# Large Language Models (LLMs) for Semantic Communication in Edge-based IoT
Networks

Alakesh Kalita,

###### Abstract

With the advent of Fifth Generation (5G) and Sixth Generation (6G)
communication technologies, as well as the Internet of Things (IoT), semantic
communication is gaining attention among researchers as current communication
technologies are approaching Shannon’s limit. On the other hand, Large
Language Models (LLMs) can understand and generate human-like text, based on
extensive training on diverse datasets with billions of parameters.
Considering the recent near-source computational technologies like Edge, in
this article, we give an overview of a framework along with its modules, where
LLMs can be used under the umbrella of semantic communication at the network
edge for efficient communication in IoT networks. Finally, we discuss a few
applications and analyze the challenges and opportunities to develop such
systems.

###### Index Terms:

Semantic Communication, Large Language Models (LLMs), Edge Computing, Internet
of Things (IoT)

##  I Introduction

The traditional communication system is mainly concerned with the accurate
transmission of the bits (by transforming into electromagnetic signal) from
one host to another host. Even though using modern encoding, decoding,
modulation, demodulation, etc., techniques bits are correctly sent to the
other end, due to the increasing number of Internet-connected devices in the
world, data generated by AR/VR devices, Internet of Things (IoT) devices,
Shannon’s capacity is already about to reach [1]. Therefore, nowadays
researchers are exploring semantic communications, which refers to the process
of exchanging information where the focus is on the meaning and interpretation
of the transmitted content, rather than just the accurate (i.e., any bit-level
error) transmission of raw data [8, 9]. In brief, traditional communication
focuses on syntax and the physical transmission of data. In contrast, semantic
communication ensures the exchanging of the inherent meaning of the
information [2, 3]. In Figure 1, we show how traditional communication is
different from semantic communication, where a person asks his smart device
(e.g., Alexa, Google Nest, Siri etc.,) to switch on his lights in a noisy
environment. In traditional communication, if the transmitted bits get
corrupted due to external noise, the receiver or the destination cannot
understand the user command. Hence, no action will be taken. However, in
semantic communication, the destination can try to understand/infer the user
command even if some of the bits get corrupted along the way and based on
retrieved meaning, action will be taken.

To transform traditional communication into semantic communication, recent
Artificial Intelligence (AI) technologies such as Deep Neural Networks (DNN),
and Recurrent Neural Networks (RNN) show exciting results in extracting and
restoring the semantic meaning of the transmitted information [7]. However,
deep generative-based Large Language Models (LLMs) are designed to understand
and generate human language. Recent LLMs, such as GPT-4, GPT-4o, and Llama 3.1
are trained on vast amounts of text data, enabling them to perform a wide
range of language tasks, including translation, summarization, and
conversation. Recently, OpenAI’s GPT series, including models like ChatGPT,
has garnered significant attention due to the advancements in LLMs which can
generate coherent and contextually relevant responses to users’ queries,
making them valuable tools in various applications like customer service,
content creation, and research. Therefore, LLMs can be a great tool for
semantic communications for both feature extractions and restorations [11].

With the development of digitization, IoT technology has become a prominent
topic in the field of networking and communication. IoT aims to connect
billions of physical devices which can be both living and non-living entities
found on the earth’s surface with sensing/actuation and communication
capacity. IoT technology enhances automation, improves efficiency, and
provides real-time insights across various sectors, including healthcare,
agriculture, transportation, and smart homes. By enabling seamless
connectivity and data exchange, IoT is transforming how we interact with the
world around us [6]. Now, the integration of semantic communication with LLMs
in IoT networks can make a huge transformation towards more intelligent,
efficient, and responsive IoT systems [10]. This convergence leverages the
natural language understanding capabilities of LLMs to enhance the semantic
communication framework in IoT, allowing devices to communicate in a more
meaningful and context-aware manner. For example, if a user says “Set up for
movie night”, semantic communication helps to set the Dim lights in the living
room, close the blinds, adjust the thermostat to a comfortable temperature,
turn on the TV and navigate to the user’s preferred streaming service. All can
be done with a single command, which requires multiple individual commands for
each actuation in the current IoT systems. Thus, LLM-based communication in
IoT can significantly improve efficiency, bandwidth utilization, and user
interactions.

Figure 1: Working of (a) Transitional communication and (b) semantic
communication. The latter ensures bit-wise (syntax) transmission, whereas
semantic communication ensures the meaning of the transmitted information and
works even if there are some bit-level errors in the receiver.

However, the devices used in IoT networks are resource-constrained in terms of
processing capacity, memory, and power. Therefore, using LLM in this kind of
device is not practically trivial even at the border router (or sometimes in
IoT gateways) of the IoT networks. Similarly, technologies like cloud
computing have their disadvantages like higher latency and cost. Therefore,
edge computing can be a suitable approach to use LLM under the umbrella of
semantic communication for IoT applications such as smart homes, smart
industries, etc. Edge computing processes data closer to the source of the
data generators rather than relying on centralized data centres at higher
propagation delay [5]. This approach reduces latency, conserves bandwidth, and
enhances real-time data processing capabilities. Hence, in the context of IoT,
edge computing plays a crucial role by enabling devices to analyze and respond
to data locally.

Therefore, in this article, we briefly discuss how LLMs can be used as a part
of semantic communication for establishing efficient communication in edge
computing-based IoT systems. In the next section, we provide basic background
information on semantic communication, LLMs, IoT, and edge computing. After
that, we discuss our proposed framework where LLMs can be used for semantic
communication in Edge-based IoT systems. Finally, before concluding our work,
we briefly discuss some of its applications, and challenges and opportunities
in such systems.

##  II Background

###  II-A Semantic Communication

Traditional communication systems focus on providing high data transmission
rates and minimizing symbol (bit) error rates. In such system, Shannon only
focused on the technical level, which deals with encoding the source message
into a bit sequence and recovering the same sequence at the receiver passing
through a noisy channel without considering the underlying meaning of the
message. In contrast, recently, a new paradigm has emerged in wireless
communications, which shifted the focus of communication towards the semantic
level, leading to the development of semantic communication. Semantic
communication targets the exchanging of semantically equivalent information
without necessarily requiring an identical match to the original transmitted
information. The core concept of semantic communications is to extract the
“meanings” or “features” from the transmitted information at the source and to
“interpret” this semantic information at the destination. However, semantic
communication requires conventional methods to encode relevant information. To
extract the meanings or features, semantic communication should have a
Knowledge Base (KB), which needs to be updated periodically based on the
information it receives from the user’s input to retrieve and interpret new
information as user inputs keep on changing. Overall, the semantic
communication system is complex, where both the source and destination need to
use the KB using advanced DNN or RNN techniques along with the functions of
traditional communication. Additionally, the semantic source and destination
can perceive their environment and operate autonomously.

###  II-B Large Language Models (LLMs)

LLMs are advanced AI-based systems designed to understand and generate human-
like text based on vast amounts of data. The training process of these kinds
of AI models requires adjusting millions or even billions of parameters
through a technique called back-propagation, and finally, optimizes the
models’ ability to understand and generate language. During training, LLMs are
exposed to large datasets containing diverse text sources, which allows them
to learn the statistical properties of language. After training, the model can
generate text by taking an initial prompt as input and predicting subsequent
words using diffusion techniques. This approach allows the model to
iteratively refine its predictions, maintaining coherence and context
throughout the generated sequence based on its learned parameters. Fine-tuning
specific tasks or domains can further enhance the model’s performance, making
it adept at specialized applications such as technical support, content
generation, or conversational agents. The ability to handle diverse linguistic
patterns and contexts makes LLMs a powerful tool for a wide range of natural
language processing tasks. Recent LLM architectures like GPT-4 and Llama 3.1,
can generate coherent and contextually relevant responses, making them
valuable tools for applications in natural language processing and human-
computer interaction.

###  II-C Edge Computing for LLMs

Edge computing is a distributed computing paradigm that brings computation and
data storage closer to the location where it is needed and where the data is
generated. Thus, the latency for processing sensory data and receiving a
response has significantly reduced compared to other distributed computing
approaches, such as cloud computing. In brief, edge computing improves
response times and also saves bandwidth by processing the data locally and not
sending the requests to the cloud. Thus, Edge computing can significantly
enhance the performance and utility of LLMs by bringing computation and data
storage closer to the data sources, thereby reducing latency and bandwidth
usage. By deploying LLMs on edge devices, it is possible to process data
locally rather than relying on centralized cloud servers. Additionally, edge
computing improves privacy and security by keeping sensitive data on local
devices rather than transmitting it over the Internet. Moreover, it enables
better scalability, as processing loads are distributed across multiple edge
nodes rather than concentrated in a central cloud, which can prevent
bottlenecks and reduce the risk of downtime of the central cloud server.

###  II-D Internet of Things (IoT)

IoT refers to a network of interconnected physical devices that communicate
and exchange data with each other and distributed computing systems such as
Edge and Cloud over the Internet. Apart from having sensors (to sense the
environment) and actuators (to act on the environment), IoT devices must have
a communication facility for connectivity. However, due to the resource-
constrained nature of IoT devices, heavy processing can not be done within
them. Therefore, IoT networks mainly collect, send, and receive data for a
wide range of applications to provide insights on data, automate processes,
and improve efficiency. The integration of IoT with edge computing further
enhances its capabilities by allowing data to be processed locally on nearby
edge devices, leading to faster decision-making and reduced dependency on
centralized cloud servers. The integration of IoT and edge computing is
crucial for applications requiring low latency, high reliability, and real-
time data processing.

Thus, semantic communication, LLMs, edge computing, and IoT networking can
work together to create intelligent, user-friendly, efficient, and quick
responsive systems. In such a system, IoT devices collect data from the
environment and transmit it to edge computing nodes for immediate processing.
After that, offline LLM in the edge device can analyze and interpret the
received data as a part of semantic communication to understand the context
and intent of the transmitted data. Hence, such an architecture can enable
real-time, context-aware responses and decisions, a user-friendly and
efficient IoT network eco-system to improve the already available IoT
applications further.

##  III A Framework for LLM-Based semantic Communication in IoT Networks

###  III-A System Model

Figure 2: A framework enabling LLM-based semantic communication in Edge-based
IoT system

IoT devices are often resource-constrained regarding processing capacity,
memory, and power. Consequently, running LLMs directly on these devices is not
technically feasible. To address this challenge, we propose a framework where
an IoT network’s border router (BR) or gateway is connected to an Edge
computing device equipped with sufficient resources to infer user requests
using LLMs. Note that, the gateway of an IoT network is the device that
connects IoT networks with different protocols such as ZigBee, Bluetooth, or
6TiSCH to the IPv6-based traditional Internet connectivity such as Ethernet,
WiFi, or 5G, facilitating basic data processing and decision-making, protocol
translation, and secure communication between IoT devices and the Internet.

In our proposed framework, LLMs are deployed at the edge of the network as
part of semantic communication, and an edge device is connected to the gateway
or BR of the IoT network. IoT devices attached to sensors collect information
from the environment and transmit their data to the gateway using multi-hop
(mesh topology) or single-hop (star topology) communication networks. The
gateway then forwards the data to the edge computing device for processing by
the LLM deployed at the edge. Note that some edge devices can also serve as
IoT gateways. Apart from the input provided by the IoT devices, users can also
provide their input directly to the LLMs in the form of text, video, or voice
commands. For example, in smart home environments, users can send voice
commands to perform different activities. The transmission of data from the
IoT devices to the gateway or edge follows traditional source and channel
encoding. Our framework assumes that feature restoration of semantic
communication is done at the receiver end, i.e., the edge.

In addition to sensory input and user input, the LLM model is fed with
historical data, as illustrated in Figure 2, for better inference. For
instance, when a user issues a voice command to set up a movie night, our
proposed framework can consider previous historical data from the user to
adjust the temperature. Thus, historical data can be used to produce better
inferences by the LLM. Apart from improving the performance of the LLM model
using historical data, we can increase the bandwidth utilization of the
overall IoT network. For this, we can consider a scenario where instead of
sending a high-resolution image from the door camera, we send only the edges
of the picture taken by the camera along with some textual information, such
as “a fair-colored man wearing a blue t-shirt with a tense face” [12]. The
receiver can generate a high-resolution and similar picture using the received
edges-based image and text. This approach requires significantly less
bandwidth to transmit a low-resolution picture and textual information
compared to sending a high-resolution picture. By reducing such data load in
the IoT network, these methods can conserve network resources and minimize
latency, leading to faster data transmission and processing times.
Additionally, it reduces the energy consumption of both the transmitting and
receiving devices along with the relay (i.e., intermediate forwarding) nodes,
which is crucial for maintaining the battery life of IoT devices and the
overall sustainability of IoT networks. Similarly, as mentioned before, users
need to send a single command instead of multiple different commands to
perform the same activities in LLM-based IoT networks. Based on the input
provided by the user or input collected from the IoT network, LLMs generate
commands for the actuators of the IoT network. This streamlined communication
improves system efficiency, enhances user experience by simplifying
interactions, and reduces the potential for command errors. Furthermore, the
ability to generate and execute complex commands through natural language
processing enhances the scalability of IoT systems, making them more adaptable
to varied applications and user needs.

To simplify our proposed framework to facilitate efficient and intelligent
interactions within IoT ecosystems, we divide it into multiple structured
modules that handle data collection, processing, prompt creation, response
handling, and actuator management. These modules are shown in Figure 3 and
discussed as follows,

IoT Data Collection Module: This module is responsible for gathering data from
various IoT devices including the command received from the users. This module
ensures real-time data collection from IoT devices and users.

Data Processing Module: Once the data is collected, it is processed to extract
relevant information and convert it into a suitable format for further
analysis. This module applies filtering, aggregation, and transformation
techniques to ensure the data is ready for the next stages.

Storage Module: This module is responsible for maintaining historical data
which can be fed to the LLM for referencing past information, providing
context and improving the accuracy of semantic communication.

Prompt Creation Module: This module generates prompts based on the processed
data from the data processing module and uses historical information stored in
the Storage Module. These prompts are generated to facilitate meaningful
queries and interactions with the LLM.

LLM Module: The core component of the framework, the LLM Module, utilizes
advanced language models to interpret prompts and generate coherent, context-
aware responses. This module leverages the capabilities of LLMs to understand
and process natural language queries effectively.

LLM Response Handling Module: This module processes the responses generated by
the LLM. It performs further processing to refine and contextualize the output
so that it can easily feed to the next module.

IoT Actuator Handling Module: The final module in the framework, the IoT
Actuator Handling Module, translates the processed responses into actionable
commands for IoT devices. It ensures that the system’s outputs are effectively
communicated within the IoT network, enabling intelligent and automated
control.

Note that these modules can be easily integrated to work with current IoT
infrastructures, enhancing their capabilities without designing the entire IoT
system from scratch. For example, an LLM can interface with existing smart
home assistants and IoT devices, using APIs and standard communication
protocols to understand and execute user commands.

Figure 3: Proposed framework is divided into multiple structured modules that
handle data collection, processing, prompt creation, response handling, and
actuator management.

###  III-B Advantages of using LLM for Semantic Communication in IoT Networks

Efficient Data Transmission: Semantic communication focuses on transmitting
the meaning or the semantics of the data rather than the raw data itself. This
can significantly reduce the amount of data that needs to be transmitted,
leading to more efficient use of bandwidth and lower energy consumption of the
senders, which is crucial for many IoT devices that operate on limited power
sources.

Enhanced Decision Making: Semantic communication can improve the quality of
decision-making processes in various IoT applications, such as smart cities,
industrial automation, and healthcare by focusing on the meaning of the data,
which can lead to more relevant and actionable information. Additionally,
semantic communication enables IoT systems to understand the context in which
data is generated and used. This context-awareness can lead to the development
of more intelligent and adaptive services that can respond dynamically to
changing conditions.

Quicker responsiveness: As LLM in semantic communication can extract
information in depth from minimal input size, therefore, it helps in quicker
decision-making as transmission timing, processing timing, and queuing timing
get reduced with the size of the data that has to be transmitted.

Enhanced User Experience Users can issue simple, natural language commands,
which the system interprets and executes accurately, thereby minimizing the
likelihood of errors. By considering historical data, the system can further
consider user preferences and adjust settings accordingly, leading to a more
personalized and intuitive experience. Thus, natural language interactions
powered by LLMs make it easier for users to communicate with IoT devices,
enhancing the overall user experience.

##  IV Use Cases

This section discusses some of the examples of how LLMs can be integrated into
Edge-based IoT systems as a part of semantic communication:

Smart Home Assistants: LLMs can make smart home assistants much more powerful
by enabling them to understand and respond to more natural and context-aware
interactions. For instance, with the help of an LLM, a smart assistant can
interpret and execute complex voice commands. If a user says, “Turn off the
lights in the living room after dinner,” the LLM can understand the context
and timing, and then control the lights accordingly. This means users can
speak more naturally and still have their commands accurately followed, making
the interaction with smart home devices smoother and more intuitive. LLMs can
also manage multiple IoT devices, like lights, thermostats, and security
systems, seamlessly integrating their functions based on user preferences and
context. Furthermore, using historical data enables the system to learn user
preferences and habits over time. For example, it can adjust temperature
settings based on past behaviour or suggest frequently used commands.
Additionally, historical data can be used to understand the context of the
command in a better way. For instance, if it knows that a user typically dims
the lights and plays soft music in the evening, it can proactively make these
adjustments without explicit commands. Overall the entire system can create a
more personalized and intuitive user experience.

Industrial IoT: In industrial IoT, IoT devices are deployed to continuously
monitor machinery and equipment. Sensor-attached IoT devices collect vast
amounts of data, including temperature readings, vibration levels, and
operational cycles, etc. By understanding the intricate patterns and contexts
within the data, LLMs can identify possible issues that may not be determined
by the existing system and accordingly, LLMs can suggest detailed maintenance
schedules and recommendations. Such preemptive replacement recommendations
before the actual failure and real-time alerts can significantly improve the
efficiency of the Industrial ecosystems by preventing unexpected downtime. The
efficiency of industries can be further improved by allowing LLMs to optimize
the overall maintenance strategy by learning from historical data. They can
identify trends such as recurring failures or seasonal variations in equipment
performance, allowing for more strategic planning. Furthermore, LLM-generated
responses can simplify complex information, making it easy for non-technical
individuals to understand and act on maintenance needs. For example, instead
of presenting raw data, the LLM can provide clear, straightforward
instructions like, “Machine A1025 needs a check-up due to unusual vibrations.
Schedule a maintenance session this week.” This helps non-technical staff also
to understand the situation and take appropriate action.

Smart Healthcare: LLMs can be seamlessly integrated with IoT devices in
healthcare to enhance real-time patient monitoring and support. For instance,
wearable sensors can continuously gather vital patient data such as heart
rate, body temperature, and activity levels. An LLM can then analyze this data
to detect any irregularities, predict potential health issues, and provide
tailored health advice. If a patient’s heart rate becomes irregular, the
system can immediately alert healthcare providers, offering insights into
possible causes and suggesting appropriate actions. This proactive approach
ensures timely medical intervention, potentially preventing more serious
health complications.

Moreover, LLM-based semantic communication can significantly improve
coordination and efficiency in medical environments such as surgical
operations. For example, in a scenario where a doctor performs a complex
surgery and uses voice commands to interact with various IoT-enabled medical
devices and communicate with the surgical team. For example, the doctor might
say, “Increase the oxygen level by 10%” and the LLM would interpret this
command, adjust the settings on the relevant device, and confirm the change.
This allows the doctor to focus on the surgery without manually adjusting
equipment.

These examples illustrate how LLM-based semantic communication can add
significant value to IoT systems by providing advanced data analysis, context-
aware interactions, and predictive capabilities.

##  V Open Challenges and Issues

Even though LLMs under the umbrella of semantic communication in Edge-based
IoT systems can revolutionize the entire communication domain, there are a
variety of challenges to be tackled before they can be used in real
applications. This section discusses the two major open issues for future
investigations.

###  V-A Privacy and Security

One of the significant challenges when integrating LLMs with semantic
communication, edge computing, and IoT systems is ensuring their dependability
and reliability. One major concern is safeguarding these models against
malicious attacks, as attacks can lead to unexpected behaviours and unreliable
outputs. Moreover, privacy issues are prominent, as LLMs can be susceptible to
revealing sensitive information embedded within the training data.
Furthermore, processing sensitive data locally on edge devices can help
enhance privacy and security, but it also poses challenges. Protecting data
from breaches is tough, especially in large-scale deployments across numerous
edge devices. Robust encryption, secure data handling practices, and regular
security updates are essential but can be resource-intensive. Hence, ensuring
these security measures is vital for the secure and reliable operation of
semantic communication systems that leverage LLMs in edge computing-based and
IoT networks. Addressing these challenges requires a combination of advanced
technical solutions, robust security measures, and ongoing optimization
efforts to make the integration of LLMs, semantic communication, and edge-
based IoT systems both practical and effective.

###  V-B Data Heterogeneity and Quality

IoT systems generate vast and diverse datasets from numerous interconnected
devices, each producing data in various formats and frequencies, which
presents significant challenges for LLMs. Data heterogeneity involves
integrating structured data from databases, unstructured data from text or
logs, time-series data from sensors, and multimedia data like images,
requiring techniques such as multi-modal learning, data fusion, and semantic
mapping to unify and interpret these varied inputs effectively. Meanwhile,
ensuring data quality is crucial, as issues like noise, missing values, and
inconsistencies can severely impact model performance. Addressing the issues
with data quality involves pre-processing methods such as data cleaning,
normalization, feature engineering, and anomaly detection, alongside robust
integration strategies like data aggregation and continuous monitoring. These
approaches are essential for maintaining high-quality, reliable LLM outputs
out of unstructured and heterogeneous IoT data.

##  VI Conclusion

In this article, we discussed how semantic communication can leverage LLMs to
enhance the efficiency of edge-based IoT systems. We proposed a framework for
designing such IoT systems and briefly covered each module involved in the
proposed framework. We also examined the benefits of using LLMs for semantic
communication in IoT networks, along with several practical use cases.
Finally, we addressed some open challenges that need to be resolved before
implementing such IoT systems.

## References

  * [1] C. E. Shannon and W. Weaver, _The Mathematical Theory of Communication_. University of Illinois Press, 1949.
  * [2] X. Luo, H. Chen, and Q. Guo, “Semantic communications: Overview, open issues, and future research directions,” _IEEE Wireless Communications_ , vol. 29, no. 1, pp. 210–219, 2022.
  * [3] E. Grassucci, J. Park, S. Barbarossa, S.-L. Kim, J. Choi, and D. Comminiello, “Generative AI meets semantic communication: Evolution and revolution of communication tasks,” arXiv:2401.06803, 2024. [Online]. Available: https://arxiv.org/abs/2401.06803
  * [4] J. Park, S. Samarakoon, A. Elgabli, J. Kim, M. Bennis, S.-L. Kim, and M. Debbah, “Communication-efficient and distributed learning over wireless networks: Principles and applications,” _Proceedings of the IEEE_ , vol. 109, no. 5, pp. 796–819, 2021.
  * [5] A. Hazra, A. Kalita, and M. Gurusamy, “Meeting the requirements of internet of things: The promise of edge computing,” _IEEE Internet of Things Journal_ , vol. 11, no. 5, pp. 7474–7498, 2024.
  * [6] A. Kalita and M. Khatua, “6TiSCH – IPv6 enabled open stack IoT network formation: A review,” _ACM Trans. Internet Things_ , vol. 3, no. 3, pp. 24:1–24:36, Jul. 2022.
  * [7] H. Xie, Z. Qin, G. Y. Li, and B.-H. Juang, “Deep learning enabled semantic communication systems,” _IEEE Transactions on Signal Processing_ , vol. 69, pp. 2663–2675, 2021.
  * [8] J. Dai, P. Zhang, K. Niu, S. Wang, Z. Si, and X. Qin, “Communication beyond transmitting bits: Semantics-guided source and channel coding,” _IEEE Wireless Communications_ , vol. 30, no. 4, pp. 170–177, 2023.
  * [9] D. Gündüz, Z. Qin, I. E. Aguerri, H. S. Dhillon, Z. Yang, A. Yener, K. K. Wong, and C.-B. Chae, “Beyond transmitting bits: Context, semantics, and task-oriented communications,” _IEEE Journal on Selected Areas in Communications_ , vol. 41, no. 1, pp. 5–41, 2023.
  * [10] Z. Lin, G. Qu, Q. Chen, X. Chen, Z. Chen, and K. Huang, “Pushing large language models to the 6G edge: Vision, challenges, and opportunities,” arXiv:2309.16739, 2024. [Online]. Available: https://arxiv.org/abs/2309.16739
  * [11] S. Barbarossa, D. Comminiello, E. Grassucci, F. Pezone, S. Sardellitti, and P. Di Lorenzo, “Semantic communications based on adaptive generative models and information bottleneck,” _IEEE Communications Magazine_ , vol. 61, no. 11, pp. 36–41, 2023.
  * [12] H. Nam, J. Park, J. Choi, M. Bennis, and S.-L. Kim, “Language-oriented communication with semantic coding and knowledge distillation for text-to-image generation,” arXiv:2309.11127, 2023. [Online]. Available: https://arxiv.org/abs/2309.11127

## References

  * [1] C. E. Shannon and W. Weaver, _The Mathematical Theory of Communication_. University of Illinois Press, 1949.
  * [2] X. Luo, H. Chen, and Q. Guo, “Semantic communications: Overview, open issues, and future research directions,” _IEEE Wireless Communications_ , vol. 29, no. 1, pp. 210–219, 2022.
  * [3] E. Grassucci, J. Park, S. Barbarossa, S.-L. Kim, J. Choi, and D. Comminiello, “Generative AI meets semantic communication: Evolution and revolution of communication tasks,” arXiv:2401.06803, 2024. [Online]. Available: https://arxiv.org/abs/2401.06803
  * [4] J. Park, S. Samarakoon, A. Elgabli, J. Kim, M. Bennis, S.-L. Kim, and M. Debbah, “Communication-efficient and distributed learning over wireless networks: Principles and applications,” _Proceedings of the IEEE_ , vol. 109, no. 5, pp. 796–819, 2021.
  * [5] A. Hazra, A. Kalita, and M. Gurusamy, “Meeting the requirements of internet of things: The promise of edge computing,” _IEEE Internet of Things Journal_ , vol. 11, no. 5, pp. 7474–7498, 2024.
  * [6] A. Kalita and M. Khatua, “6TiSCH – IPv6 enabled open stack IoT network formation: A review,” _ACM Trans. Internet Things_ , vol. 3, no. 3, pp. 24:1–24:36, Jul. 2022.
  * [7] H. Xie, Z. Qin, G. Y. Li, and B.-H. Juang, “Deep learning enabled semantic communication systems,” _IEEE Transactions on Signal Processing_ , vol. 69, pp. 2663–2675, 2021.
  * [8] J. Dai, P. Zhang, K. Niu, S. Wang, Z. Si, and X. Qin, “Communication beyond transmitting bits: Semantics-guided source and channel coding,” _IEEE Wireless Communications_ , vol. 30, no. 4, pp. 170–177, 2023.
  * [9] D. Gündüz, Z. Qin, I. E. Aguerri, H. S. Dhillon, Z. Yang, A. Yener, K. K. Wong, and C.-B. Chae, “Beyond transmitting bits: Context, semantics, and task-oriented communications,” _IEEE Journal on Selected Areas in Communications_ , vol. 41, no. 1, pp. 5–41, 2023.
  * [10] Z. Lin, G. Qu, Q. Chen, X. Chen, Z. Chen, and K. Huang, “Pushing large language models to the 6G edge: Vision, challenges, and opportunities,” arXiv:2309.16739, 2024. [Online]. Available: https://arxiv.org/abs/2309.16739
  * [11] S. Barbarossa, D. Comminiello, E. Grassucci, F. Pezone, S. Sardellitti, and P. Di Lorenzo, “Semantic communications based on adaptive generative models and information bottleneck,” _IEEE Communications Magazine_ , vol. 61, no. 11, pp. 36–41, 2023.
  * [12] H. Nam, J. Park, J. Choi, M. Bennis, and S.-L. Kim, “Language-oriented communication with semantic coding and knowledge distillation for text-to-image generation,” arXiv:2309.11127, 2023. [Online]. Available: https://arxiv.org/abs/2309.11127


# MotePy: A domain specific language for low-overhead machine learning and
data processing.

Jayaraj Poroor
JIFFY.ai
<EMAIL_ADDRESS>* This work was supported by a grant from Microsoft
Research, India.

###### Abstract

A domain specific language (DSL), named MotePy is presented. The DSL offers a
high level syntax with low overheads for ML/data processing in time
constrained or memory constrained systems. The DSL-to-C compiler has a novel
static memory allocator that tracks object lifetimes and reuses the static
memory, which we call the compiler-managed heap.

##  I INTRODUCTION

When writing code for machine learning or data processing applications we want
the convenience of a high level syntax of a language like Python [1].
Typically this convenience comes with hidden overheads like interpreted
execution (e.g., in CPython [2]) or dynamic memory allocation and garbage
collection. In many situations these overheads do not pose a concern for the
application developer.

However, in certain situations when we have tight time constraints [3] or
memory constraints [4], we cannot compromise on the need to have very low
overheads. Examples include machine learning pipelines in an OS kernel [5] or
in small embedded devices [6].

The proposed DSL, MotePy offers a high level Python-like syntax for specifying
modular data processing pipelines and performing vector/matrix operations.

##  II SYNTAX

Machine learning and data processing applications typically have a pipeline-
like execution structure. The main module of a MotePy application is a
pipeline specification of the form given in the example below (Figure 1). The
pipeline specification is a Python-like list of operations. In the example,
the first operation is acquire function defined in the module datasource and
the second operation is predict function defined in model module. MotePy’s
runtime executes the pipeline in a loop, starting from the first operation, in
sequence till the last operation. Once the last operation is complete the
pipeline is re-executed from the beginning, in a loop fashion.

pipeline.py

⬇

[

datasource.acquire,

model.predict

]

Figure 1: A sample pipeline specification

Figure 2 shows the datasource module and Figure 3 shows the model module used
in the sample pipeline in Figure 1.

datasource.py

⬇

data: float[60]

def init() -> void:

pass

@flow

def acquire() -> void:

for i in range(0, 60):

data[i] = float(rand())/RAND_MAX

printf("Next\n")

next(data)

Figure 2: The datasource module used in the pipeline specification in Figure 1

model.py

⬇

N : const int32_t = 60

M : const int32_t = 30

layer0 : float[N][M]

layer1 : float[M]

def random_weights() -> void:

for j in range(0, M):

for i in range(0, N):

layer0[i][j] =

float(rand())/RAND_MAX

layer1[j] = float(rand())/RAND_MAX

def init() -> void:

random_weights()

@flow

def predict(features: float[N]) -> void:

res: float

tmp : float[30]

printf("Prediction\n")

tmp = features * layer0

res = tmp * layer1

#res = exp(res)

printf("Result␣%f\n", res)

Figure 3: The model module used in the pipeline specification in Figure 1

Though the syntax is close to Python, the language differs from Python in its
implementation in significant ways:

  * •

MotePy is statically typed. Its type annotation syntax is based on Python’s
optional type annotation syntax, but uses MotePy’s own notation similar to
that of Java for array typing.The reason for own notation for array typing is
that type annotations are not common in Python and Java’s array notation is
familiar to most programmers.

  * •

MotePy supports only static memory allocation. Only primitive types are
allocated on the stack. Rest are allocated in the static data area. However,
unlike other languages such as C or C++ the MotePy compiler tracks lifetimes
of array objects and reuses memory. We call this the compiler-managed heap.

Every MotePy module has an init function and a special flow function. The flow
function is the operation invoked as part of the pipeline execution and is
decorated with @flow syntax. Apart from these two functions a module may
define any number of functions to perform its function.

In Figure 2 the data array is randomly initialized and then passed onto the
next stage in the pipeline by invoking a special next function. The next
function enables us to specify at what point in the execution the next stage
in pipeline must be executed. If a flow function returns without invoking the
next function the pipeline is re-executed from the beginning. For instance, if
the data acquisition is not complete, then the acquisition stage may return
without calling next. The MotePy runtime will automatically re-execute the
pipeline.

Figure 3 shows a neural-network-like processing stage. The network has two
layers of weights that are randomly initialized in the example. In an actual
system the weights will be loaded from a file. The code shows high level
syntax for vector/matrix operations, which are translated by the MotePy
compiler into low-level loop operations.

The code also shows invoking the C function printf. MotePy allows us to invoke
C/C++ functions from MotePy code without any need for a special foreign
function calling interface. The MotePy array/matrix layout is same as that of
C/C++ enabling us to pass arrays/matrices back and forth between C/C++ code
without any transformation overheads.

The MotePy compiler translates the high level DSL code to low level C/C++
code, which is then compiled using a platform C/C++ compiler such as GCC to
get the executable.

##  III COMPILER-MANAGED HEAP

Figure 4 shows a two-stage MotePy pipeline. Stage 1 initializes vec1, does
some computations on it and then invokes next for Stage 2 processing. Stage 2
initializes vec2, does some computations on it, then initializes vec3 and does
some computations on it.

Compiling and running the program gives the following output:

Address of vec1: 94473766309984
Address of vec2: 94473766309984
Address of vec3: 94473766309984

The output shows that the addresses of vec1, vec2, and vec3 are same111The
generated executable has an optional parameter for specifying the number of
pipeline iterations. In this case a value of ‘1’ was passed. If none is
passed, the pipeline is executed in an infinite loop.. The reason is that the
compiler allocates static memory for the vectors and reuses the memory based
on the variable lifetimes. It may be noted that owing to Address Space Layout
Randomization[7] the printed addresses will be different during each program
execution.

In the case of this example, the life time of vec1 is in the program region
from line numbers 8-11 in stage1.py. Life time of vec2 is from line numbers 8
- 11 in stage2.py and that of vec3 is from line numbers 12 - 15 in stage2.py.
Since the lifetimes are non-overlapping the compiler intelligently reuses the
static memory area for the three vectors.

pipeline.py

⬇

[

stage1.process,

stage2.process

]

stage1.py

⬇

1. vec1: float[60]

2.

3. def init() -> void:

4. #vec1 = 0

5. pass

6.

7. def process() -> void:

8. vec1 = 0 #initialization

9. vec1 = vec1 + vec1 #some processing

10. printf("Address␣of␣vec1:␣%ld\n",

11. uint64_t(vec1))

12. next()

stage2.py

⬇

1. vec2: float[60]

2. vec3: float[60]

3.

4. def init() -> void:

5. pass

6.

7. def process() -> void:

8. vec2 = 0 #initialization

9. vec2 = vec2 + vec2

10. printf("Address␣of␣vec2:␣%ld\n",

11. uint64_t(vec2))

12. vec3 = 0

13. vec3 = vec3 + vec3

14. printf("Address␣of␣vec3:␣%ld\n",

15. uint64_t(vec3))

Figure 4: A two-stage MotePy pipeline

Figure 5 shows a modified version of the Stage 1 of the pipeline. Here the
vec1 is initialized in the init function. The process function computes a new
value of vec1 based on the current value. Compiling and running the pipeline
with this modified Stage 1 gives the following result:

Address of vec1: 94035385520224
Address of vec2: 94035385520464
Address of vec3: 94035385520464

The output shows that the compiler allocates a separate non-overlapping memory
area for vec1 and uses the same memory area for vec2 and vec3. The compiler’s
allocator now knows that vec1’s lifetime overlaps with that of vec2 and vec3
and hence allocates separate memory for vec1.

stage1.py

⬇

1. vec1: float[60]

2.

3. def init() -> void:

4. vec1 = 0

5. #pass

6.

7. def process() -> void:

8. #vec1 = 0 #initialization

9. vec1 = vec1 + vec1 #some processing

10. printf("Address␣of␣vec1:␣%ld\n",

11. uint64_t(vec1))

12. next()

Figure 5: A modified version of Stage 1

MotePy’s static allocator allocates along the 2-dimensional space with address
as one axis and variable lifetime as the other axis. In comparison, the static
allocators of languages like C/C++ simply allocates along the address line.

The objective of the MotePy allocator is to minimize the allocated space
subject to the constraint that no two blocks overlap in this 2-dimensional
space. The current implementation uses a greedy approach, which may yield sub-
optimal results. More work on the allocator is planned in the future.

MotePy compiler code, along with sample applications is released in open
source and is available at https://github.com/jayarajporoor/motepy.

##  IV ACKNOWLEDGEMENTS

I would like to thank Microsoft Research, India for the grant support. I am
especially thankful to Dr. Sriram Rajamani, Satish Sangameswaran, and Dr.
Harsha Vardhan Simhadri for the support during the course of this work. I
thank Nandagopal, Sandesh Ghanta, and Surya Chaitanya who worked with me as
student interns during the initial development of this work.

## References

  * [1] Rossum, Guido. ”Python reference manual.” (1995).
  * [2] Roghult, Alexander. ”Benchmarking Python Interpreters: Measuring Performance of CPython, Cython, Jython and PyPy.” (2016).
  * [3] Liu, Fan, Ajit Narayanan, and Quan Bai. ”Real-time systems.” (2000).
  * [4] Lee, Edward A. ”Cyber-physical systems-are computing foundations adequate.” Position paper for NSF workshop on cyber-physical systems: research motivation, techniques and roadmap. Vol. 2. Citeseer, 2006.
  * [5] Negi, Atul, and P. Kishore Kumar. ”Applying machine learning techniques to improve linux process scheduling.” TENCON 2005-2005 IEEE Region 10 Conference. IEEE, 2005.
  * [6] Dennis, Don Kurian, et al. ”EdgeML Machine LEARNING for Resource-Constrained Edge Devices.” URL https://github. com/Microsoft/EdgeML. Retrieved January (2020).
  * [7] Jang, Yeongjin, Sangho Lee, and Taesoo Kim. ”Breaking kernel address space layout randomization with intel tsx.” Proceedings of the 2016 ACM SIGSAC Conference on Computer and Communications Security. 2016.


[mycorrespondingauthor]* Corresponding author

# Additive and multiplicative properties of Drazin inverse under a new weakly
commutativity condition

Rounak Biswas<EMAIL_ADDRESS>Falguni Roy<EMAIL_ADDRESS>National
Institute of Technology Karnataka, India

###### Abstract

Given a complex Banach space $X$, let $\mathcal{B}(X)$ be the collection of
all bounded linear operators on $X.$ For $A,B\in\mathcal{B}(X)$ we define
$A,B$ are $A$-weakly commutative if there exists $C\in\mathcal{B}(X)$
satisfying

$AB=CA\text{ and }BA=AC.$

The objective of this paper is to study the Drazin invertibility of $A+B$ and
$AB$, when $A,B\in\mathcal{B}(X)^{D}$ are $A\text{ and }B$-weakly commutative.
Consequently, this extends some of the additive and multiplicative results
established by Huanyin and Marjan Sheibani (Linear and Multilinear Algebra
70.1 (2022): 53-65) for a distinct family of elements. Moreover, we also
establish these additive and multiplicative properties for g-Drazin inverses
in a complex Banach algebra.

##  1 Introduction

Throughout this paper we consider $X$ as a complex Banach space and
$\mathcal{B}(X)$ represents the collection of all bounded linear operators on
$X.$ For $A\in\mathcal{B}(X),$ the notations $\mathcal{N}(A),\text{
}\mathcal{R}(A)$ and $\sigma(A)$ denotes the null space, range space, and
spectrum of the operator $A$, respectively. The set of all invertible
operators in $\mathcal{B}(X)$ will be denoted by $\mathcal{B}(X)^{inv}$, and
$I$ represents the identity operator on $X$. An operator $A\in\mathcal{B}(X)$
is Drazin invertible if there exists an operator $A^{D}\in\mathcal{B}(X)$ such
that

$AA^{D}=A^{D}A,\text{ }A^{D}AA^{D}=A^{D}\text{ and
}A(I-AA^{D})\in\mathcal{B}(X)^{nil},$ (1)

where $\mathcal{B}(X)^{nil}$ represent the collection of all nilpotent
operators in $\mathcal{B}(X)$. The operator $A^{D}$ satisfying (1) is the
Drazin inverse of $A$, and the least non-negative integer $k$ satisfying
$\left(A(I-AA^{D})\right)^{k}=0$ is the index of $A,$ denoted by $i(A).$
$\mathcal{B}(X)^{D}$ stands for the collection of Drazin invertible operators
in $\mathcal{B}(X)$ and for $A\in\mathcal{B}(X)^{D},$ $A^{\pi}$ denote the
spectral idemoptent $I-AA^{D}.$ If $A\in\mathcal{B}(X)$ is a Drazin invertible
operator with index $k$, then the Banach space $X$ has the decomposition
(wang2018generalized, , Theorem 12.1.2)

$X=\mathcal{R}(A^{k})\oplus\mathcal{N}(A^{k})=\mathcal{R}(AA^{D})\oplus\mathcal{N}(AA^{D}),$
(2)

and with respect to this decomposition of $X,$ $A$ have the block
formdeng2008drazin

$A=A_{1}\oplus A_{2},$

where $A_{1}$ is invertible operator on $\mathcal{R}(AA^{D})$ and $A_{2}$ is
nilpotent operator on $\mathcal{N}(AA^{D})$. A generalization of the Drazin
inverse was introduced by Koliha koliha1996generalized , known as the
generalized Drazin inverse(g-Drazin inverse). From now onwards $\mathcal{A}$
will represent an unital complex Banach algebra. An element $a\in\mathcal{A}$,
is said to be g-Drazin invertible if there exists an element
$a^{d}\in\mathcal{A}$ such that

$aa^{d}=a^{d}a,\text{ }a^{d}aa^{d}=a^{d}\text{ and
}a-a^{2}a^{d}\in\mathcal{A}^{qnil},$

where $\mathcal{A}^{qnil}$ represent the collection of all quasinilpotent
elements in $\mathcal{A}$. Here $a^{d}$ is the g-Drazin inverse of $a$, and
the collection of all g-Drazin invertible elements in $\mathcal{A}$ will be
denoted by $\mathcal{A}^{d}.$ A fascinating problem in Drazin inverse theory
revolves around examining the Drazin invertibility for the sum and product of
two Drazin invertible elements. The investigation of the Drazin inverse’s
additive and multiplicative properties is significantly influenced by the
commutative relationship between two elements. In the literature, several
results pertaining to this aspect can be found in deng2008drazin ; deng2010new
; wei2011note . Instead of taking commutativity, if we consider the weaker
relation $a^{2}b=ba^{2}$ for $a,b\in\mathcal{A}^{D}(\text{or
}\mathcal{A}^{d})$, then, in general, without any additional condition on
$a,b$, it becomes challenging to deduce any conclusions about the Drazin(or
g-Drazin) invertibility of $ab$ and $a+b$. Chen and Sheibani chen2022g
provided several equivalent conditions for the g-Drazin invertibility of
$a+b$, assuming that $\lambda a^{2}b=\lambda^{\prime}ba^{2}=aba$ and $\mu
ab^{2}=\mu^{\prime}b^{2}a=bab,$ where
$\lambda,\lambda^{\prime},\mu,\mu^{\prime}$ are non zero complex numbers.
Recently, Qin and Lu qin2023drazin  proved these results under weaker
conditions $a^{2}b=aba$ and $b^{2}a=bab$. In this context, motivated by weakly
commutativity relationpondvelivcek1975weakly , we introduce a new relation
between two elements of a Banach algebra. This relation is defined to generate
a distinct family of elements compared to the one used by Chen and
Sheibanichen2022g .

###### Definition 1.1

Two elements $a,b\in\mathcal{A}$ are said to be $a$-weakly commutative if
there exists $c\in\mathcal{A}$ satisfying $ab=ca\text{ and }ba=ac.$

Note that commutativity is a particular case of Definition 1.1, when $c=b$.

###### Remark 1.2

According to Definition 1.1, if $a,b$ are $a$-weakly commutative, then $a,c$
are also $a$-weakly commutative.

###### Definition 1.3

For $a,b\in\mathcal{A},$ if $a,b$ are $a$-weakly commutative as well as
$b$-weakly commutative also then we say $a,b$ are $\\{a,b\\}$-weakly
commutative.

###### Lemma 1.4

If two elements $a,b$ in a Banach algebra $\mathcal{A}$ are $a$-weakly
commutative then $a^{2}b=ba^{2}.$

######  Proof 1.5.

Since $a,b$ are $a$-weakly commutative therefore there exists an element
$c\in\mathcal{A}$, such that $ab=ca\text{ and }ba=ac$. Hence

$a^{2}b=aca=ba^{2}.$

######  Lemma 1.6.

If $a,b\in\mathcal{A}$ are $a$-weakly commutative and $a$ is idempotent, then
$a,b$ commutes with each other.

######  Proof 1.7.

Since $a$ is idempotent, therefore, by Lemma 1.4, we obtain $ab=ba$ as
desired.

######  Remark 1.8.

If $a\in\mathcal{A}$ then by Lemma 1.4, we have

$\displaystyle\\{b:b\in\mathcal{A}\text{, }a,b\text{ are }a\text{-weakly
commutative}\\}\subset\\{b:b\in\mathcal{A}\text{ and }a^{2}b=ba^{2}\\},$

but for any non zero complex numbers $\lambda,\lambda^{\prime}$

$\\{b:b\in\mathcal{A}\text{, }a,b\text{ are }a\text{-weakly
commutative}\\}\not\subset\\{b:b\in\mathcal{A}\text{ and }\lambda
a^{2}b=\lambda^{\prime}ba^{2}=aba\\}.$

This becomes evident in the subsequent example.

######  Example 1.9.

Let $a=\begin{bmatrix}0&1\\\ 1&0\\\ \end{bmatrix}$ and
$b=\begin{bmatrix}0&-1\\\ 0&0\\\
\end{bmatrix}\in\mathcal{A}=M_{2}(\mathbb{C}).$ Then $a,b$ are $a$-weakly
commutative but $a^{2}b=ba^{2}\neq aba.$

######  Proof 1.10.

Let $c=\begin{bmatrix}0&0\\\ -1&0\\\ \end{bmatrix}$ then $ab=ca\text{ and
}ba=ac$ therefore $a,b$ are $a$-weakly commutative. Moreover
$a^{2}b=ba^{2}=\begin{bmatrix}0&-1\\\ 0&0\\\ \end{bmatrix}$ but
$aba=\begin{bmatrix}0&0\\\ -1&0\\\ \end{bmatrix}.$

######  Remark 1.11.

For $a,b\in\mathcal{A}$, $a$-weakly commutativity of $a,b$ does not implies
$b$-weakly commutativity of $a,b$. In Example 1.9 $a,b$ are $a$-weakly
commutative, but they are not $b$-weakly commutative.

The aim of this paper is to provide several additive and multiplicative
results for Drazin and g-Drazin inverse under this newly defined relation.

The remainder of this paper is presented in the following manner. In Section
2, we establish a relation between the Drazin invertibility of $A+B$,
$AA^{D}(A+B)BB^{D}$, $AA^{D}(A+B)$, $(A+B)BB^{D}$ and $A^{2}A^{D}(I+A^{D}B)$
when $A,B\in\mathcal{B}(X)^{D}$ are $\\{A,B\\}$-weakly commutative.
Furthermore, we provide a representation for $(A+B)^{D}$ in all the cases
mentioned above. In Section 3, for $A,B\in\mathcal{B}(X)^{D}$ we investigate
the Drazin invertibility of $AB$ when $A,B$ are $\\{A,B\\}$-weakly
commutative. As an application, we establish some new additive and
multiplicative results in Section 4, for some particular types of operators,
using the results found in section 2 and section 3. Finally, in Section 5, we
extend some of the results presented in section 2 and section 3 for g-Drazin
invertibility within a complex Banach algebra.

##  2 Additive properties

The aim of this section is to establish several results related to the
addition of two Drazin invertible operators. We begin with the following
lemma.

######  Lemma 2.1.

If $A,B\in\mathcal{B}(X)^{nil}$, such as $AB=0$, then $A+B$ is nilpotent.

The following result is proven for Drazin inverse in hartwig1977group , and
the case of the g-Drazin inverse is established in gonzalez2004new .

######  Theorem 2.2.

Let $A\in\mathcal{B}(X)$ have the block form representation

$A=\begin{bmatrix}A_{1}&0\\\ A_{3}&A_{2}\end{bmatrix}.$

  1. (i)

If $A_{1},A_{2}$ are Drazin invertible then $A$ is Drazin invertible and

$A^{D}=\begin{bmatrix}A_{1}^{D}&0\\\ C&A_{2}^{D}\\\ \end{bmatrix},$

where

$C=\sum_{n=0}^{i(A_{1})-1}(A_{2}^{D})^{n+2}A_{3}A_{1}^{n}(I-A_{1}A_{1}^{D})+\sum_{n=0}^{i(A_{2})-1}(I-A_{2}A_{2}^{D})A_{2}^{n}A_{3}(A_{1}^{D})^{n+2}-A_{2}^{D}A_{3}A_{1}^{D}.$

  2. (ii)

If $A\text{ and }A_{1}(\text{or }A_{2})$ are Drazin invertible then
$A_{2}(\text{or }A_{1})$ is also Drazin invertible.

Now, we present the following beneficial outcome.

######  Lemma 2.3.

If $A,\text{}B\in\mathcal{B}(X)$ are $A$-weakly commutative operators and
$A\in\mathcal{B}(X)^{nil}$ then $AB\in\mathcal{B}(X)^{nil}$. Moreover, if
$A,B$ are $B$-weakly commutative and $B\in\mathcal{B}(X)^{nil}$, then
$A+B\in\mathcal{B}(X)^{nil}$.

######  Proof 2.4.

Since $A,B$ are $A$-weakly commutative operators then there exists
$C_{1}\in\mathcal{B}(X)$ satisfying $AB=C_{1}A\text{ and }BA=AC_{1}$. Hence
for $k\in\mathbb{N}$ we have

$\displaystyle(AB)^{k}$ $\displaystyle=\begin{cases}A^{k}BC_{1}B\cdots
B\text{, if $k$ is odd,}\\\ A^{k}C_{1}BC_{1}\cdots B\text{, if $k$ is
even}.\end{cases}$

Therefore, if $A$ is nilpotent, then $AB$ is also nilpotent. Moreover $A,B$
are $B$-weakly commutative then $AB=BC_{2},\text{ }BA=C_{2}B$ for some
operator $C_{2}\in\mathcal{B}(X)$. Let $k\geq\text{max}\\{k_{1},k_{2}\\}$,
where $k_{1},k_{2}$ are the nilpotency index of $A$ and $B$, respectively.
Then $(A+B)^{3k}$ is the sum of the some monomials of the form

$A^{p_{1}}B^{q_{1}}A^{p_{2}}B^{q_{2}}\cdots A^{p_{m}}B^{q_{m}},\text{ for
}1\leq m\leq 3k$ (3)

where $p_{i},q_{j}\in\\{0,1,2\cdots 3k\\}$ for $1\leq i,j\leq m,$ such that
$\displaystyle\sum_{i=1}^{m}(p_{i}+q_{i})=3k.$ In fact in (3) at least one of
$\displaystyle\sum_{i=1}^{m}p_{i}$, $\displaystyle\sum_{i=1}^{m}q_{i}$ is
larger than $k$. Without loss of generality, let us assume
$\displaystyle\sum_{i=1}^{m}q_{i}\geq k.$ Then using $B$-weakly commutativity
of $A,B$ we obtain

$\displaystyle A^{p_{1}}B^{q_{1}}A^{p_{2}}B^{q_{2}}\cdots A^{p_{m}}B^{q_{m}}$
$\displaystyle=B^{k}T,\text{ for some }T\in\mathcal{B}(X)$
$\displaystyle=0,\text{ for }1\leq m\leq 3k.$

Hence, each monomial in the expansion of $(A+B)^{3k}$ is $0$. Thus, we
conclude that $A+B$ is nilpotent.

######  Remark 2.5.

In Lemma 2.3, if we take $A^{2}B=BA^{2}$ and $AB^{2}=B^{2}A$, instead of
taking $\\{A,B\\}$-weakly commutativity, then, in general, $A+B$ may not be
nilpotent. This can be observed in the following example.

######  Example 2.6.

Let

$A=\begin{bmatrix}0&0\\\ 1&0\end{bmatrix}\text{, }B=\begin{bmatrix}0&1\\\
0&0\end{bmatrix}\in M_{2}(\mathbb{C}).$

Then $A,\text{ }B$ are nilpotent and $A^{2}B=BA^{2}\text{, }AB^{2}=B^{2}A$,
but $A+B=\begin{bmatrix}0&1\\\ 1&0\end{bmatrix}$, which is not nilpotent.

The following lemma is important for our main results.

######  Lemma 2.7.

If $A\in\mathcal{B}(X)^{inv}$, $B\in\mathcal{B}(X)^{nil}$ such as $A,B$ are
$B$-weakly commutative, and $A^{2}B=BA^{2},$ then
$A+B\in\mathcal{B}(X)^{inv}$.

######  Proof 2.8.

Let $B\in\mathcal{B}(X)^{nil}$ and $A,B$ are $B$-weakly commutative then by
Lemma 2.3, $AB$ is nilpotent. Furthermore since $A\in\mathcal{B}(X)^{inv}$,
therefore we have

$A+B=A(I+A^{-1}B).$

Now $A$ is invertible and $A^{2}B=BA^{2}$ implies $A^{-2}(AB)=(AB)A^{-2}$.
Then we have

$(I+A^{-1}B)^{-1}=(I+A^{-2}AB)^{-1}=I-A^{-2}AB+(A^{-2})^{2}(AB)^{2}+\cdots+(-1)^{m-1}(A^{-2})^{m-1}(AB)^{m-1},$

where $m\text{ is the nilpotency index of }AB$. Hence we obtain
$A+B=A(I+A^{-1}B)$ is invertible.

######  Remark 2.9.

In Lemma 2.7, if $A,\text{}B$ is not $B$-weakly commutative, then $A+B$ may
not be invertible; the next example describes that.

######  Example 2.10.

Let $A=\begin{bmatrix}0&1\\\ 1&0\end{bmatrix}$ and $B=\begin{bmatrix}0&-1\\\
0&0\end{bmatrix}\in M_{2}(\mathbb{C})$. Here $A,\text{ }B$ is not $B$-weakly
commutative but $A^{2}B=BA^{2}.$ Moreover $A$ is invertible, $B$ is nilpotent,
but $A+B$ is not invertible.

######  Corollary 2.11.

If $A\in\mathcal{B}(X)^{inv}$, $B\in\mathcal{B}(X)^{nil}$ such as $A,B$ are
$\\{A,B\\}$-weakly commutative. Then $A+B\in\mathcal{B}(X)^{inv}$.

######  Proof 2.12.

If $A,B$ are $A$-weakly commutative, then by Lemma 1.4, $A^{2}B=BA^{2}.$
Therefore the required result follows from Lemma 2.7.

Next, we demonstrate one of the main theorems of this section.

######  Theorem 2.13.

If $A,\text{}B\in\mathcal{B}(X)^{D}$ such as $A,B$ are $\\{A,B\\}$-weakly
commutative, then the followings are equivalent:

  1. (i)

$A+B\in\mathcal{B}(X)^{D};$

  2. (ii)

$AA^{D}(A+B)BB^{D}\in\mathcal{B}(X)^{D};$

  3. (iii)

$AA^{D}(A+B)\in\mathcal{B}(X)^{D};$

  4. (iv)

$(A+B)BB^{D}\in\mathcal{B}(X)^{D}.$

Before proceeding with the proof of Theorem 2.13, we must establish the
following result, which holds significance in the context of this theorem:

######  Theorem 2.14.

Let $A\in\mathcal{B}(X)^{D}$, $B\in\mathcal{B}(X)^{nil}$ such as $A,B$ are
$\\{A,B\\}$-weakly commutative. Then $A+B\in\mathcal{B}(X)^{D}$ and

$\displaystyle(A+B)^{D}$ $\displaystyle=(I+A^{D}B)^{-1}A^{D}$
$\displaystyle=\left(\sum_{n=0}^{i(A)-1}(A^{D})^{2n}(-AB)^{n}\right)A^{D}.$

######  Proof 2.15.

Since $A\in\mathcal{B}(X)^{D}$, then $X$ has the decomposition
$X=\mathcal{R}(AA^{D})\oplus\mathcal{N}(AA^{D})$ and corresponding to this
decompositon of $X$, $A$ have the block form

$A=A_{1}\oplus A_{2},$

where $A_{1}$ is invertible operator on $\mathcal{R}(AA^{D})$ and $A_{2}$ is
nilpotent operator on $\mathcal{N}(AA^{D}).$ Moreover $A,B$ are
$\\{A,B\\}$-weakly commutative means $AB=C_{1}A,\text{ }BA=AC_{1}$ and
$AB=BC_{2},\text{ }BA=C_{2}B$ for two operators $C_{1},\text{
}C_{2}\in\mathcal{B}(X)$. Since $AB=C_{1}A,\text{ }BA=AC_{1}$, then both the
spaces $\mathcal{R}(AA^{D})\text{ and }\mathcal{N}(AA^{D})$ are invariant
under $B$ and $C_{1}.$ Therefore relative to the decomposition
$X=\mathcal{R}(AA^{D})\oplus\mathcal{N}(AA^{D})$, $B$ have the block
representation

$B=B_{1}\oplus B_{2},$

where both $B_{1}\text{ and }B_{2}$ are nilpotent operators on
$\mathcal{R}(AA^{D})$ and $\mathcal{N}(AA^{D})$, respectively. Similarly we
have

$C_{1}=C_{11}\oplus C_{12},$

where $C_{11}\text{ , }C_{12}$ are two operators on $\mathcal{R}(AA^{D})$ and
$\mathcal{N}(AA^{D}),$ respectively. Using this block forms of $A,B\text{ and
}C_{1}$ in $AB=C_{1}A,\text{ }BA=AC_{1}$ we obtain

$\displaystyle A_{1}B_{1}=C_{11}A_{1},\text{ }B_{1}A_{1}=A_{1}C_{11}\text{ and
}A_{2}B_{2}=C_{12}A_{2},\text{ }B_{2}A_{2}=A_{2}C_{12}.$

Again using this block forms in $AB=BC_{2},\text{ }BA=C_{2}B$ we get

$A_{1}B_{1}=B_{1}C_{21},\text{ }B_{1}A_{1}=C_{21}B_{1}\text{ and
}A_{2}B_{2}=B_{2}C_{22},\text{ }B_{2}A_{2}=C_{22}B_{2},$

where $\begin{bmatrix}C_{21}&C_{23}\\\ C_{24}&C_{22}\end{bmatrix}$ is the
block representation of $C_{2}$ corresponding to the space decomposition
$X=\mathcal{R}(AA^{D})\oplus\mathcal{N}(AA^{D}).$ Hence we get that
$A_{1},B_{1}$ are $\\{A_{1},B_{1}\\}$-weakly commutative and $A_{2},B_{2}$ are
$\\{A_{2},B_{2}\\}$-weakly commutative. Therefore, by Lemma 2.3, $A_{2}+B_{2}$
is nilpotent, and $A_{1}+B_{1}$ is invertible by Corollary 2.11. Hence we
obtain $A+B=(A_{1}+B_{1})\oplus(A_{2}+B_{2})\in\mathcal{B}(X)^{D}$ by Theorem
2.2, and

$\displaystyle(A+B)^{D}$ $\displaystyle=(A_{1}+B_{1})^{-1}\oplus 0$
$\displaystyle=(I+A^{D}B)^{-1}A^{D}$
$\displaystyle=\left(\sum_{n=0}^{i(A)-1}(A^{D})^{2n}(-AB)^{n}\right)A^{D},$

as desired.

######  Proof 2.16 (Proof of Theorem 2.13).

Since $A,B$ are $\\{A,B\\}$-weakly commutative, then similar to the proof of
Theorem 2.14, we get

$A=A_{1}\oplus A_{2}\text{ and }B=B_{1}\oplus B_{2},$

where $A_{1}$ is an invertible operator on $\mathcal{R}(AA^{D})$, $A_{2}$ is
an nilpotent operator on $\mathcal{N}(AA^{D})$ and $B_{1},B_{2}$ are two
Drazin invertible operators on $\mathcal{R}(AA^{D})$ and
$\mathcal{N}(AA^{D})$, respectively. Moreover, similar to the proof of Theorem
2.14 we also obtain $A_{1},B_{1}$ are $\\{A_{1},B_{1}\\}$-weakly commutative
and $A_{2},B_{2}$ are $\\{A_{2},B_{2}\\}$-weakly commutative. Since $A_{2}$ is
nilpotent, and $B_{2}$ is Drazin invertible therefore, by Theorem 2.14,
$A_{2}+B_{2}$ is Drazin invertible. Thus
$A+B=(A_{1}+B_{1})\oplus(A_{2}+B_{2})\in\mathcal{B}(X)^{D}$ if and only if
$A_{1}+B_{1}$ is Drazin invertible, hence (i)$\iff$(iii). Again since
$A_{1},\text{ }B_{1}$ are $A_{1}\text{ and }B_{1}$-weakly commutative, then
corresponding to the decomposition

$\mathcal{R}(AA^{D})=\mathcal{R}(B_{1}B_{1}^{D})\oplus\mathcal{N}(B_{1}B_{1}^{D}),$

$B_{1}\text{ and }A_{1}$ has the block representation $B_{1}=B_{11}\oplus
B_{12}$ and $A_{1}=A_{11}\oplus A_{12}$, respectively. Furthermore
$B_{11},\text{ }A_{11}$ are $\\{A_{11},B_{11}\\}$-weakly commutative and
$B_{12},\text{}A_{12}$ are $\\{B_{12},A_{12}\\}$-weakly commutative. Since
$A_{12}$ is invertible and $B_{12}$ is nilpotent, then by Corollary 2.11, we
get $A_{12}+B_{12}$ is invertible. Hence by Theorem 2.2,
$A_{1}+B_{1}=(A_{11}+B_{11})\oplus(A_{12}+B_{12})$ is Drazin invertible if and
only if $A_{11}+B_{11}$ is Drazin invertible, thus we obtain (i)$\iff$ (ii),
(i)$\iff$ (iv).

Now, we present the following finding, which includes an expression for
$(A+B)^{D}$ as well.

######  Theorem 2.17.

Let $A,B\in\mathcal{B}(X)^{D}$, such as $A,B$ are $\\{A,B\\}$-weakly
commutative and $A^{2}A^{D}(I+A^{D}B)\in\mathcal{B}(X)^{D}$. Then
$A+B\in\mathcal{B}(X)^{D}$, and

$\displaystyle(A+B)^{D}$
$\displaystyle{}=\left(A^{2}A^{D}(I+A^{D}B)\right)^{D}+(I-AA^{D})B^{D}\left(A(I-AA^{D})B^{D}+I\right)^{-1}$
$\displaystyle=\left(A^{2}A^{D}(I+A^{D}B)\right)^{D}+(I-AA^{D})B^{D}\left(\sum_{n=0}^{i(A)-1}(-AB)^{n-1}(B^{D})^{2n}\right).$

######  Proof 2.18.

Since here $A,B$ are $\\{A,B\\}$-weakly commutative, therefore as in the proof
of Theorem 2.14, we have

$A=A_{1}\oplus A_{2}\text{ and }B=B_{1}\oplus B_{2},$

where $A_{1},B_{1}$ are $\\{A_{1},B_{1}\\}$-weakly commutative and
$A_{2},B_{2}$ are $\\{A_{2}$,$B_{2}\\}$-weakly commutative. Here
$A^{2}A^{D}(I+A^{D}B)\in\mathcal{B}(X)^{D}$, therefore
$(A_{1}+B_{1})\in\mathcal{B}(\mathcal{R}(AA^{D}))^{D}$ and by Theorem 2.2, we
have

$(A_{1}+B_{1})^{D}\oplus 0=\left(A^{2}A^{D}(I+A^{D}B)\right)^{D}.$ (4)

Furthermore $A_{2},B_{2}$ are $\\{A_{2},B_{2}\\}$-weakly commutative and
$A_{2}$ is nilpotent, $B_{2}$ is Drazin invertible in $\mathcal{N}(AA^{D}).$
Therefore by Theorem 2.14, $A_{2}+B_{2}\in(\mathcal{N}(AA^{D}))^{D}$ and

$\displaystyle 0\oplus(A_{2}+B_{2})^{D}$
$\displaystyle=(I-AA^{D})B^{D}\left(A(I-AA^{D})B^{D}+I\right)^{-1}$ (5)
$\displaystyle=(I-AA^{D})B^{D}\left(\sum_{n=0}^{i(A)-1}(-AB)^{n-1}(B^{D})^{2n}\right).$
(6)

Hence the required result follows from Theorem 2.2, equation (4), (5) and (6).

Similar to the previous result, using Theorem 2.13 and Theorem 2.14 one can
establish the following corollary, which provides representations for
$(A+B)^{D}$.

######  Corollary 2.19.

Let $A,B\in\mathcal{B}(X)^{D},$ such as $A.B$ are $\\{A,B\\}$-weakly
commutative.

  1. (i)

If $AA^{D}(A+B)BB^{D}\in\mathcal{B}(X)^{D}$ then

$\displaystyle(A+B)^{D}={}$
$\displaystyle\left(AA^{D}(A+B)BB^{D}\right)^{D}+(I-AA^{D})B^{D}\left(\sum_{n=0}^{i(A)-1}(-AB)^{n}(B^{D})^{2n}\right)$
$\displaystyle+\left(\sum_{n=0}^{i(B)-1}(A^{D})^{2n}(-AB)^{n}\right)A^{D}(I-BB^{D})$

  2. (ii)

If $AA^{D}(A+B)\in\mathcal{B}(X)^{D},$ then

$(A+B)^{D}=\left(AA^{D}(A+B)\right)^{D}+(I-AA^{D})B^{D}\left(\sum_{n=0}^{i(A)-1}(-AB)^{n}(B^{D})^{2n}\right),$

  3. (iii)

If $(A+B)BB^{D}\in\mathcal{B}(X)^{D}$, then

$(A+B)^{D}=\left((A+B)BB^{D}\right)^{D}+\left(\sum_{n=0}^{i(B)-1}(A^{D})^{2n}(-AB)^{n}\right)A^{D}(I-BB^{D}).$

The subsequent outcome presents a broader perspective of Theorem 2.14.

######  Theorem 2.20.

Let $A\in\mathcal{B}(X)^{D}\text{, }B\in\mathcal{B}(X)^{nil}$ such as $A,B$
are $B$-weakly commutative. If $AA^{D}(A^{2}B-BA^{2})=0$,
$(A^{2}B-BA^{2})AA^{D}=0$, $A^{\pi}(AB-CA)=0$ and $A^{\pi}(BA-AC)=0$ for some
operator $C\in\mathcal{B}(X)$, then $A+B\in\mathcal{B}(X)^{D}.$

######  Proof 2.21.

Since $A\in\mathcal{B}(X)^{D}$, therefore $A$ have the block representation

$A=A_{1}\oplus A_{2},$

where $A_{1}\in\mathcal{B}(\mathcal{R}(AA^{D}))^{inv},$ and
$A_{2}\in\mathcal{B}(\mathcal{N}(AA^{D}))^{nil}.$ Again since
$AA^{D}(A^{2}B-BA^{2})=0$ and $(A^{2}B-BA^{2})AA^{D}=0$ then $B$ also have the
block representation

$B=B_{1}\oplus B_{2},$

where $B_{1},B_{2}$ both are nilpotent operators on $\mathcal{R}(AA^{D})$ and
$\mathcal{N}(AA^{D}),$ respectvely. Moreover $A,B$ are $B$-weakly commutative
then $A_{1},B_{1}$ are $B_{1}$-weakly commutative and $A_{2},B_{2}$ are
$B_{2}$-weakly commutative. Furthermore using these block representations of
$A$ and $B$ on the condition $A^{\pi}(AB-CA)=0$ and $A^{\pi}(BA-AC)=0$ we
obtain $A_{2},B_{2}$ are $A_{2}$-weakly commutative also. Hence we have
$A_{2},B_{2}$ are $\\{A_{2},B_{2}\\}$-weakly commutative, therefore by Lemma
2.3, $A_{2}+B_{2}\in\mathcal{B}(\mathcal{N}(BB^{D}))^{nil}.$ Again from
$AA^{D}(A^{2}B-BA^{2})=0$ we obtain $A_{1}^{2}B_{1}=B_{1}A_{1}^{2}$, therefore
by Lemma 2.7, $A_{1}+B_{1}\in\mathcal{B}(\mathcal{R}(AA^{D}))^{inv}.$ Hence
using Theorem 2.2, we obtain

$A+B=(A_{1}+B_{1})\oplus(A_{2}+B_{2})\in\mathcal{B}(X)^{D},$

as desired.

DS Djordjević and Y Wei djordjevic2002additive  established that if
$A,B\in\mathcal{B}(X)^{d}$ and $AB=0$ then $A+B\in\mathcal{B}(X)^{d}.$ This
same conclusion holds in situations involving the Drazin inverse also.
Consequently, utilizing this finding leads us to the following theorem.

######  Theorem 2.22.

Let $A,B\in\mathcal{B}(X)^{D}$ such as $A,B$ are $A$-weakly commutative, and
$A^{\pi}A,A^{\pi}B$ are $A^{\pi}B$-weakly commutative. If $AA^{D}AB=0$, then
$A+B\in\mathcal{B}(X)^{D}.$

######  Proof 2.23.

Similar to the proof of Theorem 2.14, we have

$A=A_{1}\oplus A_{2}\text{ and }B=B_{1}\oplus B_{2},$

where $A_{1}\in\mathcal{B}(\mathcal{R}(AA^{D}))^{inv}$,
$B_{1}\in\mathcal{B}(\mathcal{R}(AA^{D}))^{D}$,
$A_{2}\in\mathcal{B}(\mathcal{N}(AA^{D}))^{nil}$ and
$B_{2}\in\mathcal{B}(\mathcal{N}(AA^{D}))^{D}.$ It follows from the $A$-weakly
commutativity of $A,B$ that $A_{1},B_{1}$ are $A_{1}$-weakly commutative and
$A_{2},B_{2}$ are $A_{2}$-weakly commutative. Furthermore, from
$A^{\pi}B$-weakly commutativity of $A^{\pi}A,A^{\pi}B$ we obtain $A_{2},B_{2}$
are $B_{2}$-weakly commutative. Therefore using Theorem 2.14, we get
$A_{2}+B_{2}\in\mathcal{B}(\mathcal{N}(AA^{D}))^{D}.$ Moreover $AA^{D}AB=0$
implies $A_{1}B_{1}=0.$ Therefore
$A_{1}+B_{1}\in\mathcal{B}(\mathcal{R}(AA^{D}))^{D}.$ Hence using Theorem 2.2,
we get $A+B=(A_{1}+B_{1})\oplus(A_{2}+B_{2})\in\mathcal{B}(X)^{D}$.

##  3 Multiplicative properties

In this section, we investigate the Drazin invertibility of $AB$ for a pair of
Drazin invertible operators $A\text{ and }B$, where $A,\text{ }B$ are
$\\{A,B\\}$-weakly commutative. We start by presenting some results that hold
significance for the main theorem of this section.

######  Theorem 3.1.

If $A\in\mathcal{B}(X)^{D}$ such that $AB=CA$ and $BA=AC$ for some
$B,C\in\mathcal{B}(X)$. Then

  1. (i)

$AA^{D}B=BAA^{D};$

  2. (ii)

$AA^{D}C=CAA^{D};$

  3. (iii)

$A^{D}B=CA^{D};$

  4. (iv)

$A^{D}C=BA^{D}.$

######  Proof 3.2.

  1. (i)

Let $A\in\mathcal{B}(X)^{D}$, then $A^{k}(I-AA^{D})=0$ for $k\geq i(A).$ Since
$AB=CA$ and $BA=AC$ implies $A^{2}B=BA^{2}$, then we have

$\displaystyle AA^{D}B-AA^{D}BAA^{D}$ $\displaystyle=AA^{D}B(I-AA^{D})$
$\displaystyle=(A^{D})^{2k}A^{2k}B(I-AA^{D})$
$\displaystyle=(A^{D})^{2k}BA^{2k}(I-AA^{D})=0.$

Therefore $AA^{D}B=AA^{D}BAA^{D},$ similarly we get $BAA^{D}=AA^{D}BAA^{D}.$
Hence $AA^{D}B=BAA^{D}.$

  2. (ii)

Since $AB=CA$ and $BA=AC$ implies $A^{2}C=CA^{2}.$ Therefore this is similar
to (i).

  3. (iii)

Now

$\displaystyle A^{D}B=(A^{D})^{2}AB$ $\displaystyle=A^{D}BAA^{D}$
$\displaystyle=A^{D}ACA^{D}$ $\displaystyle=CA^{D}.$

  4. (iv)

Follows similarly as (iii).

The subsequent corollary follows from Theorem 3.1.

######  Corollary 3.3.

Let $A,B\in\mathcal{B}(X)$ are $A$-weakly commutative operators and $A$ is
Drazin invertible then $AA^{D}B=BAA^{D}.$

Now, we are ready to look over the Drazin invertibility of product of two
Drazin invertible operators.

######  Theorem 3.4.

Let $A,B\in\mathcal{B}(X)^{D}$. If $A,B$ are $\\{A,B\\}$-weakly commutative,
then $AB$ is Drazin invertible and $(AB)^{D}=B^{D}A^{D}.$

######  Proof 3.5.

Since $A,B$ are $\\{A,B\\}$-weakly commutative, then

$AB=C_{1}A,\text{ }BA=AC_{1}\text{ and }AB=BC_{2},\text{ }BA=C_{2}B,$

for two operators $C_{1},C_{2}\in\mathcal{B}(X).$ Let $X=B^{D}A^{D}$. Then,
using Corollary 3.3, we can ascertain that $ABX=XAB$ and $XABX=X$. Moreover we
have

$\displaystyle AB-ABXAB$ $\displaystyle=AB-ABB^{D}A^{D}AB$
$\displaystyle=AB-A^{2}A^{D}B^{D}B^{2}$
$\displaystyle=A^{2}A^{D}(B-B^{2}B^{D})+(A-A^{2}A^{D})B.$

Since $A-A^{2}A^{D}$ is nilpotent and

$(A-A^{2}A^{D})B=C_{1}(A-A^{2}A^{D}),\text{
}B(A-A^{2}A^{D})=(A-A^{2}A^{D})C_{1}$

therefore by Lemma 2.3, $(A-A^{2}A^{D})B$ is nilpotent. Similarly since
$B-B^{2}B^{D}$is nilpotent and

$A^{2}A^{D}(B-B^{2}B^{D})=(B-B^{2}B^{D})C_{2}AA^{D},\text{
}(B-B^{2}B^{D})A^{2}A^{D}=C_{2}AA^{D}(B-B^{2}B^{D})$

therefore $A^{2}A^{D}(B-B^{2}B^{D})$ is also nilpotent. Furthermore, we have

$(A-A^{2}A^{D})BA^{2}A^{D}(B-B^{2}B^{D})=0\text{ and
}(A-A^{2}A^{D})A^{D}ABA(B-B^{2}B^{D})=0.$

Therefore by Lemma 2.1, we get $AB-ABB^{D}A^{D}AB$ is nilpotent.

######  Example 3.6.

Let $\mathcal{B}(X)=\mathcal{M}_{3}(\mathbb{C})$,

$A=\begin{bmatrix}0&x&0\\\ x&0&x\\\ 0&x&0\end{bmatrix}\text{ and
}B=\begin{bmatrix}y&0&0\\\ 0&-y&0\\\ 0&0&y\end{bmatrix}\in M_{3}(\mathbb{C}),$

where $x,y\in\mathbb{C}-\\{0\\}.$ Therefore $A,B$ are $\\{A,B\\}$-weakly
commutative, and hence $(AB)^{D}=B^{D}A^{D}$.

######  Proof 3.7.

Let $C_{1}=-B$ and $C_{2}=-A$ then we have $AB=C_{1}A,$ $BA=AC_{1}$ and
$AB=BC_{2}$, $BA=C_{2}B.$ Moreover we have

$A^{D}=\begin{bmatrix}0&\frac{1}{2x}&0\\\ \frac{1}{2x}&0&\frac{1}{2x}\\\
0&\frac{1}{2x}&0\end{bmatrix}\text{, }B^{D}=\begin{bmatrix}\frac{1}{y}&0&0\\\
0&-\frac{1}{y}&0\\\ 0&0&\frac{1}{y}\end{bmatrix}\text{ and
}(AB)^{D}=\begin{bmatrix}0&\frac{1}{2xy}&0\\\
-\frac{1}{2xy}&0&-\frac{1}{2xy}\\\ 0&\frac{1}{2xy}&0\end{bmatrix}.$

Hence we obtain $(AB)^{D}=B^{D}A^{D}$.

######  Remark 3.8.

The results found in (campbell1991generalized, , Theorem 7.8.4) followed as a
specific case of Theorem 3.4 and Theorem 3.1.

Utilizing the preceding result, we obtain the subsequent corollary.

######  Corollary 3.9.

If $A,B\in\mathcal{B}(X)^{D}$ such as $A,B$ are $A$-weakly commutative and
$AA^{D}A,AA^{D}B$ are $AA^{D}B$-weakly commutative. Then
$AB\in\mathcal{B}(X)^{D}$ and $(AB)^{D}=B^{D}A^{D}.$

######  Proof 3.10.

Let $A\in\mathcal{B}(X)^{D}$ and $A,B$ are $A$-weakly commutative therefore
$A\text{ and }B$ has the block form representation

$A=A_{1}\oplus A_{2}\text{, }B=B_{1}\oplus B_{2},$

where $A\in\mathcal{B}(\mathcal{R}(AA^{D}))^{inv}$ and
$A_{2}\in\mathcal{B}(\mathcal{N}(AA^{D}))^{nil}$. Moreover from $A$-weakly
commutativity of $A,B$ it follows that $A_{1},B_{1}$ are $A_{1}$-weakly
commutative and $A_{2},B_{2}$ are $A_{2}$ weakly commutative. Then, by Lemma
2.3, $A_{2}B_{2}$ is nilpotent since $A_{2}$ is nilpotent. Furthermore from
the $AA^{D}B$-weakly commutativity of $AA^{D}A$ and $AA^{D}B$ we obtain
$B_{1}$-weakly commutativity of $A_{1},B_{1}$. Therefore $A_{1},B_{1}$ are
$\\{A_{1},B_{1}\\}$-weakly commutative, then by Theorem 3.4,
$A_{1}B_{1}\in\mathcal{B}(\mathcal{R}(AA^{D}))$ and
$(A_{1}B_{1})^{D}=B_{1}^{D}A_{1}^{D}.$ Hence using Theorem 2.2, we get

$AB=A_{1}B_{1}\oplus A_{2}B_{2}\in\mathcal{B}(X)^{D},\text{ and
}(AB)^{D}=B^{D}A^{D}.$

##  4 Applications

The aim of this section is to explore new additive and multiplicative
properties of Drazin inverse for some specific types of operators by using
results found in section 2 and section 3. We begin with the subsequent lemma.

######  Lemma 4.1.

If $A,B,C\in\mathcal{B}(X),$ such as $AB=CA,$ $BA=AC$ and $A$ is invertible.
Then $A+B\in\mathcal{B}(X)^{D}$ if and only if $A+C\in\mathcal{B}(X)^{D}$.

######  Proof 4.2.

Since $A$ is invertible then $A^{D}=A^{-1}$ and from $AB=CA$ we obtain
$B=A^{-1}CA$. Moreover we have

$A+B=A+A^{-1}CA=(I+A^{-1}C)A.$

Therefore by Cline’s formula cline1965application ,
$(I+A^{-1}C)A\in\mathcal{B}(X)^{D}$ if and only if $A(I+A^{-1}C)=A+C$ is
Drazin invertible. Hence, we get $A+B\in\mathcal{B}(X)^{D}$ if and only if
$A+C\in\mathcal{B}(X)^{D}$.

By applying Theorem 3.4, we obtain the following result, which provides a
sufficient condition for the Drazin invertibility of the sum of two involutary
operators.

######  Theorem 4.3.

Let $A,B\in\mathcal{B}(X)$ such that $A^{2}=B^{2}=I$ and $A-B=BAB-ABA$. If
$I+AB\in\mathcal{B}(X)^{D}$ then $A+B\in\mathcal{B}(X)^{D}$, and

$(A+B)^{D}=(I+AB)^{D}A.$

######  Proof 4.4.

If $A^{2}=B^{2}=I$ then we have $A+B=A(I+AB).$ Moreover

$A(I+AB)=(I+BA)A\text{ and }(I+AB)A=A(I+BA).$

Therefore $A,I+AB$ are $A$-weakly commutative. Furthermore, we have

$A(I+AB)=(I+AB)B$

and from $A-B=BAB-ABA$ it follows that $(I+AB)A=B(I+AB)$, hence $A,I+AB$ are
$I+AB$-weakly commutative. Since $A$ is invertible, therefore, by Theorem 3.4,
if $I+AB\in\mathcal{B}(X)^{D}$, then $A(I+AB)=A+B$ is Drazin invertible where

$\displaystyle(A+B)^{D}=$ $\displaystyle(I+AB)^{D}A^{D}$ $\displaystyle=$
$\displaystyle(I+AB)^{D}A.$

Now, we give an example to illustrate the preceding theorem.

######  Example 4.5.

Let $T,S\in\mathcal{B}(\ell^{2}(\mathbb{C})),$ define by

$\displaystyle T(x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},\cdots)={}$
$\displaystyle(-x_{1},x_{2}-10x_{3},-x_{3},-x_{4},x_{5}-10x_{6},-x_{6},\cdots)\text{
and }$ $\displaystyle S(x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},\cdots)={}$
$\displaystyle(-x_{1}+2x_{3},-x_{2}-10x_{3},x_{3},-x_{4}+2x_{6},-x_{5}-10x_{6},x_{6},\cdots).$

Then one can easily verify that $T^{2}=I$, $S^{2}=I$ and $T-S=STS-TST.$
Therefore by Theorem 4.3, if $I+TS$ is Drazin invertible then $T+S$ is Drazin
invertible.

Next, using Lemma 2.3, we prove that if in Definition 1.1, we restrict the
choice of $B$ and $C$ to the collection of invertible elements only, then the
Drazin invertibility of $AA^{D}(A+B)$ ensure the Drazin invertibility of
$A+B$, when $A,B$ are $A$-weakly commutative.

######  Theorem 4.6.

Let $A\in\mathcal{B}(X)^{D}$ and $B,C\in\mathcal{B}(X)^{inv}$ such as $AB=CA$
and $BA=AC$. If $AA^{D}(A+B)\in\mathcal{B}(X)^{D},$ then
$A+B\in\mathcal{B}(X)^{D}$.

######  Proof 4.7.

Since $A\in\mathcal{B}(X)^{D}$, therefore $A$ have the block representation

$A=A_{1}\oplus A_{2},$

where $A_{1}\in\mathcal{B}(\mathcal{R}(AA^{D}))^{inv},$ and
$A_{2}\in\mathcal{B}(\mathcal{N}(AA^{D}))^{nil}.$ Furthermore since $AB=CA$,
$BA=AC$ then $B$ also have the block representation

$B=B_{1}\oplus B_{2},$

where $B_{1}$ and $B_{2}$ are invertible operators in
$\mathcal{B}(\mathcal{R}(AA^{D}))$ and $\mathcal{B}(\mathcal{N}(AA^{D}))$,
respectively. Moreover from $AA^{D}(A+B)\in\mathcal{B}(X)^{D}$ we obtain that
$A_{1}+B_{1}\in\mathcal{B}(\mathcal{R}(AA^{D}))^{D}$. Now since $B,C$ are
invertible, therefore $AB=CA$ and $BA=AC$ implies

$AB^{-1}=C^{-1}A\text{ and }B^{-1}A=AC^{-1}.$

Hence $A,B^{-1}$ are $A$-weakly commutative, moreover $A_{2},B_{2}^{-1}$ are
also $A_{2}$-weakly commutative. Now since
$A_{2}\in\mathcal{B}(\mathcal{N}(AA^{D}))^{nil},$ then by Lemma 2.3,
$A_{2}B_{2}^{-1}\in\mathcal{B}(\mathcal{N}(AA^{D}))^{nil}$. Therefore
$I+A_{2}B_{2}^{-1}\in\mathcal{B}(\mathcal{N}(AA^{D}))^{inv},$ hence
$A_{2}+B_{2}=(A_{2}B_{2}^{-1}+I)B_{2}\in\mathcal{B}(\mathcal{N}(AA^{D}))^{inv}.$
Therefore by Theorem 2.2, $A+B\in\mathcal{B}(X)^{D},$ since
$A_{1}+B_{1}\in\mathcal{B}(\mathcal{R}(AA^{D}))^{D}$ and
$A_{2}+B_{2}\in\mathcal{B}(\mathcal{N}(AA^{D}))^{inv}.$

######  Proposition 4.8.

If $A\in(\mathcal{B}(X))^{D},$ such that $AB=CA$ and $BA=AC$ for two commuting
operators $B,\text{}C\in\mathcal{B}(X)^{inv}$, then $A^{2}B=ACA$ and
$ABA=A^{2}C$ are Drazin invertible.

######  Proof 4.9.

Let $M=AB=CA$ and $N=BA=AC$, then by (barraa2019drazin, , Proposition 2.5.)
both $M\text{ and }N$ are Drazin invertible. Furthermore we have

$AM=(AC)A=M(B^{-1}M)\text{ and }MA=A(BA)=(MC^{-1})M,$

where $B^{-1}M=MC^{-1}$ since $BC=CB.$ Therefore $A,M$ are $\\{A,M\\}$-weakly
commutative. Similarly, $A,N$ are also $\\{A,N\\}$-weakly commutative.
Therefore, in light of Theorem 3.4, we obtain both $AN\text{ and }AM$ are
Drazin invertible, as desired.

##  5 For g-Drazin inverse in a Banach algebra

In this section, we extend the scope of certain additive and multiplicative
results presented in Section 2 and 3 to the case of g-Drazin inverse within a
complex Banach algebra. Let $\mathcal{A}$ be an unital complex Banach algebra
with the unit $1$. If $p\in\mathcal{A}$ is an idempotent, then each
$x\in\mathcal{A}$ has a unique matrix representation

$\begin{bmatrix}x_{11}&x_{12}\\\ x_{21}&x_{22}\\\ \end{bmatrix}_{p},\text{
where }x_{ij}=p_{i}xp_{j},\text{ }p_{1}=p,\text{ }p_{2}=1-p.$

Now, we begin with the following result presented in cvetkovic2006additive .

######  Lemma 5.1.

cvetkovic2006additive  If $a,b\in\mathcal{A}^{qnil}$ such as $ab=0$ or
$ab=ba$, then $a+b\in\mathcal{A}^{qnil}.$

######  Lemma 5.2.

If $a,b\in\mathcal{A}$ are $a$-weakly commutative such that
$a\in\mathcal{A}^{qnil}$ then $ab\in\mathcal{A}^{qnil}.$ Moreover, if $a,b$
are $b$-weakly commutative and $b\in\mathcal{A}^{qnil}$, then
$a+b\in\mathcal{A}^{qnil}.$

######  Proof 5.3.

Since $a,b$ are $a$-weakly commutative then $ab=c_{1}a\text{ and }ba=ac_{1}$
for some $c_{1}\in\mathcal{A}.$ Let
$\alpha=\text{max}\\{1,\|a\|+\|b\|+\|c_{1}\|\\},$ and take
$a_{1}=\frac{a}{\alpha},\text{ }b_{1}=\frac{b}{\alpha}\text{ and
}c_{1}^{\prime}=\frac{c_{1}}{\alpha},$ then
max$\\{\|a_{1}\|,\|b_{1}\|,\|c_{1}^{\prime}\|\\}\leq 1$, furthermore
$ab\in\mathcal{A}^{qnil}$ if and only if $a_{1}b_{1}\in\mathcal{A}^{qnil}.$
Thus without loss of generality we can assume that
max$\\{\|a\|,\|b\|,\|c_{1}\|\\}\leq 1$. Now for $n\in\mathbb{N}$ we have

$\displaystyle(ab)^{n}$ $\displaystyle=\begin{cases}a^{n}bc_{1}b\cdots
b,\text{ if $n$ is odd,}\\\ a^{n}c_{1}bc_{1}\cdots b,\text{ if $n$ is
even}.\end{cases}$

Therefore

$\lim_{n\to\infty}\|(ab)^{n}\|^{\frac{1}{n}}\leq\lim_{n\to\infty}\|a^{n}\|^{\frac{1}{n}}=0,$

hence we get $ab\in\mathcal{A}^{qnil}.$ In order to prove
$a+b\in\mathcal{A}^{qnil}$, its enough to prove that
$(a+b)^{2}\in\mathcal{A}^{qnil}.$ Now

$(a+b)^{2}=a^{2}+ab+ba+b^{2},$

where $a^{2},b^{2},ab,ba\in\mathcal{A}^{qnil}$, since
$b\in\mathcal{A}^{qnil}$. Moreover, if $a,b$ are $\\{a,b\\}$-weakly
commutative, then by Lemma 1.4, $a^{2},ab,ba,b^{2}$ all commutes with each
other. Therefore in light of Lemma 5.1, we obtain
$(a+b)^{2}\in\mathcal{A}^{qnil}.$

######  Theorem 5.4.

Let $a\in\mathcal{A}^{d}$ and $ab=ca\text{, }ba=ac$ for some
$b,c\in\mathcal{A}$. Then the following are true:

  1. (i)

$aa^{d}b=baa^{d}$;

  2. (ii)

$aa^{d}c=caa^{d}$;

  3. (iii)

$a^{d}b=ca^{d}$;

  4. (iv)

$a^{d}c=ba^{d}$.

######  Proof 5.5.

Since $a\in\mathcal{A}^{d}$, therefore $a(1-aa^{d})\in\mathcal{A}^{qnil},$
therefore
$\displaystyle\lim_{n\to\infty}\|\left(a-a^{2}a^{d}\right)^{n}\|^{\frac{1}{n}}=0.$

  1. (i)

Now we have

$\displaystyle aa^{d}b-aa^{d}baa^{d}$ $\displaystyle=aa^{d}b(1-aa^{d})$
$\displaystyle=(a^{d})^{2n}a^{2n}b(1-aa^{d})$
$\displaystyle=(a^{d})^{2n}ba^{2n}(1-aa^{d})$
$\displaystyle=(a^{d})^{2n}b(a-a^{2}a^{d})^{2n}.$

Hence we obtain

$\|aa^{d}-aa^{d}baa^{d}\|^{\frac{1}{2n}}\leq\|a^{d}\|\|b\|^{\frac{1}{2n}}\|(a-a^{2}a^{d})^{{2n}}\|^{\frac{1}{2n}}.$

Therefore

$\lim_{n\to\infty}\|aa^{d}-aa^{d}baa^{d}\|^{\frac{1}{2n}}=0,$

and we get $aa^{d}b-aa^{d}baa^{d}=0.$ Similarly, $baa^{d}-aa^{d}baa^{d}=0$,
thus we have $aa^{d}b=baa^{d},$ as required.

  2. (ii)

Similar like (i), follows from the fact that $a^{2}c=ca^{2}.$

(iii) and (iv) follows similarly as Theorem 3.1.

The previous lemma yields the following result.

######  Corollary 5.6.

Let $a,b\in\mathcal{A}$ are $a$-weakly commutative. If $a$ is g-Drazin
invertible then $aa^{d}b=baa^{d}.$

In accordance with Theorem 3.4 using Theorem 5.4 and Lemma 5.2, we obtain the
following result.

######  Theorem 5.7.

If $a,b\in\mathcal{A}^{d}$, are $\\{a,b\\}$-weakly commutative then
$ab\in\mathcal{A}^{d}$ and $(ab)^{d}=b^{d}a^{d}.$

######  Theorem 5.8.

If $a\in\mathcal{A}^{inv}$ and $b\in\mathcal{A}^{qnil}$ such as $a,b$ are
$\\{a,b\\}$-weakly commutative, then $a+b\in\mathcal{A}^{inv}.$

######  Proof 5.9.

Since $a,b$ are $\\{a,b\\}$-weakly commutative, then by Lemma 1.4, we have
$a^{2}b=ba^{2}$ and by Lemma 5.2, $ab\in\mathcal{A}^{qnil}.$ Moreover

$a+b=a(1+a^{-1}b)=a(1+a^{-2}ab).$

Now since $a\in\mathcal{A}^{inv}$, then $a^{2}b=ba^{2}$ implies
$a^{-2}(ab)=(ab)a^{-2}$ but $ab\in\mathcal{A}^{qnil}$ therefore
$a^{-2}ab\in\mathcal{A}^{qnil}$, so $1+a^{-2}ab\in\mathcal{A}^{inv}.$ Hence
$a+b\in\mathcal{A}^{inv}$ and

$(a+b)^{-1}=\left(a(1+a^{-2}ab)\right)^{-1}=(1+a^{-2}ab)^{-1}a^{-1}.$

######  Theorem 5.10.

Let $a\in\mathcal{A}^{d}$ and $b\in\mathcal{A}^{qnil}$, where $a,b$ are
$\\{a,b\\}$-weakly commutative, then $a+b\in\mathcal{A}^{d}.$

######  Proof 5.11.

Since $a,b$ are $\\{a,b\\}$-weakly commutative, then we have
$ab=c_{1}a,ba=ac_{1}$ and $ab=bc_{2},ba=c_{2}b$ for some
$c_{1},c_{2}\in\mathcal{A}$. Let $p=aa^{d}$ and
$\mathcal{A}_{1}=p\mathcal{A}p,\mathcal{A}_{2}=(1-p)\mathcal{A}(1-p)$. Then
corresponding to the idempotent $p$, $a$ has the matrix representation

$a=\begin{bmatrix}a_{1}&0\\\ 0&a_{2}\end{bmatrix}_{p},\text{ where
}a_{1}\in\mathcal{A}_{1}^{inv}\text{ and }a_{2}\in\mathcal{A}_{2}^{qnil}.$ (7)

Furthermore by Theorem 5.4, $b,c_{1}$ also have a representation corresponding
to (7), given by

$c_{1}=\begin{bmatrix}c_{11}&0\\\ 0&c_{12}\end{bmatrix}_{p}\text{ and
}b=\begin{bmatrix}b_{1}&0\\\ 0&b_{2}\end{bmatrix}_{p},$

where $b_{1}\in\mathcal{A}_{1}^{qnil},\text{ }b_{2}\in\mathcal{A}_{2}^{qnil}.$
Then $ab=c_{1}a,ba=ac_{1}$ gives

$a_{1}b_{1}=c_{11}a_{1},\text{ }b_{1}a_{1}=a_{1}c_{11}\text{ and
}a_{2}b_{2}=c_{12}a_{2},\text{ }b_{2}a_{2}=c_{12}a_{2}.$

Moreover if $c_{2}=\begin{bmatrix}c_{21}&c_{22}\\\
c_{23}&c_{24}\end{bmatrix}_{p}$ then $ab=bc_{2},ba=c_{2}b$ gives

$a_{1}b_{1}=b_{1}c_{21},\text{ }b_{1}a_{1}=c_{21}b_{1}\text{ and
}a_{2}b_{2}=b_{2}c_{24},\text{ }b_{2}a_{2}=c_{24}b_{2}.$

Therefore by Theorem 5.8, we get $a_{1}+b_{1}\in\mathcal{A}_{1}^{inv}$ and by
Lemma 5.2, $a_{2}+b_{2}\in\mathcal{A}_{2}^{qnil}.$ Hence
$a+b\in\mathcal{A}^{d}.$

Using Theorem 5.8 and Theorem 5.10, we obtain the following results similar to
Theorem 2.13.

######  Theorem 5.12.

Let $a,b\in\mathcal{A}^{d}$ such that $a,b$ are $\\{a,b\\}$-weakly
commutative, then the following are equivalent:

  1. (i)

$a+b\in\mathcal{A}^{d};$

  2. (ii)

$aa^{d}(a+b)bb^{d}\in\mathcal{A}^{d};$

  3. (iii)

$aa^{d}(a+b)\in\mathcal{A}^{d};$

  4. (iv)

$(a+b)bb^{d}\in\mathcal{A}^{d}.$

## Declarations

The authors have no conflicts of interest to declare.

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