilities in the vicinity of the users.

WebAssembly could be a key enabler for serverless at the edge. WASM permit
developers to stay focus on their application without worrying about available
architectures. Then, every application could be deployed at the edge.

Serverless is an important paradigm for edge computing. Many use cases can be
solved with FaaS, e.g., aggregating and processing data from IoT devices.

###  IV-D Automation

In order to save time and reduce cost, automation has a role to play.

####  IV-D1 Software development and deployment

Practices from DevOps workflow should be adapted to edge computing.

The applications should be agnostic of where there will be executed, and this
should not affect the development. In addition, developers should be able to
access hardware accelerators easily. A simple interface should be accessible
to defines the needs of the applications. Then, the scheduler will find the
best nodes that achieve these needs.

Operating an edge node should be as simple as a traditional node in the cloud,
but there are significant architectural differences. An application executed
at the edge needs to be able to run on the infrastructure of most providers;
in some cases, due to the node location, the user will have only one choice of
provider. Application deployment needs to be multi-cloud friendly. A seamless
and provider agnostic deployment at the edge should be possible.

Techniques of Continuous Integration and Continuous Delivery (CI/CD) have to
be adapted. The advent of a highly heterogeneous environment raises the need
for an execution standard. It will not be possible for a team to test every
kind of architecture available on edge nodes to get continuous deployment.

GitOps, a set of practices based on version control and declarative
specification (e.g., yaml files), permits the quick deployment of
infrastructures. Edge infrastructures could take advantage of GitOps to
facilitate their management.

####  IV-D2 Infrastructure automation

Edge computing needs infrastructure automation to be resilient. With GKE
Autopilot [2], Google proposes a ”Kubernetes serverless”. The idea is to
automate the management nodes and pool of nodes, not only the pods. This
improvement will bring flexibility and scalability for clusters. Virtual
clusters and virtual networking can make infrastructure deployment easier.
Automated infrastructure management and multi-cloud manager will help deal
with structural heterogeneity in the edge cloud.

##  V Conclusion

Most of the data will be processed at the edge in the following years.
Proposing a seamless cloud experience is becoming more and more important and
raises new challenges.

Architectures have to be adapted to become more flexible. In response to the
mobility of devices, topology needs to be dynamic and resilient. The topology
should also reflect the heterogeneity of the devices at the edge. In addition,
these devices have special needs in term of energy consumption. Networking is
a keystone for a seamless cloud experience, this is essential for the proper
functioning of the applications. Virtualisation and cloud-native technologies
help improve networking, and networking technologies help edge computing
deployment.

The major challenge in execution at the edge is being able to run applications
on every node seamlessly without worrying about its platform or architecture.
WebAssembly is one of the best candidates to manage this heterogeneity. With a
strong community behind its development, current limitations should be solved
soon. As long as CPUs remain limited in computation power and energy
efficiency, hardware accelerator support is a priority. We believe that higher
abstraction interfaces are a way to make accelerators available without the
need to make them mandatory. It is also essential to have an overview of the
needs of accelerators to permit efficient scheduling.

Orchestration of computation from device to data centre needs to rely on
technologies similar to Kubernetes. Some improvements are needed to make
Kubernetes truly ready for the edge: managing heterogeneous devices, working
with non-reliable connection and handling node churn and resources volatility.
Access to accelerators is also important to consider.

MEC allows joining telco networks and edge computing which makes it an
important paradigm to consider, 5G is one of the significant edge computing
use cases. With a seamless cloud experience, deploying application at the edge
should be as easy as using serverless functions.

## References

  * [1] A. C. Baktir, A. Ozgovde, and C. Ersoy, “How can edge computing benefit from software-defined networking: A survey, use cases, and future directions,” _IEEE Communications Surveys Tutorials_ , vol. 19, no. 4, pp. 2359–2391, Fourthquarter 2017.
  * [2] D. Bradstock, “Introducing gke autopilot: a revolution in managed kubernetes,” Feb 2021. [Online]. Available: https://cloud.google.com/blog/products/containers-kubernetes/introducing-gke-autopilot
  * [3] R. A. Cooke and S. A. Fahmy, “A model for distributed in-network and near-edge computing with heterogeneous hardware,” _Future Generation Computer Systems_ , vol. 105, pp. 395–409, 2020.
  * [4] A. J. Ferrer, J. M. Marques, and J. Jorba, “Ad-hoc edge cloud: A framework for dynamic creation of edge computing infrastructures,” in _2019 28th International Conference on Computer Communication and Networks (ICCCN)_ , July 2019, pp. 1–7.
  * [5] A. J. Ferrer, J. Panadero, J.-M. Marques, and J. Jorba, “Admission control for ad-hoc edge cloud,” _Future Generation Computer Systems_ , vol. 114, pp. 548–562, 2021.
  * [6] P. K. Gadepalli, S. McBride, G. Peach, L. Cherkasova, and G. Parmer, “Sledge: A serverless-first, light-weight wasm runtime for the edge,” in _Proceedings of the 21st International Middleware Conference_ , ser. Middleware ’20. New York, NY, USA: Association for Computing Machinery, 2020, p. 265–279. [Online]. Available: https://doi-org.proxy.bnl.lu/10.1145/3423211.3425680
  * [7] F. Giust, X. Costa-Perez, and A. Reznik, “Multi-access edge computing: An overview of etsi mec isg,” _IEEE 5G Tech Focus_ , vol. 1, no. 4, p. 4, 2017\.
  * [8] C.-H. HONG and B. VARGHESE, “Resource management in fog/edge computing: A survey on architectures, infrastructure, and algorithms,” _ACM Computing Survey_ , vol. 52, no. 5, 2019. [Online]. Available: https://doi.org/10.1145/3326066
  * [9] N. Kazemifard and V. Shah-Mansouri, “Minimum delay function placement and resource allocation for open ran (o-ran) 5g networks,” _Computer Networks_ , vol. 188, p. 107809, 2021.
  * [10] W. Z. Khan, E. Ahmed, S. Hakak, I. Yaqoob, and A. Ahmed, “Edge computing: A survey,” _Future Generation Computer Systems_ , vol. 97, pp. 219–235, 08 2019.
  * [11] A. Khandelwal, A. Kejariwal, and K. Ramasamy, “Le taureau: Deconstructing the serverless landscape & a look forward,” in _Proceedings of the 2020 ACM SIGMOD International Conference on Management of Data_ , ser. SIGMOD ’20. New York, NY, USA: Association for Computing Machinery, 2020, p. 2641–2650. [Online]. Available: https://doi-org.proxy.bnl.lu/10.1145/3318464.3383130
  * [12] F. Liu, G. Tang, Y. Li, Z. Cai, X. Zhang, and T. Zhou, “A survey on edge computing systems and tools,” _Proceedings of the IEEE_ , vol. 107, no. 8, pp. 1537–1562, Aug 2019.
  * [13] A. Randazzo and I. Tinnirello, “Kata containers: An emerging architecture for enabling mec services in fast and secure way,” in _2019 Sixth International Conference on Internet of Things: Systems, Management and Security (IOTSMS)_ , 10 2019, pp. 209–214.
  * [14] B. Xu, S. Wu, J. Xiao, H. Jin, Y. Zhang, G. Shi, T. Lin, J. Rao, L. Yi, and J. Jiang, “Sledge: Towards efficient live migration of docker containers,” in _2020 IEEE 13th International Conference on Cloud Computing (CLOUD)_. IEEE, 2020, pp. 321–328.
  * [15] A. Yousefpour, C. Fung, T. Nguyen, K. Kadiyala, F. Jalali, A. Niakanlahiji, J. Kong, and J. P. Jue, “All one needs to know about fog computing and related edge computing paradigms: A complete survey,” _Journal of Systems Architecture_ , vol. 98, pp. 289–330, 09 2019.
  * [16] Z. Zhu, J. Zhang, J. Zhao, J. Cao, D. Zhao, G. Jia, and Q. Meng, “A hardware and software task-scheduling framework based on cpu+fpga heterogeneous architecture in edge computing,” _IEEE Access_ , vol. 7, pp. 148 975–148 988, 2019.


# Perturbation analysis for the generalized inverses with prescribed
idempotents in Banach algebras

Jianbing Cao
Department of Mathematics, East China Normal University,
Shanghai 200241, P.R. China
Department of mathematics, Henan Institute of Science and Technology
Xinxiang, Henan, 453003, P.R. China
Email<EMAIL_ADDRESS>Yifeng Xue
Department of Mathematics, East China Normal University,
Shanghai 200241, P.R. China Email<EMAIL_ADDRESS>Corresponding
author

###### Abstract

In this paper, we first study the perturbations and expressions for the
generalized inverses $a^{(2)}_{p,q}$, $a^{(1,2)}_{p,q}$, $a^{(2,l)}_{p,q}$ and
$a^{(l)}_{p,q}$ with prescribed idempotents $p$ and $q$. Then, we investigate
the general perturbation analysis and error estimate for some of these
generalized inverses when $p,\,q$ and $a$ also have some small perturbations.

2010 Mathematics Subject Classification: 15A09; 46L05

Key words: gap function, idempotent element, generalized inverse, perturbation

##  1 Introduction

Let $\mathcal{R}$ be a unital ring and let $\mathcal{R}^{\bullet}$ denote the
set of all idempotent elements in $\mathcal{R}$. Given
$p,\,q\in\mathcal{R}^{\bullet}$. Recall that an element $a\in\mathcal{R}$ has
the $(p,q)$–outer generalized inverse $b=a^{(2)}_{p,q}\in\mathcal{R}$ if
$bab=b$, $ba=p$ and $1-ab=q$. If $b=a^{(2)}_{p,q}$ also satisfies the equation
$aba=a$, then we say $a$ has the $(p,q)$–generalized inverse $b$, in this
case, written $b=a^{(1,2)}_{p,q}$. If an outer generalized inverse with
prescribed idempotents exists, it is necessarily unique (cf. [6]). According
to this definition, obviously, we see that the Moore–Penrose inverses in a
$C^{*}$-algebra and (generalized) Drazin inverses in a Banach algebra can be
expressed by some $(p,q)$–outer generalized inverses (cf. [6, 5, 1]).

Based on some results of Djordjević and Wei in [6], Ilic, Liu and Zhong gave
some equivalent conditions for the existence of the $(p,q)$–outer generalized
inverse in a Banach algebra in [5]. But in our recent paper [1], we find that
Theorem 1.4 of [5] is wrong. In [1], we first present a counter–example to [5,
Theorem 1.4], then based on our counter–example, we define a new type of
generalized inverse with prescribed idempotents in a Banach algebra as
follows:

######  Definition 1.1 (see [1]).

Let $a\in\mathscr{A}$ and $p,\;q\in\mathscr{A}^{\bullet}$. An element
$b\in\mathscr{A}$ satisfying

$bab=b,\quad R_{r}(b)=R_{r}(p),\quad K_{r}(b)=R_{r}(q),$

will be called the $(p,q,l)$–outer generalized inverse of $a$, written as
$a^{(2,l)}_{p,q}=b.$

In addition, if $a^{(2,l)}_{p,q}$ satisfies $a=aa^{(2,l)}_{p,q}a$, we call
$a^{(2,l)}_{p,q}$ is the $(p,q,l)$–generalized inverse of $a$, denoted by
$a^{(l)}_{p,q}$.

Perturbation analysis of the generalized inverses is very important in both
theory and applications. In recent years, there are many fruitful results
concerning the perturbation analysis for various types generalized inverses of
operators on Hilbert spaces or Banach spaces. The concept of stable
perturbation of an operator on Hilbert spaces and Banach spaces is introduced
by Chen and Xue in [1]. Later the notation is generalized to the set of Banach
algebras by the second author in [13] and to the set of Hilbert
$C^{*}$–modules by Xu, Wei and Gu in [16]. Using the notation “stable
perturbation”, many important results in perturbation analyses for
Moore–Penrose inverses on Hilbert spaces and Drazin inverses on Banach spaces
or in Banach algebras have been obtained. Please see [2, 3, 4, 14, 13, 15] for
detail.

Let $X,Y$ be Banach spaces over complex field $\mathbb{C}$. Let $T$ (resp.
$S)$ be a given closed space in $X$ (resp, $Y$). Let $A$ be a bounded linear
operator from $X$ to $Y$ such that $A_{T,S}^{(2)}$ exists. The perturbation
analysis of $A_{T,S}^{(2)}$ for small perturbation of $T$, $S$ and $A$ has
been done in [7, 8]. Motivated by some recent results concerning the
perturbation analysis for the generalized inverses of operators, in this
paper, we mainly study the perturbations and expressions for various types of
generalized inverses with prescribed idempotents in Banach algebras. We first
consider the stable perturbation characterizations for $a^{(2)}_{p,q}$,
$a^{(1,2)}_{p,q}$, $a^{(2,l)}_{p,q}$ and $a^{(l)}_{p,q}$ with prescribed
idempotents $p$ and $q$. Then, by using stable perturbation characterizations,
we can investigate the general perturbation analysis and error estimate for
some of these generalized inverses when $p,\,q$ and $a$ also have some small
perturbations. The results obtained in this paper extend and improve many
recent results in this area.

##  2 Preliminaries

In this section, we give some notations in this paper, we also list some
preliminary results which will be frequently used in our main sections.
Throughout the paper, $\mathscr{A}$ is always a complex Banach algebra with
the unit 1.

Let $a\in\mathscr{A}$. If there is $b\in\mathscr{A}$ such that $aba=a$ and
$bab=b$, then $a$ is called to be generalized invertible and $b$ is called the
generalized inverse of $a$, denoted by $b=a^{+}$. Let $Gi(\mathscr{A})$ denote
the set of all generalized invertible elements in
$\mathscr{A}\backslash\\{0\\}$. Let $\mathscr{A}^{\bullet}$ denote the set of
all idempotent elements in $\mathscr{A}$. If $a\in Gi(\mathscr{A})$, then
$a^{+}a$ and $1-aa^{+}$ are all idempotent elements. For $a\in\mathscr{A}$,
set

$\displaystyle K_{r}(a)$ $\displaystyle=\\{x\in\mathscr{A}\;|\;ax=0\\},$
$\displaystyle\quad R_{r}(a)$ $\displaystyle=\\{ax\;|\;x\in\mathscr{A}\\};$
$\displaystyle K_{l}(a)$ $\displaystyle=\\{x\in\mathscr{A}\;|\;xa=0\\},$
$\displaystyle\quad R_{l}(a)$ $\displaystyle=\\{xa\;|\;x\in\mathscr{A}\\}.$

Clearly, if $p\in\mathscr{A}^{\bullet}$, then $\mathscr{A}$ has the direct sum
decompositions:

$\mathscr{A}=K_{r}(p)\dotplus R_{r}(p)\quad
or\quad\mathscr{A}=K_{l}(p)\dotplus R_{l}(p).$

The following useful and well–known lemma can be easily proved.

######  Lemma 2.1.

Let $x\in\mathscr{A}$ and $p\in\mathscr{A}^{\bullet}$. Then

$K_{r}(p)$ and $R_{r}(p)$ are all closed and $K_{r}(p)=R_{r}(1-p),\
R_{r}(p)\mathscr{A}\subset R_{r}(p);$

$px=x$ if and only if $R_{r}(x)\subset R_{r}(p)$ or $K_{l}(p)\subset
K_{l}(x);$

$xp=x$ if and only if $K_{r}(p)\subset K_{r}(x)$ or $R_{l}(x)\subset
R_{l}(p).$

We list some of the necessary and sufficient conditions for the existence of
$a^{(2,l)}_{p,q}$ in the following lemma, which will be frequently used in the
paper. Here we should indicate that $a^{(2,l)}_{p,q}$ is unique if it exists.
Please see [1] for the proofs and more information.

######  Lemma 2.2.

Let $a\in\mathscr{A}$ and $p,q\in\mathscr{A}^{\bullet}$. Then the following
statements are equivalent:

$a^{(2,l)}_{p,\,q}$ exists;

There exists $b\in\mathscr{A}$ such that $bab=b$, $R_{r}(b)=R_{r}(p)$ and
$K_{r}(b)=R_{r}(q)$;

$K_{r}(a)\cap R_{r}(p)=\\{0\\}$ and $\mathscr{A}=aR_{r}(p)\dotplus R_{r}(q)$;

There exists $b\in\mathscr{A}$ satisfying $b=pb$, $p=bap$, $b(1-q)=b$,
$1-q=(1-q)ab$;

$p\in R_{l}((1-q)ap)=\\{x(1-q)ap\;|\;x\in\mathscr{A}\\}$ and $1-q\in
R_{r}((1-q)ap)$;

There exist some $s,t\in\mathscr{A}$ such that $p=t(1-q)ap$, $1-q=(1-q)aps$.

The following lemma gives some equivalent conditions about the existence of
$a^{(l)}_{p,\,q}$. See [1] for more information.

######  Lemma 2.3.

Let $a\in\mathscr{A}$ and $p,q\in\mathscr{A}^{\bullet}$. Then the following
conditions are equivalent:

$a^{(l)}_{p,\,q}$ exists, i.e.,there exists some $b\in\mathscr{A}$ such that

$aba=a,\quad bab=b,\quad R_{r}(b)=R_{r}(p),\quad K_{r}(b)=R_{r}(q),$

$\mathscr{A}=R_{r}(a)\dotplus R_{r}(q)=K_{r}(a)\dotplus R_{r}(p),$

$\mathscr{A}=aR_{r}(p)\dotplus R_{r}(q),\;R_{r}(a)\cap
R_{r}(q)=\\{0\\},\;K_{r}(a)\cap R_{r}(p)=\\{0\\}.$

Let $X$ be a complex Banach space. Let $M,\,N$ be two closed subspaces in $X$.
Set

$\delta(M,N)=\begin{cases}\sup\\{{\rm dist}(x,N)\,|\,x\in
M,\,\|x\|=1\\},\quad&M\not=\\{0\\}\\\ 0\quad&M=\\{0\\}\end{cases},$

where ${\rm dist}(x,N)=\inf\\{\|x-y\|\,|\,y\in N\\}$. The gap
$\hat{\delta}(M,N)$ of $M,\,N$ is given by
$\hat{\delta}(M,N)=\max\\{\delta(M,N),\delta(N,M)\\}$. For convenience, we
list some properties about $\delta(M,N)$ and $\hat{\delta}(M,N)$ which come
from [10] as follows.

######  Proposition 2.4 ([10]).

Let $M,\,N$ be closed subspaces in a Banach space $X$. Then

$\delta(M,N)=0$ if and only if $M\subset N$;

$\hat{\delta}(M,N)=0$ if and only if $M=N$;

$\hat{\delta}(M,N)=\hat{\delta}(N,M)$;

$0\leq\delta(M,N)\leq 1$, $0\leq\hat{\delta}(M,N)\leq 1$.

##  3 Stable perturbations for the (p, q)–generalized inverses

Let $a\in Gi(\mathscr{A})$ and let $\bar{a}=a+\delta a\in\mathscr{A}$. Recall
from [14] that $\bar{a}$ is a stable perturbation of $a$ if
$R_{r}(\bar{a})\cap K_{r}(a^{+})=\\{0\\}$. Obviously, we can define the stable
perturbation for various kind of generalized inverses. In this section, we
concern the stable perturbation problem for various types of
$(p,q)$–generalized inverses in a Banach algebra.

######  Lemma 3.1 ([9, Lemma 2.2]).

Let $a,\,b\in\mathscr{A}$. If $1+ab$ is left invertible, then so is $1+ba$.

######  Lemma 3.2.

Let $a,\delta a\in\mathscr{A}$ and $p,\,q\in\mathscr{A}^{\bullet}$ such that
$a^{(2,l)}_{p,\,q}$ exists. Put $\bar{a}=a+\delta a$. If $1+\delta
aa^{(2,l)}_{p,\,q}$ is invertible, $w=a^{(2,l)}_{p,\,q}(1+\delta
aa^{(2,l)}_{p,\,q})^{-1}$. Then $\bar{a}^{(2,l)}_{p,\,q}$ exists and
$w=\bar{a}^{(2,l)}_{p,\,q}$ .

###### Proof.

We prove our result by showing that
$waw=w,R_{r}(w)=R_{r}(p),K_{r}(w)=R_{r}(q)$. It is easy to check that

$w=a^{(2,l)}_{p,\,q}(1+\delta
aa^{(2,l)}_{p,\,q})^{-1}=(1+a^{(2,l)}_{p,\,q}\delta a)^{-1}a^{(2,l)}_{p,\,q}.$

Then, by using these two equalities, we can show
$R_{r}(w)=R_{r}(a^{(2,l)}_{p,\,q})=R_{r}(p)$ and
$K_{r}(w)=K_{r}(a^{(2,l)}_{p,\,q})=R_{r}(q)$. We can also compute

$\displaystyle w\bar{a}w$ $\displaystyle=a^{(2,l)}_{p,\,q}(1+\delta
aa^{(2,l)}_{p,\,q})^{-1}\bar{a}a^{(2,l)}_{p,\,q}(1+\delta
aa^{(2,l)}_{p,\,q})^{-1}$ $\displaystyle=a^{(2,l)}_{p,\,q}(1+\delta
aa^{(2,l)}_{p,\,q})^{-1}[(aa^{(2,l)}_{p,\,q}-1)+(1+\delta
aa^{(2,l)}_{p,\,q})](1+\delta aa^{(2,l)}_{p,\,q})^{-1}$
$\displaystyle=a^{(2,l)}_{p,\,q}(1+\delta
aa^{(2,l)}_{p,\,q})^{-1}(aa^{(2,l)}_{p,\,q}-1)(1+\delta
aa^{(2,l)}_{p,\,q})^{-1}+w$ $\displaystyle=w.$

By Definition 1.1 and the uniqueness of $a^{(2,l)}_{p,\,q}$, we see
$\bar{a}^{(2,l)}_{p,\,q}$ exists and $w=\bar{a}^{(2,l)}_{p,\,q}$ . ∎

Obviously, from the proof of Lemma 3.2, we see that if $a^{(2,l)}_{p,\,q}$
exists and $1+a^{(2,l)}_{p,\,q}\delta a$ is invertible, set
$v=(1+a^{(2,l)}_{p,\,q}\delta a)^{-1}a^{(2,l)}_{p,\,q}$, then we also have
$v=\bar{a}^{(2,l)}_{p,\,q}$. In order to prove the main results about the
stable perturbation, we need one more characterizations of the existence of
$a^{(2,l)}_{p,\,q}$.

For an element $a\in\mathscr{A}$ and $p,\;q\in\mathscr{A}^{\bullet}$. Let
$R_{a}\colon\mathscr{A}\to\mathscr{A}$ be the right multiplier on
$\mathscr{A}$(i.e., $R_{a}(x)=xa$ for any $x\in\mathscr{A}$). Then it easy to
see that $a^{(2,l)}_{p,q}$ exists in $\mathscr{A}$ if and only if
$(R_{a})^{(2)}_{\mathscr{A}(1-q),\mathscr{A}(1-p)}$ exists in the Banach
algebra $B(\mathscr{A})$. So from the equivalences of (1), (2) and (3) in
Lemma 2.2, dually, we can get the following equivalent conditions for the
existence of $a^{(2,l)}_{p,q}$.

######  Proposition 3.3.

Let $a\in\mathscr{A}$ and $p,\;q\in\mathscr{A}^{\bullet}$. Then the following
statements are equivalent:

$a^{(2,l)}_{p,\,q}$ exists;

There exists $c\in\mathscr{A}$ such that $cac=c$, $R_{l}(c)=R_{l}(1-q)$ and
$K_{l}(c)=R_{l}(1-p)$;

$K_{l}(a)\cap R_{l}(1-q)=\\{0\\}$ and $\mathscr{A}=R_{l}(1-q)a\dotplus
R_{l}(1-p)$.

###### Proof.

$(1)\Leftrightarrow(2)$ Suppose that $a^{(2,l)}_{p,\,q}$ exists. Let
$c=a^{(2,l)}_{p,\,q}$. Then from Definition 1.1, we know that $cac=c$, and
then $ca,\,ac\in\mathscr{A}^{\bullet}$, $R_{r}(ca)=R_{r}(c)=R_{r}(p)$,
$K_{r}(ac)=K_{r}(c)=R_{r}(q)$. Thus, it follows from Lemma 2.1 that

$\displaystyle cap=p,\quad pca=ca,\quad ac(1-q)=ac,\quad(1-q)ac=1-q.$

Then, by using Lemma 2.1 again, we have

$\displaystyle K_{l}(ca)\subset K_{l}(p)\subset K_{l}(ca),\quad
R_{l}(ac)\subset R_{l}(1-q)\subset R_{l}(ac).$ (3.1)

By using $cac=c$, we have $K_{l}(ca)=K_{l}(c)$ and $R_{r}(ac)=R_{r}(c)$. Thus
from Eq. (3.1) we see that (2) holds. If (2) holds, similarly, by using
Definition 1.1 and Lemma 2.1, we can obtain $a^{(2,l)}_{p,\,q}$ exists.

$(2)\Leftrightarrow(3)$ By our remark above this lemma, we see these hold
simply from the equivalences of (2) and (3) in Lemma 2.2. Note that we can
also prove these equivalences directly by using the right multiplier $R_{a}$
on $\mathscr{A}$. Here we omit the detail. ∎

Now we can present one of our main results about the stable perturbation of
the generalized inverse $a^{(2,l)}_{p,\,q}$.

######  Theorem 3.4.

Let $a,\,\delta a\in\mathscr{A}$ and $p,\,q\in\mathscr{A}^{\bullet}$ such that
$a^{(2,l)}_{p,\,q}$ exists. Put $\bar{a}=a+\delta a$. Then the following
statements are equivalent:

$1+\delta aa^{(2,l)}_{p,\,q}$ is invertible;

$1+a^{(2,l)}_{p,\,q}\delta a$ is invertible;

$\bar{a}^{(2,l)}_{p,\,q}$ exists. In this case, we have
$\bar{a}^{(2,l)}_{p,\,q}=a^{(2,l)}_{p,\,q}(1+\delta
aa^{(2,l)}_{p,\,q})^{-1}=(1+a^{(2,l)}_{p,\,q}\delta a)^{-1}a^{(2,l)}_{p,\,q}.$

###### Proof.

$(1)\Leftrightarrow(2)$ follows from the well-known spectral theory in Banach
algebras.

$(2)\Rightarrow(3)$ We prove our result by using Lemma 2.2. Let $x\in
K_{r}(\bar{a})\cap R_{r}(p)=\\{0\\}$. Since
$R_{r}(p)=R_{r}(a^{(2,l)}_{p,\,q})$, then there exists some $t\in\mathscr{A}$
satisfying $x=a^{(2)}_{p,\,q}t$ and $\bar{a}t=0$. Thus we have

$\displaystyle(1+a^{(2,l)}_{p,\,q}\delta a)a^{(2,l)}_{p,\,q}t$
$\displaystyle=a^{(2,l)}_{p,\,q}t+a^{(2,l)}_{p,\,q}\delta aa^{(2,l)}_{p,\,q}t$
$\displaystyle=a^{(2,l)}_{p,\,q}(a+\delta a)a^{(2,l)}_{p,\,q}t$
$\displaystyle=a\bar{a}t=0.$

Since $1+a^{(2,l)}_{p,\,q}\delta a$ is invertible, it follows that
$x=a^{(2,l)}_{p,\,q}t=0$. Therefore,

$\displaystyle K_{r}(\bar{a})\cap R_{r}(p)=\\{0\\}$ (3.2)

Let $s\in\bar{a}R_{r}(p)\cap R_{r}(q)$. Since
$R_{r}(p)=R_{r}(a^{(2,l)}_{p,\,q})$ and $R_{r}(q)=K_{r}(a^{(2,l)}_{p,\,q})$,
then there exists some $z\in\mathscr{A}$ such that
$s=\bar{a}a^{(2,l)}_{p,\,q}z$ and $a^{(2,l)}_{p,\,q}s=0$. Similar to the proof
of Eq. (3.2), we can get $s=a^{(2,l)}_{p,\,q}t=0$, i.e, $\bar{a}R_{r}(p)\cap
R_{r}(q)=\\{0\\}$. Since $1+a^{(2,l)}_{p,\,q}\delta a$ is invertible, then for
any $w\in\mathscr{A}$ there is some $v\in\mathscr{A}$ such that
$a^{(2,l)}_{p,\,q}w=(1+a^{(2,l)}_{p,\,q}\delta a)v$. From $\bar{a}=a+\delta
a$, we have

$(1-a^{(2,l)}_{p,\,q}a)v=a^{(2,l)}_{p,\,q}(w-\bar{a}v)\in
R_{r}(a^{(2,l)}_{p,\,q}a)\cap K_{r}(a^{(2,l)}_{p,\,q}a)=\\{0\\}.$

Thus, $w-\bar{a}v\in K_{r}(a^{(2,l)}_{p,\,q})$ and $v=a^{(2,l)}_{p,\,q}av\in
R_{r}(a^{(2,l)}_{p,\,q})$. Since for $w\in\mathscr{A}$, we also have
$w=\bar{a}v+(w-\bar{a}v)\in\bar{a}R_{r}(p)\dotplus R_{r}(q).$ Thus, we have

$\displaystyle\mathscr{A}=\bar{a}R_{r}(p)\dotplus R_{r}(q).$ (3.3)

Now, from Eqs. (3.2) and (3.3), by using Lemma 2.2, we see that
$\bar{a}^{(2,l)}_{p,\,q}$ exists.

$(3)\Rightarrow(1)$ Suppose that $\bar{a}^{(2,l)}_{p,\,q}$ exists, we want to
prove $1+\delta aa^{(2,l)}_{p,\,q}$ is both left and right invertible. Since
$\bar{a}^{(2,l)}_{p,\,q}$ exists, then from Lemma 2.2,
$\mathscr{A}=\bar{a}R_{r}(p)\dotplus
R_{r}(q)=\bar{a}R_{r}(a^{(2,l)}_{p,\,q})\dotplus K_{r}(a^{(2,l)}_{p,\,q})$.
Thus, for any $x\in\mathscr{A}$, we can write
$x=\bar{a}a^{(2,l)}_{p,\,q}t_{1}+t_{2}$, where $t_{1}\in\mathscr{A}$ and
$t_{2}\in K_{r}(a^{(2,l)}_{p,\,q})$. Set $s=aa^{(2,l)}_{p,\,q}t_{1}+t_{2}$,
then

$\displaystyle(1+\delta aa^{(2,l)}_{p,\,q})s$ $\displaystyle=(1+\delta
aa^{(2,l)}_{p,\,q})(aa^{(2,l)}_{p,\,q}t_{1}+t_{2})$
$\displaystyle=\bar{a}a^{(2,l)}_{p,\,q}t_{1}+t_{2}=x.$

Since $x\in\mathscr{A}$ is arbitrary, let $x=1$, then we see that $1+\delta
aa^{(2,l)}_{p,\,q}$ is right invertible. Now we prove that $1+\delta
aa^{(2,l)}_{p,\,q}$ is also left invertible. In fact, from Proposition 3.3, we
also have $\mathscr{A}=R_{l}(1-q)\bar{a}\dotplus
R_{l}(1-p)=R_{l}(a^{(2,l)}_{p,\,q})\bar{a}\dotplus K_{l}(a^{(2,l)}_{p,\,q})$
for $\bar{a}^{(2,l)}_{p,\,q}$ exists. Then for any $z\in\mathscr{A}$, we can
write $z=s_{1}a^{(2,l)}_{p,\,q}\bar{a}+s_{2}$, where $s_{1}\in
R_{l}(a^{(2,l)}_{p,\,q}a)$ and $s_{2}\in K_{l}(a^{(2,l)}_{p,\,q})$. Let
$t=s_{1}+s_{2}$, then we have

$\displaystyle t(1+a^{(2,l)}_{p,\,q}\delta a)$
$\displaystyle=(s_{1}+s_{2})(1+a^{(2,l)}_{p,\,q}\delta a)$
$\displaystyle=s_{1}+s_{2}+s_{1}a^{(2,l)}_{p,\,q}(\bar{a}-a)$
$\displaystyle=s_{1}a^{(2,l)}_{p,\,q}\bar{a}+s_{2}+s_{1}(1-a^{(2,l)}_{p,\,q}a)$
$\displaystyle=z.$

Since $z\in\mathscr{A}$ is arbitrary, let $z=1$, then we get that
$1+a^{(2,l)}_{p,\,q}\delta a$ is left invertible. But from Lemma 3.1 we see
$1+\delta aa^{(2,l)}_{p,\,q}$ is also left invertible. Thus, $1+\delta
aa^{(2,l)}_{p,\,q}$ is invertible.

Now, from Lemma 3.2, $\bar{a}^{(2,l)}_{p,\,q}=a^{(2,l)}_{p,\,q}(1+\delta
aa^{(2,l)}_{p,\,q})^{-1}=(1+a^{(2,l)}_{p,\,q}\delta a)^{-1}a^{(2,l)}_{p,\,q}.$
This completes the proof. ∎

######  Lemma 3.5.

Let $a,\,\delta a\in\mathscr{A}$ and $p,\,q\in\mathscr{A}^{\bullet}$ such that
$a^{(2,l)}_{p,\,q}$ exists. If $1+a^{(2,l)}_{p,\,q}\delta a$ is invertible.
Put $\bar{a}=a+\delta a$ and $f=(1+a^{(2,l)}_{p,\,q}\delta
a)^{-1}(1-a^{(2,l)}_{p,\,q}a)$. Then

$f\in\mathscr{A}^{\bullet}$ with $K_{r}(\bar{a})\subset R_{r}(f)$;

$K_{r}(\bar{a})=R_{r}(f)$ if and only if $R_{r}(\bar{a})\cap
R_{r}(q)=\\{0\\}$.

###### Proof.

(1) Since $(1-a^{(2,l)}_{p,\,q}a)(1+a^{(2,l)}_{p,\,q}\delta
a)=1-a^{(2,l)}_{p,\,q}a$ and $1+a^{(2,l)}_{p,\,q}\delta a$ is invertible, we
have $1-a^{(2,l)}_{p,\,q}a=(1-a^{(2,l)}_{p,\,q}a)(1+a^{(2,l)}_{p,\,q}\delta
a)^{-1}.$ Thus,

$f^{2}=(1+a^{(2,l)}_{p,\,q}\delta
a)^{-1}(1-a^{(2,l)}_{p,\,q}a)(1+a^{(2,l)}_{p,\,q}\delta
a)^{-1}(1-a^{(2,l)}_{p,\,q}a)=f.$

Now for any $x\in\mathscr{A}$, from
$(1-aa^{(2,l)}_{p,\,q})x=(1+a^{(2,l)}_{p,\,q}\delta
a)x-a^{(2,l)}_{p,\,q}\bar{a}x$, we have

$fx=(1+a^{(2,l)}_{p,\,q}\delta
a)^{-1}(1-a^{(2,l)}_{p,\,q}a)x=x-(1+a^{(2,l)}_{p,\,q}\delta
a)^{-1}a^{(2,l)}_{p,\,q}\bar{a}x.$ (3.4)

Eq. (3.4) implies that $K_{r}(\bar{a})\subset R_{r}(f)$.

(2) $(\Rightarrow)$ Let $t\in R_{r}(\bar{a})\cap R_{r}(q)=\\{0\\}$. Since
$R_{r}(q)=K_{r}(a^{(2,l)}_{p,\,q})$, then there is some $x\in\mathscr{A}$ such
that $t=\bar{a}x$ and $a^{(2,l)}_{p,\,q}\bar{a}x=a^{(2,l)}_{p,\,q}t=0$. Thus,
$x=fx$ by Eq. (3.4). So, $x\in R_{r}(f)=K_{r}(\bar{a})$ and $t=\bar{a}x=0$,
i.e., $R_{r}(\bar{a})\cap R_{r}(q)=\\{0\\}$.

$(\Leftarrow)$ Thanks to (1), we need only to prove $R_{r}(f)\subset
K_{r}(\bar{a})$. Let $t\in R_{r}(f)$. Since $f\in\mathscr{A}^{\bullet}$, we
have $t=ft$. So by Eq. (3.4), we get $(1+a^{(2,l)}_{p,\,q}\delta
a)^{-1}a^{(2,l)}_{p,\,q}\bar{a}t=0$ and then $a^{(2,l)}_{p,\,q}\bar{a}t=0$.
Hence $\bar{a}t\in R_{r}(\bar{a})\cap R_{r}(q)=\\{0\\}$ and
$K_{r}(\bar{a})=R_{r}(f)$. ∎

Similar to [13, Proposition 2.2] or [14, Theorem 2.4.7], but by using some of
our characterizations for $a^{(2,l)}_{p,\,q}$ and $a^{(l)}_{p,\,q}$, we can
obtain the following results about the stable perturbations for these two
kinds of generalized inverses.

######  Theorem 3.6.

Let $a,\,\delta a\in\mathscr{A}$ and $p,\,q\in\mathscr{A}^{\bullet}$ such that
$a^{(2,l)}_{p,\,q}$ exists. Suppose that $1+a^{(2,l)}_{p,\,q}\delta a$ is
invertible. Put $\bar{a}=a+\delta a$ and $w=(1+a^{(2,l)}_{p,\,q}\delta
a)^{-1}a^{(2,l)}_{p,\,q}$. Then the following statements are equivalent:

$w=\bar{a}^{(l)}_{p,\,q}$, i.e.,
$\bar{a}^{(2,l)}_{p,\,q}=\bar{a}^{(l)}_{p,\,q}$;

$R_{r}(\bar{a})\cap R_{r}(q)=\\{0\\}$, i.e., $\bar{a}$ is a stable
perturbation of $a$;

$\bar{a}(1+a^{(2,l)}_{p,\,q}\delta a)^{-1}(1-a^{(2,l)}_{p,\,q}a)=0$;

$(1-aa^{(2,l)}_{p,\,q})(1+\delta aa^{(2,l)}_{p,\,q})^{-1}\bar{a}=0$.

###### Proof.

The implication $(1)\Leftrightarrow(2)$ comes from Lemma 2.2, Lemma 2.3 and
Theorem 3.4. The implication $(2)\Leftrightarrow(3)$ comes from Lemma 3.5.

$(3)\Leftrightarrow(4)$ we can compute in the following way,

$\displaystyle\bar{a}(1+a^{(2,l)}_{p,\,q}\delta a)^{-1}(1-a^{(2,l)}_{p,\,q}a)$
$\displaystyle=\bar{a}(1+a^{(2,l)}_{p,\,q}\delta
a)^{-1}[(1+a^{(2,l)}_{p,\,q}\delta a)-a^{(2,l)}_{p,\,q}\bar{a}]$
$\displaystyle=\bar{a}-\bar{a}a^{(2,l)}_{p,\,q}(1+\delta
aa^{(2,l)}_{p,\,q})^{-1}\bar{a}$ $\displaystyle=\bar{a}-[(1+\delta
aa^{(2,l)}_{p,\,q})-(1-aa^{(2,l)}_{p,\,q})](1+\delta
aa^{(2,l)}_{p,\,q})^{-1}\bar{a}$
$\displaystyle=(1-aa^{(2,l)}_{p,\,q})(1+\delta
aa^{(2,l)}_{p,\,q})^{-1}\bar{a}.$

This completes the proof. ∎

Furthermore, by using the above theorem, we have the following results.

######  Corollary 3.7.

Let $a,\,\delta a\in\mathscr{A}$ and $p,\,q\in\mathscr{A}^{\bullet}$ such that
$a^{(2,l)}_{p,\,q}$ exists. Put $\bar{a}=a+\delta a$. If
$1+a^{(2,l)}_{p,\,q}\delta a$ is invertible. Then the following statements are
equivalent:

$R_{r}(\bar{a})\cap R_{r}(q)=\\{0\\}$, i.e., $\bar{a}$ is stable perturbation
of $a$;

$(1+a^{(2,l)}_{p,\,q}\delta a)^{-1}K_{r}(a^{(2,l)}_{p,\,q}a)=K_{r}(\bar{a})$;

$(1+\delta aa^{(2,l)}_{p,\,q})^{-1}R_{r}(\bar{a})=R_{r}(aa^{(2,l)}_{p,\,q})$.

###### Proof.

Note that we have $K_{r}(a^{(2,l)}_{p,\,q}a)=R_{r}(1-a^{(2,l)}_{p,\,q}a)$ and
$R_{r}(aa^{(2,l)}_{p,\,q})=K_{r}(1-aa^{(2,l)}_{p,\,q})$. So we can get the
assertions by using Theorem 3.6. ∎

######  Theorem 3.8.

Let $a,\,\delta a\in\mathscr{A}$ and $p,\,q\in\mathscr{A}^{\bullet}$ such that
$a^{(l)}_{p,\,q}$ exists. Put $\bar{a}=a+\delta a$. Then the following
statements are equivalent:

$1+a^{(l)}_{p,\,q}\delta a$ is invertible, $R_{r}(\bar{a})=K_{r}(q)$ and
$\bar{a}^{(l)}_{p,\,q}=a^{(l)}_{p,\,q}(1+\delta aa^{(l)}_{p,\,q})^{-1}$.

$R_{r}(\bar{a})\cap R_{r}(q)=\\{0\\}$, $K_{r}(\bar{a})\cap R_{r}(p)=\\{0\\}$
and $\bar{a}R_{r}(p)=K_{r}(q)$.

###### Proof.

$(1)\Rightarrow(2)$ Suppose that (1) holds. Since $a^{(l)}_{p,\,q}$ exists, we
obtain that $a^{(2,l)}_{p,\,q}$ exists and
$a^{(2,l)}_{p,\,q}=a^{(l)}_{p,\,q}$. Thus, from our assumption, by using
Theorem 3.6 and Lemma 2.3, we have

$R_{r}(\bar{a})\cap R_{r}(q)=\\{0\\},\quad K_{r}(\bar{a})\cap
R_{r}(p)=\\{0\\}.$

Now we need to show $\bar{a}R_{r}(p)=K_{r}(q)$. But since
$R_{r}(\bar{a})=K_{r}(q)$, so we can prove our result by showing that
$\bar{a}R_{r}(p)=R_{r}(\bar{a})$. Obviously, $\bar{a}R_{r}(p)\subset
R_{r}(\bar{a})$. On the other hand, since $\bar{a}^{(l)}_{p,\,q}$ exists, then
by Lemma 2.3 again, we have $\mathscr{A}=\bar{a}R_{r}(p)\dotplus R_{r}(q)$.
Now for any $x\in R_{r}(\bar{a})$, we can write $x=x_{1}+x_{2}$ with
$x_{1}\in\bar{a}R_{r}(p)$ and $x_{2}\in R_{r}(q)$. From
$\bar{a}R_{r}(p)\subset R_{r}(\bar{a})$, we get $x_{1}\in R_{r}(\bar{a})$.
Thus,

$x_{2}=x-x_{1}\in R_{r}(\bar{a})\cap R_{r}(q)=\\{0\\}.$

Therefore, $x_{2}=0$ and then $x=x_{1}\in\bar{a}R_{r}(p)$. Hence,
$\bar{a}R_{r}(p)=R_{r}(\bar{a})=K_{r}(q)$.

$(2)\Rightarrow(1)$ Since $q\in\mathscr{A}^{\bullet}$ and
$\bar{a}R_{r}(p)=K_{r}(q)$, we can write $\mathscr{A}=K_{r}(q)\dotplus
R_{r}(q)=\bar{a}R_{r}(p)\dotplus R_{r}(q)$. Note that $R_{r}(\bar{a})\cap
R_{r}(q)=\\{0\\}$, $K_{r}(\bar{a})\cap R_{r}(p)=\\{0\\}$, then by using Lemma
2.3, we get $a^{(l)}_{p,\,q}$ exists, then $a^{(2,l)}_{p,\,q}$ also exists and
$a^{(2,l)}_{p,\,q}=a^{(l)}_{p,\,q}$. Thus, from Theorem 3.4, we see
$1+a^{(l)}_{p,\,q}\delta a$ is invertible and
$\bar{a}^{(l)}_{p,\,q}=a^{(l)}_{p,\,q}(1+\delta aa^{(l)}_{p,\,q})^{-1}$. Now,
by using Lemma 2.3, we can get $\mathscr{A}=\bar{a}R_{r}(p)\dotplus R_{r}(q)$.
Similarly, as in $(1)\Rightarrow(2)$, by using $R_{r}(\bar{a})\cap
R_{r}(q)=\\{0\\}$, we can show that $\bar{a}R_{r}(p)=R_{r}(\bar{a})$ and then
$R_{r}(\bar{a})=K_{r}(q)$. This completes the proof. ∎

The first result in the following lemma has been proved for generalized
inverse $a^{+}$ by the second author (see [13, Proposition 2.5]). By using the
same method, we can prove the following results for the general inverse
$a^{(l)}_{p,\,q}$.

######  Lemma 3.9 ([13, Proposition 2.5]).

Let $a,\delta a\in\mathscr{A}$ and $p,\,q\in\mathscr{A}^{\bullet}$ such that
$a^{(l)}_{p,\,q}$ exists. Put $\bar{a}=a+\delta a$.

If $\delta(R_{r}(\bar{a}),R_{r}(a))<\|1-aa^{(l)}_{p,\,q}\|^{-1}$, then
$R_{r}(\bar{a})\cap R_{r}(q)=\\{0\\}$;

If $\delta(K_{r}(\bar{a}),K_{r}(a))<\|a^{(l)}_{p,\,q}a\|^{-1}$, then
$K_{r}(\bar{a})\cap R_{r}(p)=\\{0\\}$.

By using Theorem 3.6, Theorem 3.8 and Lemma 3.9, we have the following:

######  Corollary 3.10.

Let $a,\delta a\in\mathscr{A}$ and $p,\,q\in\mathscr{A}^{\bullet}$ such that
$a^{(l)}_{p,\,q}$ exists. Put $\bar{a}=a+\delta a$. If one of the following
condition holds,

$\delta(K_{r}(\bar{a}),K_{r}(a))<\|a^{(l)}_{p,\,q}a\|^{-1}$,
$\delta(R_{r}(\bar{a}),R_{r}(a))<\|1-aa^{(l)}_{p,\,q}\|^{-1}$ and
$\bar{a}R_{r}(p)=K_{r}(q)$.

$1+a^{(l)}_{p,\,q}\delta a$ is invertible and
$\delta(R_{r}(\bar{a}),R_{r}(a))<\|1-aa^{(l)}_{p,\,q}\|^{-1}$. Then
$\bar{a}^{(l)}_{p,\,q}$ exists and
$\bar{a}^{(l)}_{p,\,q}=a^{(l)}_{p,\,q}(1+\delta aa^{(l)}_{p,\,q})^{-1}$.

Finally, we present some perturbation results for $a^{(2)}_{p,\,q}$.

######  Theorem 3.11.

Let $a,\,\delta a\in\mathscr{A}$ and $p,\,q\in\mathscr{A}^{\bullet}$ such that
$a^{(2)}_{p,\,q}$ exists. Put $\bar{a}=a+\delta a$. If
$1+a^{(2)}_{p,\,q}\delta a$ is invertible. Then the following statements are
equivalent:

$\bar{a}^{(2)}_{p,\,q}$ exists and
$\bar{a}^{(2)}_{p,\,q}=a^{(2)}_{p,\,q}(1+\delta aa^{(2)}_{p,\,q})^{-1}$;

$\bar{a}p=(1-q)\bar{a}$;

$\bar{a}a^{(2)}_{p,\,q}=(1-q)\bar{a}a^{(2)}_{p,\,q}$ and
$a^{(2)}_{p,\,q}\bar{a}=a^{(2)}_{p,\,q}\bar{a}p$.

###### Proof.

$(1)\Leftrightarrow(2)$ comes from [6, Theorem 4.1]. We show that (2) and (3)
are equivalent. If $\bar{a}p=(1-q)\bar{a}$, then

$\bar{a}a^{(2)}_{p,\,q}=\bar{a}pa^{(2)}_{p,\,q}=(1-q)\bar{a}a^{(2)}_{p,\,q}\quad
and\quad
a^{(2)}_{p,\,q}\bar{a}=a^{(2)}_{p,\,q}(1-q)\bar{a}=a^{(2)}_{p,\,q}\bar{a}p.$

Conversely, if (3) holds, then
$\bar{a}p=\bar{a}a^{(2)}_{p,\,q}a=(1-q)\bar{a}a^{(2)}_{p,\,q}a=aa^{(2)}_{p,\,q}\bar{a}p=(1-q)\bar{a}$.
∎

######  Corollary 3.12.

Let $a,\,\delta a\in\mathscr{A}$ and $p,\,q\in\mathscr{A}^{\bullet}$ such that
$a^{(2)}_{p,\,q}$ exists. Put $\bar{a}=a+\delta a$. If
$1+a^{(2)}_{p,\,q}\delta a$ is invertible and $\delta a=(1-q)\delta a=\delta
ap$. Then $\bar{a}^{(2)}_{p,\,q}$ exists and

$\bar{a}^{(2)}_{p,\,q}=a^{(2)}_{p,\,q}(1+\delta
aa^{(2)}_{p,\,q})^{-1}=(1+a^{(2)}_{p,\,q}\delta a)^{-1}a^{(2)}_{p,\,q}.$

###### Proof.

If $\delta a=(1-q)\delta a=\delta ap$, then it is easy to check that
$\bar{a}p=(1-q)\bar{a}$. Thus, Theorem 3.11 shows that our results hold. ∎

##  4 Perturbation analysis for the $(p,q)$–generalized inverses

In this section, we mainly investigate the general perturbations problem for
the $(p,q)$–generalized inverses $a^{(2,l)}_{p,\,q}$ and $a^{(1,2)}_{p,\,q}$.
Let $\kappa=\|a\|\|a^{(2,l)}_{p,\,q}\|$, which is the generalized condition
number of the generalized inverse $a^{(2,l)}_{p,\,q}$.

######  Lemma 4.1.

Let $a\in\mathscr{A}$ and $p\in\mathscr{A}^{\bullet}$ with
$R_{r}(p)=R_{r}(a)$. Let $c\in\mathscr{A}$ with $R_{r}(c)$ closed and
$\hat{\delta}(R_{r}(c),\,R_{r}(a))<\dfrac{1}{1+\|p\|}$. Then
$\mathscr{A}=R_{r}(c)\dotplus K_{r}(p)$.

###### Proof.

Let $L_{p}\,x=p\,x$, $\forall\,x\in\mathscr{A}$. Then $L_{p}$ is an idempotent
operator on $\mathscr{A}$ with $\|L_{p}\|=\|p\|$ and
$\\{L_{p}\,x|\,x\in\mathscr{A}\\}=R_{r}(p)$. By [11, Theorem 11] or [14, Lemma
4.4.4], $\hat{\delta}(R_{r}(c),\,R_{r}(a))<\dfrac{1}{1+\|L_{p}\|}$ implies
that $\mathscr{A}=R_{r}(c)\dotplus K_{r}(p)$. ∎

######  Lemma 4.2 ([13, Lemma 2.4]).

For any $p,\,q\in\mathscr{A}^{\bullet}$ we have
$\hat{\delta}(R_{r}(p),R_{r}(q))\leq\|p-q\|$.

######  Lemma 4.3.

Let $a\in\mathscr{A}$ and $p,\,q\in\mathscr{A}^{\bullet}$ such that
$a^{(2,l)}_{p,\,q}$ exists. Suppose that $p^{\prime}\in\mathscr{A}^{\bullet}$
satisfying $\|p-p^{\prime}\|<\dfrac{1}{1+\kappa}$. Then

$\hat{\delta}(aR_{r}(p),aR_{r}(p^{\prime}))\leq\dfrac{\kappa\|p-p^{\prime}\|}{1-(1+\kappa)\|p-p^{\prime}\|}$;

$aR_{r}(p^{\prime})\subset\mathscr{A}$ is closed and $K_{r}(a)\cap
R_{r}(p^{\prime})=\\{0\\}$.

###### Proof.

(1) Set $b=a^{(2,l)}_{p,\,q}$. For any $t^{\prime}\in R_{r}(p^{\prime})$, we
have

$\displaystyle{\text{dist}}(at^{\prime},aR_{r}(p))$ $\displaystyle=\inf_{t\in
R_{r}(p)}\|at^{\prime}-at\|\leq\|a\|\inf_{t\in R_{r}(p)}\|t-t^{\prime}\|$
$\displaystyle\leq\|a\|{\text{dist}}(t^{\prime},R_{r}(p))$
$\displaystyle\leq\|a\|\|t^{\prime}\|\hat{\delta}(R_{r}(p^{\prime}),R_{r}(p)).$

Thus, we get

$\displaystyle\delta(at^{\prime},aR_{r}(p))\leq\|a\|\|t^{\prime}\|\hat{\delta}(R_{r}(p^{\prime}),R_{r}(p)).$
(4.1)

But for any $t^{\prime}\in R_{r}(p^{\prime})$ and $t\in R_{r}(p)$, we have

$\displaystyle\|b\|\|at^{\prime}\|$
$\displaystyle=\|b\|\|a(t^{\prime}-t+t)\|\geq\|b\|\|at\|-\|b\|\|a\|\|t^{\prime}-t\|$
$\displaystyle\geq\|t\|-\|b\|\|a\|\|t^{\prime}-t\|$
$\displaystyle\geq\|t^{\prime}\|-(1+\|b\|\|a\|)\|t^{\prime}-t\|.$

Thus, $\|t^{\prime}\|-\|b\|\|at^{\prime}\|\leq(1+\|b\|\|a\|)\|t^{\prime}-t\|,$
and then

$\displaystyle\|t^{\prime}\|-\|b\|\|at^{\prime}\|$
$\displaystyle\leq(1+\|b\|\|a\|){\text{dist}}(t^{\prime},R_{r}(p))$
$\displaystyle\leq(1+\|b\|\|a\|)\|t^{\prime}\|\hat{\delta}(R_{r}(p^{\prime}),R_{r}(p)).$

Therefore, we have

$\displaystyle\|t^{\prime}\|\leq\frac{\|b\|\|at^{\prime}\|}{1-(1+\|b\|\|a\|)\hat{\delta}(R_{r}(p^{\prime}),R_{r}(p))}.$
(4.2)

Then by Eq. (4.1) and Eq. (4.2), we get

$\displaystyle\delta(at^{\prime},aR_{r}(p))$
$\displaystyle\leq\frac{\|b\|\|a\|\|at^{\prime}\|\hat{\delta}(R_{r}(p^{\prime}),R_{r}(p))}{1-(1+\|b\|\|a\|)\hat{\delta}(R_{r}(p^{\prime}),R_{r}(p))}.$

Now, by Lemma 4.2 and the Definition of gap–function, we have

$\displaystyle\delta(aR_{r}(p^{\prime}),aR_{r}(p))$
$\displaystyle\leq\frac{\kappa\|p-p^{\prime}\|}{1-(1+\kappa)\|p-p^{\prime}\|}.$
(4.3)

On the other hand, for any $t\in R_{r}(p)$, by Lemma 2.2, we have
$t=pt=bapt=bat,$ then

$\displaystyle{\rm dist}(at,aR_{r}(p^{\prime}))$ $\displaystyle=\inf_{s\in
R_{r}(p^{\prime})}\|at-as\|\leq\|a\|\inf_{s\in R_{r}(p^{\prime})}\|t-s\|$
$\displaystyle=\|a\|{\text{dist}}(t,R_{r}(p^{\prime}))\leq\|a\|\|t\|\delta(R_{r}(p),R_{r}(p^{\prime}))$
$\displaystyle=\|a\|\|bat\|\delta(R_{r}(p),R_{r}(p^{\prime}))$
$\displaystyle\leq\|a\|\|b\|\|at\|\delta(R_{r}(p),R_{r}(p^{\prime})).$

Thus we have
$\delta(aR_{r}(p),aR_{r}(p^{\prime}))\leq\kappa\hat{\delta}(R_{r}(p),R_{r}(p^{\prime}))$.
So by Lemma 4.2,

$\displaystyle\delta(aR_{r}(p),aR_{r}(p^{\prime}))\leq\kappa\|p-p^{\prime}\|.$
(4.4)

Consequently, from Eq. (4.3) and Eq. (4.4), we have

$\displaystyle\hat{\delta}(aR_{r}(p),aR_{r}(p^{\prime}))$
$\displaystyle=\max\\{\delta(aR_{r}(p^{\prime}),aR_{r}(p)),\delta(aR_{r}(p),aR_{r}(p^{\prime}))\\}$
$\displaystyle\leq\dfrac{\kappa\|p-p^{\prime}\|}{1-(1+\kappa)\|p-p^{\prime}\|}.$

(2) Obviously, by Eq. (4.2), we get $aR_{r}(p^{\prime})\subset\mathscr{A}$ is
closed and $K_{r}(a)\cap R_{r}(p^{\prime})=\\{0\\}.$ This completes the proof.
∎

Now we can give the following perturbation result for $a^{(2,l)}_{p,\,q}$ when
$p$ has a small perturbation.

######  Theorem 4.4.

Let $a\in\mathscr{A}$ and $p,\,q\in\mathscr{A}^{\bullet}$ such that
$a^{(2,l)}_{p,\,q}$ exists. Suppose that $p^{\prime}\in\mathscr{A}^{\bullet}$
with $\|p-p^{\prime}\|<\dfrac{1}{(1+\kappa)^{2}}$. Then
$a_{p^{\prime},q}^{(2,l)}$ exists and

$\frac{\|a^{(2,l)}_{p^{\prime},q}-a^{(2,l)}_{p,\,q}\|}{\|a^{(2,l)}_{p,\,q}\|}\leq\dfrac{(1+\kappa)\|p^{\prime}-p\|}{1-(1+\kappa)\|p^{\prime}-p\|}\
\text{and}\
\|a^{(2,l)}_{p^{\prime},q}\|\leq\frac{\|a^{(2,l)}_{p,\,q}\|}{1-(1+\kappa)\|p^{\prime}-p\|}.$

###### Proof.

Let $b=a^{(2,l)}_{p,\,q}$, then by Definition 1.1, we know that
$ab\in\mathscr{A}^{\bullet}$ and $\kappa=\|b\|\|a\|\geq\|ab\|$. By using Lemma
4.3 and note that $\|p-p^{\prime}\|<\dfrac{1}{(1+\kappa)^{2}}$, we have

$\hat{\delta}(aR_{r}(p),aR_{r}(p^{\prime}))\leq\dfrac{\kappa\|p-p^{\prime}\|}{1-(1+\kappa)\|p-p^{\prime}\|}<\frac{1}{1+\|ab\|}.$

From Lemma 4.3 (2), we know that $K_{r}(a)\cap R_{r}(p^{\prime})=\\{0\\}$ and
$aR_{r}(p^{\prime})\subset\mathscr{A}$ is closed. Thus, by Lemma 4.1,
$aR_{r}(p^{\prime})$ is complemented and
$\mathscr{A}=aR_{r}(p^{\prime})\dotplus R_{r}(q)$. Therefore, by Lemma 2.2, we
know $a^{(2)}_{p^{\prime},\,q}$ exists.

For convenience we write $a^{(2,l)}_{p^{\prime},q}=b^{\prime}$. Since we have
proved that $a^{(2,l)}_{p^{\prime},q}$ exists, then by Theorem 2.2,
$\mathscr{A}=aR_{r}(p^{\prime})\dotplus R_{r}(q)=aR_{r}(p^{\prime})\dotplus
K_{r}(b^{\prime})$ and $K_{r}(b^{\prime})=R_{r}(q)=K_{r}(b)$. Thus for any
$x\in\mathscr{A}$, we can write $x=t+t^{\prime}$ with $t=ab^{\prime}z$ for
some $z\in\mathscr{A}$ and $t^{\prime}\in R_{r}(q)$.

Since
dist$(b^{\prime}z,R_{r}(p))\leq\|b^{\prime}z\|\delta(R_{r}(p^{\prime}),R_{r}(p))\leq|b^{\prime}z\|\|p^{\prime}-p\|$,
then for every $\epsilon>0$, we can choose $y\in\mathscr{A}$ such that
$\|b^{\prime}z-by\|<\|b^{\prime}z\|\|p^{\prime}-p\|+\epsilon.$ Put $s=aby$.
Then we have

$\displaystyle\|t-s\|$
$\displaystyle=\|ab^{\prime}z-aby\|<\|a\|\|b^{\prime}z\||p^{\prime}-p\|+\|a\|\epsilon.$
$\displaystyle\|(b^{\prime}-b)x\|$
$\displaystyle=\|(b^{\prime}-b)(t+t^{\prime})\|=\|(b^{\prime}-b)t\|$
$\displaystyle\leq\|b^{\prime}ab^{\prime}t-babs\|+\|bs-bt\|$
$\displaystyle\leq\|b^{\prime}z-by\|+\|b\|\|s-t\|$
$\displaystyle\leq(1+\kappa)\|b^{\prime}z\|\|p^{\prime}-p\|+(1+\kappa)\epsilon.$
(4.5)

From $t=ab^{\prime}z$, we get $b^{\prime}t=b^{\prime}z$ and therefore

$\|b^{\prime}z\|=\|b^{\prime}t\|=\|(b^{\prime}-b)x+bx\|\leq\|(b^{\prime}-b)x\|+\|b\|\|x\|.$
(4.6)

Thus by using Eq. (4) and Eq. (4.6), we get

$\|(b^{\prime}-b)x\|\leq(1+\kappa)(\|b\|\|x\|+\|(b^{\prime}-b)x\|)\|p^{\prime}-p\|+(1+\kappa)\epsilon.$
(4.7)

Letting $\epsilon\to 0^{+}$ in Eq. (4.7), we can get

$\frac{\|a^{(2,l)}_{p^{\prime},q}-a^{(2,l)}_{p,\,q}\|}{\|a^{(2,l)}_{p,\,q}\|}\leq\dfrac{(1+\kappa)\|p^{\prime}-p\|}{1-(1+\kappa)\|p^{\prime}-p\|},\quad\|a^{(2,l)}_{p^{\prime},q}\|\leq\frac{\|a^{(2,l)}_{p,\,q}\|}{1-(1+\kappa)\|p^{\prime}-p\|}.$

This completes the proof. ∎

Some representations for the generalized inverse $a^{(2,l)}_{p,q}$ have been
presented in [1]. The following result gives a representation of
$a^{(2,l)}_{p,q}$ based on (1,5) inverse. Note that this result is also an
improvement of the group inverses representation of $a^{(2,l)}_{p,q}$(see
[1]), which removes the existence of the group inverses of $wa$ or $aw$.

######  Lemma 4.5 ([1, Theorem 5.6]).

Let $a,w\in\mathscr{A}$ and $p,\;q\in\mathscr{A}^{\bullet}$ such that
$R_{r}(w)=R_{r}(p)$ and $K_{r}(w)=R_{r}(q)$. Then the following statements are
equivalent:

$a^{(2,l)}_{p,q}$ exists;

$(aw)^{(1,5)}$ exists and $K_{r}(a)\cap R_{r}(w)=\\{0\\};$

$(wa)^{(1,5)}$ exists and $R_{r}(w)=R_{r}(wa).$ In this case, $waw$ is inner
regular and

$a^{(2,l)}_{p,\,q}=(wa)^{(1,5)}w=w(aw)^{(1,5)}=w(waw)^{-}w.$

Now we give the result when $q$ has a small perturbation. By using our above
lemma 4.5, we can also give a new representation for the generalized inverse
of the perturbed operator.

######  Theorem 4.6.

Let $a\in\mathscr{A}$ and $p,\,q\in\mathscr{A}^{\bullet}$ such that
$a^{(2,l)}_{p,\,q}$ exists. Suppose that $q^{\prime}\in\mathscr{A}^{\bullet}$
with $\|q-q^{\prime}\|<\dfrac{1}{2+\kappa}$. Then $a_{p,\,q\prime}^{(2,l)}$
exists and

$\dfrac{\|(a^{(2,l)}_{p,\,q^{\prime}}-a^{(2,l)}_{p,\,q})\|}{\|a^{(2,l)}_{p,\,q}\|}\leq\dfrac{(1+\kappa)\|q-q^{\prime}\|}{1-\kappa\|q-q^{\prime}\|}$
and
$\|a^{(2,l)}_{p,\,q^{\prime}}\|\leq\dfrac{1+\|q^{\prime}-q\|}{1-\kappa\|q^{\prime}-q\|}\,\|a^{(2,l)}_{p,\,q}\|$.

If there are some $w,\,v\in\mathscr{A}$ with $R_{r}(w)=R_{r}(p)$,
$K_{r}(w)=K_{r}(v)=R_{r}(q)$ and $R_{r}(v)=R_{r}(q^{\prime})$. Then

$a_{p,\,q^{\prime}}^{(2,l)}=a^{(2,l)}_{p,\,q}+a^{(2,l)}_{p,\,q}(av)^{(1,5)}a(v-w)(1-aa^{(2,l)}_{p,\,q}).$

###### Proof.

Since $1+\|1-ab\|\leq 2+\kappa$, then by Lemma 4.2, we have

$\hat{\delta}(R_{r}(q),R_{r}(q^{\prime}))\leq\|q-q^{\prime}\|<\frac{1}{2+\kappa}\leq\frac{1}{1+\|1-ab\|}.$

So $\mathscr{A}=aR_{r}(p)\dotplus R_{r}(q^{\prime})$ by Lemma 4.1. Note that
$K_{r}(a)\cap R_{r}(p)=\\{0\\}$. So $a^{(1,2)}_{p,\,q^{\prime}}$ exists.

(1) From
$a^{(2,l)}_{p,\,q^{\prime}}=pa^{(2,l)}_{p,\,q^{\prime}}=a^{(2,l)}_{p,\,q}aa^{(2,l)}_{p,\,q^{\prime}}$,
we get

$a^{(2,l)}_{p,\,q^{\prime}}-a^{(2,l)}_{p,\,q}=a^{(2,l)}_{p,\,q}(aa^{(2,l)}_{p,\,q^{\prime}}-aa^{(2,l)}_{p,\,q}).$

Since $(1-aa^{(2,l)}_{p,\,q^{\prime}})x\in R_{r}(q^{\prime})$ for any
$x\in\mathscr{A}$, so we have

$\displaystyle{\rm dist}((1-aa^{(2,l)}_{p,\,q^{\prime}})x,R_{r}(q))$
$\displaystyle\leq\hat{\delta}(R_{r}(q^{\prime}),R_{r}(q))\|(1-aa^{(2,l)}_{p,\,q^{\prime}})x\|$
$\displaystyle\leq\|q-q^{\prime}\|\|(1-aa^{(2,l)}_{p,\,q^{\prime}})x\|.$

Since $K_{r}(a^{(2,l)}_{p,\,q})=R_{r}(q)$, then, for any $\epsilon>0$, there
is $z\in\mathscr{A}$ such that

$\|(1-aa^{(2,l)}_{p,\,q^{\prime}})x-(1-aa^{(2,l)}_{p,\,q})z\|\leq\|q-q^{\prime}\|\|(1-aa^{(2,l)}_{p,\,q^{\prime}})x\|+\epsilon.$

and then we have

$\displaystyle\|(a^{(2,l)}_{p,\,q^{\prime}}-a^{(2,l)}_{p,\,q})x\|$
$\displaystyle=\|a^{(2,l)}_{p,\,q}(aa^{(2,l)}_{p,\,q^{\prime}}-aa^{(2,l)}_{p,\,q})x\|$
$\displaystyle\leq\|a^{(2,l)}_{p,\,q}\\{(1-aa^{(2,l)}_{p,\,q^{\prime}})x-(1-aa^{(2,l)}_{p,\,q})z\\}\|$
$\displaystyle\leq\left(\|q-q^{\prime}\|\|(1-aa^{(2,l)}_{p,\,q^{\prime}})x\|+\epsilon\right)\|a^{(2,l)}_{p,\,q}\|.$
(4.8)

Since we also have

$\displaystyle\|(1-aa^{(2,l)}_{p,\,q^{\prime}})x\|$
$\displaystyle\leq\|x\|+\|a\|\|a^{(2,l)}_{p,\,q}x-(a^{(2,l)}_{p,\,q^{\prime}}-a^{(2,l)}_{p,\,q})x\|$
$\displaystyle\leq(1+\kappa)\|x\|+\|a\|\|(a^{(2,l)}_{p,\,q^{\prime}}-a^{(2,l)}_{p,\,q})x\|.$
(4.9)

Now from Eqs. (4) and (4), we can compute

$\|(a^{(2,l)}_{p,\,q^{\prime}}-a^{(2,l)}_{p,\,q})x\|\leq\dfrac{(\|q-q^{\prime}\|(1+\kappa)\|x\|+\epsilon)\|a^{(2,l)}_{p,\,q}\|}{1-\kappa\|q-q^{\prime}\|}.$

Let $\epsilon\to 0^{+}$ in the above inequality, we obtain that

$\frac{\|a^{(2,l)}_{p,\,q^{\prime}}-a^{(2,l)}_{p,\,q}\|}{\|a^{(2,l)}_{p,\,q}\|}\leq\dfrac{(1+\kappa)\|q^{\prime}-q\|}{1-\kappa\|q^{\prime}-q\|},\quad\|a^{(2,l)}_{p,\,q^{\prime}}\|\leq\frac{1+\|q^{\prime}-q\|}{1-\kappa\|q^{\prime}-q\|}\,\|a^{(2,l)}_{p,\,q}\|.$

(2) By Lemma 4.5, $a^{(2,l)}_{p,q^{\prime}}=(va)^{(1,5)}v=v(av)^{(1,5)}$. For
convenience, we write $b=a^{(2,l)}_{p,\,q}$,
$b^{\prime}=a_{p,\,q^{\prime}}^{(2,l)}$ and
$x=a^{(2,l)}_{p,\,q}+a^{(2,l)}_{p,\,q}(av)^{(1,5)}a(v-w)(1-aa^{(2,l)}_{p,\,q})$.
Now we prove that $x=b^{\prime}$ by using Lemma 2.2 (4). Obviously, we have
$px=x$. Note that $p=a^{(2,l)}_{p,\,q}ap$, so

$\displaystyle xap$
$\displaystyle=\\{a^{(2,l)}_{p,\,q}+a^{(2,l)}_{p,\,q}(av)^{(1,5)}a(v-w)(1-aa^{(2,l)}_{p,\,q})\\}ap$
$\displaystyle=a^{(2,l)}_{p,\,q}ap+a^{(2,l)}_{p,\,q}(av)^{(1,5)}a(v-w)(ap-ap)$
$\displaystyle=p.$

Since $b=w(aw)^{(1,5)}$. we have

$\displaystyle xq^{\prime}$
$\displaystyle=\\{a^{(2,l)}_{p,\,q}+a^{(2,l)}_{p,\,q}(av)^{(1,5)}a(v-w)(1-aa^{(2,l)}_{p,\,q})\\}q^{\prime}$
$\displaystyle=bq^{\prime}+\\{b(av)^{(1,5)}av-b(av)^{(1,5)}avab-b(av)^{(1,5)}aw+b(av)^{(1,5)}awab\\}q^{\prime}.$
$\displaystyle=bq^{\prime}+bab^{\prime}q^{\prime}-bab^{\prime}abq^{\prime}-b(av)^{(1,5)}awq^{\prime}+b(av)^{(1,5)}awabq^{\prime}.$
$\displaystyle=bq^{\prime}+0-bq^{\prime}-b(av)^{(1,5)}awq^{\prime}+b(av)^{(1,5)}awaw(aw)^{(1,5)}q^{\prime}.$
$\displaystyle=0.$

Thus, we have $x(1-q^{\prime})=x$. Finally, since $aw=awaw(aw)^{(1,5)}=awab$,
we have

$\displaystyle(1-q^{\prime})ax$
$\displaystyle=(1-q^{\prime})a\\{a^{(2,l)}_{p,\,q}+a^{(2,l)}_{p,\,q}(av)^{(1,5)}a(v-w)(1-aa^{(2,l)}_{p,\,q})\\}$
$\displaystyle=(1-q^{\prime})ab^{\prime}a\\{a^{(2,l)}_{p,\,q}+a^{(2,l)}_{p,\,q}(av)^{(1,5)}a(v-w)(1-aa^{(2,l)}_{p,\,q})\\}$
$\displaystyle=(1-q^{\prime})\\{ab^{\prime}ab+ab^{\prime}ab(av)^{(1,5)}(av-w)(1-ab)\\}$
$\displaystyle=(1-q^{\prime})(ab+ab^{\prime}-ab-
ab^{\prime}ab(av)^{(1,5)}aw+ab^{\prime}ab(av)^{(1,5)}awab)$
$\displaystyle=(1-q^{\prime})ab^{\prime}$ $\displaystyle=1-q^{\prime}.$

Therefore by Lemma 2.2 and the uniqueness of $a^{(2,l)}_{p,\,q^{\prime}}$, we
get that $x=a^{(2,l)}_{p,\,q^{\prime}}$. ∎

When the idempotents $p$ and $q$ both have some small perturbations, we have
the following result.

######  Theorem 4.7.

Let $a\in\mathscr{A}$ and
$p,\,q,\,p^{\prime},\,q^{\prime}\in\mathscr{A}^{\bullet}$ such that
$a^{(2,l)}_{p,\,q}$ exists. If $\|p-p^{\prime}\|<\dfrac{1}{(1+\kappa)^{2}}$
and $\|q-q^{\prime}\|<\dfrac{1}{3+\kappa}$. Then
$a_{p^{\prime},\,q^{\prime}}^{(2,l)}$ exists and

$\displaystyle\frac{\|a^{(2,l)}_{p^{\prime},\,q^{\prime}}-a^{(2,l)}_{p,\,q}\|}{\|a_{p,\,q}^{(2,l)}\|}$
$\displaystyle\leq\frac{(1+\kappa)(\|p-p^{\prime}\|+\|q-q^{\prime}\|)}{1-(1+\kappa)\|p-p^{\prime}\|-\kappa\|q-q^{\prime}\|}$
$\displaystyle\|a_{p^{\prime},\,q^{\prime}}^{(2,l)}\|$
$\displaystyle\leq\dfrac{(1+\|q-q^{\prime}\|)\|a_{p,\,q}^{(2,l)}\|}{1-(1+\kappa)\|p-p^{\prime}\|-\kappa\|q-q^{\prime}\|}.$

###### Proof.

By Theorem 4.4, $a_{p^{\prime},\,q}^{(2,l)}$ exists when
$\|p-p^{\prime}\|<\dfrac{1}{(1+\kappa)^{2}}$ and in this case,

$\|a^{(2,l)}_{p^{\prime},q}\|\leq\frac{\|a^{(2,l)}_{p,\,q}\|}{1-(1+\kappa)\|p^{\prime}-p\|}\leq\frac{1+\kappa}{\kappa}\|a^{(2,l)}_{p,\,q}\|.$

So
$\|q-q^{\prime}\|<\dfrac{1}{3+\kappa}<\dfrac{1}{2+\|a\|\|a^{(2,l)}_{p^{\prime},\,q}\|}$
and consequently, $a^{(2,l)}_{p^{\prime},\,q^{\prime}}$ exists by Theorem 4.6.
Finally, by Theorem 4.4 and Theorem 4.6, we have

$\displaystyle\|a_{p^{\prime},\,q^{\prime}}^{(2,l)}-a_{p,\,q}^{(2,l)}\|$
$\displaystyle\leq\|a_{p^{\prime},\,q^{\prime}}^{(2,l)}-a_{p^{\prime},\,q}^{(2,l)}\|+\|a_{p^{\prime},\,q}^{(2,l)}-a^{(2,l)}_{p,\,q}\|$
$\displaystyle\leq\frac{(1+\|a\|\|a_{p^{\prime},\,q}^{(2,l)})\|q-q^{\prime}\|}{1-\|a\|\|a_{p^{\prime},\,q}^{(2,l)}\|\|q-q^{\prime}\|}\|a_{p^{\prime},\,q}^{(2,l)}\|+\frac{(1+\kappa)\|p-p^{\prime}\|}{1-(1+\kappa)\|p-p^{\prime}\|}\|a_{p^{\prime},\,q}^{(2,l)}\|$
$\displaystyle\leq\frac{(1+\kappa)(1-\|p-p^{\prime}\|)\|q-q^{\prime}\|}{1-(1+\kappa)\|p-p^{\prime}\|-\kappa\|q-q^{\prime}\|}\frac{\|a_{p^{\prime},\,q}^{(2,l)}\|}{1-(1+\kappa)\|p-p^{\prime}\|}$
$\displaystyle\
+\frac{(1+\kappa)\|p-p^{\prime}\|}{1-(1+\kappa)\|p-p^{\prime}\|}\|a_{p^{\prime},\,q}^{(2,l)}\|$
$\displaystyle=\frac{(1+\kappa)(\|p-p^{\prime}\|+\|q-q^{\prime}\|)}{1-(1+\kappa)\|p-p^{\prime}\|-\kappa\|q-q^{\prime}\|}\|a_{p,\,q}^{(2,l)}\|$

and
$\|a_{p^{\prime},\,q^{\prime}}^{(2,l)}\|\leq\|a_{p^{\prime},\,q^{\prime}}^{(2,l)}-a_{p,\,q}^{(2,l)}\|+\|a_{p,\,q}^{(2,l)}\|\leq\dfrac{(1+\|q-q^{\prime}\|)\|a_{p,\,q}^{(2,l)}\|}{1-(1+\kappa)\|p-p^{\prime}\|-\kappa\|q-q^{\prime}\|}.$
∎

Now, we consider the case when the elements $a,p,q\in\mathscr{A}^{\bullet}$
all have some small perturbations.

######  Theorem 4.8.

Let $a,\delta a\in\mathscr{A}$ and
$p,\,q,\,p^{\prime},\,q^{\prime}\in\mathscr{A}^{\bullet}$ such that
$a^{(2,l)}_{p,\,q}$ exists. If $\|p-p^{\prime}\|<\dfrac{1}{(\kappa+1)^{2}}$,
$\|q-q^{\prime}\|<\dfrac{1}{\kappa+3}$ and $\|a^{(2,l)}_{p,\,q}\|\|\delta
a\|<\dfrac{2\kappa}{(\kappa+1)(\kappa+4)}$. Then
$\bar{a}_{p^{\prime},\,q^{\prime}}^{(2,l)}$ exists and

$\displaystyle\|\bar{a}_{p^{\prime},\,q^{\prime}}^{(2,l)}\|$
$\displaystyle\leq\frac{(1+\|q-q^{\prime}\|)\|a_{p,\,q}^{(2,l)}\|}{1-(1+\kappa)\|p-p^{\prime}\|-\kappa\|q-q^{\prime}\|-(1+\|q-q^{\prime}\|)\|a_{p,\,q}^{(2,l)}\|\|\delta
a\|}$
$\displaystyle\|\bar{a}_{p^{\prime},\,q^{\prime}}^{(2,l)}-a_{p,q}^{(2,l)}\|$
$\displaystyle\leq\frac{\|a_{p,\,q}^{(2,l)}\|}{1-(1+\kappa)\|p-p^{\prime}\|-\kappa\|q-q^{\prime}\|}\Big{[}(1+\kappa)(\|p-p^{\prime}\|+\|q-q^{\prime}\|)$
$\displaystyle\ +\frac{(1+\|q-q^{\prime}\|)^{2}\|\delta
a\|\|a_{p,\,q}^{(2,l)}\|}{1-(1+\kappa)\|p-p^{\prime}\|-\kappa\|q-q^{\prime}\|-(1+\|q-q^{\prime}\|)\|a_{p,\,q}^{(2,l)}\|\|\delta
a\|}\Big{]}.$

###### Proof.

Theorem 4.7 indicates that $a_{p^{\prime},\,q^{\prime}}^{(2,l)}$ exists and

$\|a^{(2,l)}_{p^{\prime}\,,q^{\prime}}\|\leq\dfrac{(1+\|q-q^{\prime}\|)\|a_{p,\,q}^{(2,l)}\|}{1-(1+\kappa)\|p-p^{\prime}\|-\kappa\|q-q^{\prime}\|}<\dfrac{(1+\kappa)(4+\kappa)}{2\kappa}\|a^{(2,l)}_{p,\,q}\|.$

Thus, $\|a^{(2,l)}_{p^{\prime},q^{\prime}}\|\|\delta a\|<1$ and hence
$1+a^{(2,l)}_{p^{\prime},q^{\prime}}\delta a$ is invertible. Therefore,
$a^{(2,l)}_{p^{\prime},q^{\prime}}$ exists and
$\bar{a}^{(2,l)}_{p^{\prime},q^{\prime}}=a^{(2,l)}_{p^{\prime},q^{\prime}}(1+\delta
aa^{(2,l)}_{p^{\prime},q^{\prime}})^{-1}$ by Theorem 3.4. Now by Theorem 4.7,
we have

$\|\bar{a}_{p^{\prime},q^{\prime}}^{(2,l)}\|\\!\leq\\!\dfrac{\|a_{p^{\prime},q^{\prime}}^{(2,l)}\|}{1-\|a_{p^{\prime},q^{\prime}}^{(2,l)}\|\|\delta
a\|}\\!\leq\\!\frac{[1+\|q-q^{\prime}\|]\|a_{p,\,q}^{(2,l)}\|}{1-[1+\kappa]\|p-p^{\prime}\|-\kappa\|q-q^{\prime}\|-[1+\|q-q^{\prime}\|]\|a_{p,\,q}^{(2,l)}\|\|\delta
a\|}$

and

$\displaystyle\|\bar{a}_{p^{\prime},\,q^{\prime}}^{(2,l)}-a_{p,q}^{(2,l)}\|$
$\displaystyle\leq\|(1+{a}_{p^{\prime},\,q^{\prime}}^{(2,l)}\delta
a)^{-1}{a}_{p^{\prime},\,q^{\prime}}^{(2,l)}-a_{p^{\prime},\,q^{\prime}}^{(2,l)}\|+\|a_{p^{\prime},\,q^{\prime}}^{(2,l)}-a_{p,\,q}^{(2,l)}\|$
$\displaystyle\leq\frac{\|\delta
a\|\|a_{p^{\prime},\,q^{\prime}}^{(2,l)}\|^{2}}{1-\|\delta
a\|\|a_{p^{\prime},\,q^{\prime}}^{(2,l)}\|}+\|a_{p^{\prime},\,q^{\prime}}^{(2,l)}-a_{p,\,q}^{(2,l)}\|$
$\displaystyle\leq\frac{\|a_{p,\,q}^{(2,l)}\|}{1-(1+\kappa)\|p-p^{\prime}\|-\kappa\|q-q^{\prime}\|}\Big{[}(1+\kappa)(\|p-p^{\prime}\|+\|q-q^{\prime}\|)$
$\displaystyle\ +\frac{(1+\|q-q^{\prime}\|)^{2}\|\delta
a\|\|a_{p,\,q}^{(2,l)}\|}{1-(1+\kappa)\|p-p^{\prime}\|-\kappa\|q-q^{\prime}\|-(1+\|q-q^{\prime}\|)\|a_{p,\,q}^{(2,l)}\|\|\delta
a\|}\Big{]}.$

This completes the proof. ∎

By using perturbation theorems for the generalized inverse
$a^{(2,l)}_{p,\,q}$, we can also investigate the perturbation analysis for the
generalized inverse $a^{(1,2)}_{p,\,q}$ under some conditions.

######  Corollary 4.9.

Let $a\in\mathscr{A}$ and $p,\,q\in\mathscr{A}^{\bullet}$ such that
$a^{(1,2)}_{p,\,q}$ exists. Suppose that $p^{\prime}\in\mathscr{A}^{\bullet}$
with $\|p-p^{\prime}\|<\dfrac{1}{(1+\kappa)^{2}}$ and $ap^{\prime}=a$. Then
$a_{p^{\prime},\,q}^{(1,2)}$ exists and

$\frac{\|a^{(1,2)}_{p^{\prime},q}-a^{(1,2)}_{p,\,q}\|}{\|a^{(2,l)}_{p,\,q}\|}\leq\dfrac{(1+\kappa)\|p^{\prime}-p\|}{1-(1+\kappa)\|p^{\prime}-p\|}\
\text{and}\
\|a^{(1,2)}_{p^{\prime},q}\|\leq\frac{\|a^{(1,2)}_{p,\,q}\|}{1-(1+\kappa)\|p^{\prime}-p\|}.$

###### Proof.

Set $b=a^{(1,2)}_{p,\,q}$. Then $bab=b,\ aba=a,\ ba=p,\ 1-ab=q$ and
$a^{(2,l)}_{p,\,q}=a^{(1,2)}_{p,\,q}=b$. By Theorem 4.4,
$a^{(2,l)}_{p^{\prime},\,q}$ is exists and

$\frac{\|a^{(2,l)}_{p^{\prime},q}-a^{(1,2)}_{p,\,q}\|}{\|a^{(2,l)}_{p,\,q}\|}\leq\dfrac{(1+\kappa)\|p^{\prime}-p\|}{1-(1+\kappa)\|p^{\prime}-p\|}\
\text{and}\
\|a^{(2,l)}_{p^{\prime},q}\|\leq\frac{\|a^{(1,2)}_{p,\,q}\|}{1-(1+\kappa)\|p^{\prime}-p\|}.$

We need only to show that
$a^{(1,2)}_{p^{\prime},\,q}=a^{(2,l)}_{p^{\prime},\,q}$ in this case. Put
$b^{\prime}=a^{(2,l)}_{p^{\prime},\,q}$. Then
$b^{\prime}ab^{\prime}=b^{\prime}$,
$(b^{\prime}a)\mathscr{A}=p^{\prime}\mathscr{A}$,
$(1-ab^{\prime})\mathscr{A}=q\mathscr{A}$. Thus, $(1-q)(1-ab^{\prime})=0$ and
hence $1-q=(1-q)ab^{\prime}=abab^{\prime}=ab^{\prime}$. Furthermore,
$ab^{\prime}a=(1-q)a=aba=a$. From
$(b^{\prime}a)\mathscr{A}=p^{\prime}\mathscr{A}$, we get that
$(1-b^{\prime}a)p^{\prime}=0$ and
$p^{\prime}=b^{\prime}ap^{\prime}=b^{\prime}a$. Therefore,
$b^{\prime}=a^{(1,2)}_{p^{\prime},\,q}$. ∎

We need the following easy representation lemma for $a^{(1,2)}_{p,\,q}$.

######  Lemma 4.10.

Let $a\in\mathscr{A}$ and $p,\,q\in\mathscr{A}^{\bullet}$ such that
$a^{(1,2)}_{p,\,q}$ exists. Let $w\in\mathscr{A}$ such that $wa=p$ and
$aw=1-q$. Then $a^{(1,2)}_{p,\,q}=(wa)^{\\#}w=w(aw)^{\\#}$.

###### Proof.

Obviously, $wa,\ aw\in\mathscr{A}^{g}$ for $wa=p$ and $aw=1-q$. We also have
$(wa)^{\\#}=p$ and $(aw)^{\\#}=1-q$. Then by using the uniqueness of
$a^{(1,2)}_{p,\,q}$, we can prove our lemma by simple computation. ∎

######  Corollary 4.11.

Let $a\in\mathscr{A}$ and $p,\,q\in\mathscr{A}^{\bullet}$ such that
$a^{(1,2)}_{p,\,q}$ exists. Suppose that $q^{\prime}\in\mathscr{A}^{\bullet}$
with $\|q-q^{\prime}\|<\dfrac{1}{2+\kappa}$ and $a=(1-q^{\prime})a$. Then
$a^{(1,2)}_{p,q^{\prime}}$ exists and

$\dfrac{\|(a^{(1,2)}_{p,\,q^{\prime}}-a^{(1,2)}_{p,\,q})\|}{\|a^{(1,2)}_{p,\,q}\|}\leq\dfrac{(1+\kappa)\|q-q^{\prime}\|}{1-\kappa\|q-q^{\prime}\|}$
and
$\|a^{(1,2)}_{p,\,q^{\prime}}\|\leq\dfrac{1+\|q^{\prime}-q\|}{1-\kappa\|q^{\prime}-q\|}\,\|a^{(1,2)}_{p,\,q}\|$.

If there are some $w,\,v\in\mathscr{A}$ with $wa=p=va$, $aw=1-q$ and
$av=1-q^{\prime}$. Then

$a_{p,\,q^{\prime}}^{(1,2)}=a^{(1,2)}_{p,\,q}+a^{(1,2)}_{p,\,q}(av)^{\\#}a(v-w)q.$

###### Proof.

$a^{(2,l)}_{p,\,q^{\prime}}$ exists by Theorem 4.6. From $a=(1-q^{\prime})a$
and Lemma 2.2, we can obtain that $a^{(1.2)}_{p,\,q^{\prime}}$ exists and
$a^{(2,l)}_{p,\,q^{\prime}}=a^{(1.2)}_{p,\,q^{\prime}}$.

Now the estimates in (1) and the representation for
$a^{(1,2)}_{p,\,q^{\prime}}$ in (2) follow from Theorem 4.6 and Lemma 4.10. ∎

Finally, by Corollary 4.9, Corollary 4.11 and Theorem 4.7, we have

######  Corollary 4.12.

Let $a\in\mathscr{A}$ and
$p,\,q,\,p^{\prime},\,q^{\prime}\in\mathscr{A}^{\bullet}$ with
$a^{(1,2)}_{p,\,q}$ exists. If $\|p-p^{\prime}\|<\dfrac{1}{(1+\kappa)^{2}}$,
$\|q-q^{\prime}\|<\dfrac{1}{3+\kappa}$ and $ap^{\prime}=a=(1-q^{\prime})a$.
Then $a_{p^{\prime},\,q^{\prime}}^{(1,2)}$ exists and

$\displaystyle\frac{\|a^{(1,2)}_{p^{\prime},\,q^{\prime}}-a^{(1,2)}_{p,\,q}\|}{\|a_{p,\,q}^{(1,2)}\|}$
$\displaystyle\leq\frac{(1+\kappa)(\|p-p^{\prime}\|+\|q-q^{\prime}\|)}{1-(1+\kappa)\|p-p^{\prime}\|-\kappa\|q-q^{\prime}\|}$
$\displaystyle\|a_{p^{\prime},\,q^{\prime}}^{(1,2)}\|$
$\displaystyle\leq\dfrac{(1+\|q-q^{\prime}\|)\|a_{p,\,q}^{(1,2)}\|}{1-(1+\kappa)\|p-p^{\prime}\|-\kappa\|q-q^{\prime}\|}.$

## References

  * [1] J. Cao, Y. Xue, The characterizations and representations for the generalized inverses with prescribed idempotents in Banach algebras, (Submitted).
  * [2] G. Chen, Y. Wei and Y. Xue, Perturbation analysis of the least square solution in Hilbert spaces, Linear Algebra Appl., 244 (1996), 69–80.
  * [3] G. Chen and Y. Xue, Perturbation analysis for the operator equation $Tx=b$ in Banach spaces, J. Math. Anal. Appl., 212 (1997), 107–125.
  * [4] G. Chen and Y. Xue. The expression of the generalized inverse of the perturbed operator under Type I perturbation in Hilbert spaces, Linear Algebra Appl., 285 (1998), 1–6.
  * [5] D. Cvetkovic-Ilić, X. Liu and J. Zhong, On the $(p,q)$-outer generalized inverse in Banach Algebras, Appl. Math. Comput., 209 (2) (2009), 191–196.
  * [6] D. S. Djordjević and Y. Wei, Outer generalized inverses in rings, Comm. Algebra, 33 (2005), 3051–3060.
  * [7] F. Du and Y. Xue, Perturbation analysis of $A^{(2)}_{T,S}$ on Banach spaces, Electronic J. Linear Algebra, 23 (2012), 586–598.
  * [8] F. Du and Y. Xue, Perturbation analysis of $A^{(2)}_{T,S}$ on Hilbert spaces, Funct. Anal. Approx. Comput., (2013).
  * [9] F. Du and Y. Xue, The perturbation of the group inverse under the stable perturbation in a unital ring, Filomat, 27 (2013), 61–70.
  * [10] T. Kato, Perturbation Theory for Linear Operators, Springer–Verlag, New York, 1984.
  * [11] V. Müller, Spectral Theory of Linear Operators and spectral systems in Banach algebras. Birkhäuser Verlag AG, 2nd Edition, (2007).
  * [12] M. Z. Nashed, Inner, outer and generalized inverses in Banach and Hilbert spaces. Numer. Funct. Anal. Optim., 9 (1987), 261–325.
  * [13] Y. Xue, Stable perturbation in Banach spaces, J. Aust. Math. Soc., 83 (2007), 1–14.
  * [14] Y. Xue, Stable Perturbations of Operators and Related Topics, World Scientific, 2012.
  * [15] Y. Xue and G. Chen, Some equivalent conditions of stable perturbation of operators in Hilbert spaces, Appl. Math. Comput., 147 (2004), 765–772.
  * [16] Q. Xu, Y. Wei and Y. Gu, Sharp norm–estimation for Moore–Penrose inverses of stable perturbations of Hilbert $C^{*}$–module operators, SIAM J. Numer. Anal., 47 (2010), 4735–4758.