INTERNATIONAL JOURNAL OF RESEARCH AND SCIENTIFIC INNOVATION (IJRSI)  
ISSN No. 2321-2705 | DOI: 10.51244/IJRSI |Volume XII Issue XI November 2025  
N-Power Stable and Robust Operator Models in Computational  
Spaces  
Dr. N. Sivamani, Professor, Dr. P. Selvanayaki, V. Aarthika  
Department of Mathematics, Suguna College of Engineering, Coimbatore-Tamil Nadu, India.  
Assistant Professor, Department of Mathematics, Sri Ramakrishna College of Arts and Science for  
Women, Coimbatore  
Department of Computer Science and Engineering, Jai Shriram Engineering College, Tirupur  
DOI: https://doi.org/10.51244/IJRSI.2025.12110126  
Received: 24 November 2025; Accepted: 01 December 2025; Published: 18 December 2025  
ABSTRACT  
In this article, n-power stable, n-power robust, quasi-stable, and quasi-robust operator models are characterized  
in computational spaces. These classes of operators, originally studied in mathematical Fock spaces, are  
extended to applications in Computer Technology. In particular, we establish how such operator conditions  
contribute to the stability of iterative algorithms, normalization in machine learning, bounded mappings in signal  
and image processing and operator evolution in quantum computing. The analysis shows that the operator-  
theoretic framework ensures convergence, robustness and error control in modern computational pipelines.  
ACM/IEEE Classification  
Theory of computation → Operator models in computing  
Computing methodologies → Machine learning theory, Quantum computing, Signal processing  
Keywords-Composition operator, Data transformation, Machine learning stability, Quantum computing, Robust  
operators, n-power models  
INTRODUCTION  
In Computer Science, operator models arise naturally in diverse areas such as machine learning, quantum  
computing, computer vision and signal processing. For instance, the composition of transformation functions  
in neural networks parallels composition operators studied in mathematics. Stability of such operators ensures  
that iterative applications of transformations (e.g., deep layers or repeated quantum gates) remain bounded and  
converge.  
Earlier studies in mathematics have focused on Hardy, Bergman, and Fock spaces. In this paper, we reinterpret  
the mathematical framework of n-power posinormal and paranormal operators in the context of computational  
spaces. This allows us to provide a unified perspective on algorithmic stability, robustness, and  
boundedness of transformation operators widely used in Computer Technology.  
Preliminaries  
Let X denote a computational space such as a feature space in ML, signal space in DSP, or quantum state  
space.  
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A composition operator is defined as Cϕ(f)=f∘ϕ where ϕ is a mapping function (e.g., a data transformation,  
a neural network layer or a quantum gate).  
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INTERNATIONAL JOURNAL OF RESEARCH AND SCIENTIFIC INNOVATION (IJRSI)  
ISSN No. 2321-2705 | DOI: 10.51244/IJRSI |Volume XII Issue XI November 2025  
●
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Bounded operator → transformation that converges under repeated application (ensures algorithmic  
stability).  
Compact operator → transformation that compresses information and improves generalization (important  
in ML).  
Definitions for CS context:  
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n-power stable: Repeated application of an operator Tn*Tn preserves boundedness (ensures deep model  
convergence).  
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Quasi-stable: Stability holds up to a scaling factor, useful in iterative optimization algorithms.  
n-power robust: Ensures error does not amplify across multiple transformations.  
Quasi-robust: Ensures robustness in noisy or approximate computations.  
MAIN RESULTS  
Proposition  
An operator on a computational space is n-power stable if and only if repeated compositions maintain bounded  
transformation outputs.  
Proof Sketch: Repeated operator action must satisfy boundedness inequality analogous to Tn*Tn ≤ CTnTn*, which  
ensures stability across deep learning layers or iterative signal filters.  
Proposition  
An operator is n-power robust if and only if it resists amplification of computational errors when applied  
iteratively in noisy environments (e.g., quantum gates under decoherence).  
Corollary  
If C=1, n-power stable and robust operators coincide with normal operators, which directly model reversible  
transformations such as unitary operators in quantum computing.  
Proposition  
An operator is quasi-stable if and only if boundedness holds after rescaling, ensuring gradient stability in  
iterative ML optimizers.  
Proposition  
An operator is robust (paranormal equivalent) if and only if it ensures  
∥Tnx∥2≤⟨T2nx, x⟩ which corresponds to error contraction properties in algorithms.  
Proposition  
An operator is quasi-robust if and only if robustness is maintained in approximate computational models  
(important in edge AI and resource-limited systems).  
Proposition  
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INTERNATIONAL JOURNAL OF RESEARCH AND SCIENTIFIC INNOVATION (IJRSI)  
ISSN No. 2321-2705 | DOI: 10.51244/IJRSI |Volume XII Issue XI November 2025  
Class-A operators coincide with robust operators, providing a direct operator-theoretic foundation for robust  
deep learning architectures.  
APPLICATIONS IN COMPUTER TECHNOLOGY  
Machine Learning  
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Operator boundedness ensures convergence of deep networks.  
Quasi-stable operators model optimizers that work under adaptive scaling.  
Robust operator’s parallel normalization techniques (BatchNorm, LayerNorm).  
Quantum Computing  
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Fock space analogues imples quantum state space.  
n-power stable operators correspond to repeated quantum gates that remain unitary.  
Robustness ensures error-resilient quantum circuits.  
Signal and Image Processing  
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Composition operators correspond to filters.  
Stability ensures repeated filtering does not amplify noise.  
Quasi-robustness allows approximation in real-time processing.  
Data Transformation Pipelines  
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Operator theory explains why transformations must be bounded to avoid overflow.  
Ensures reliability of ETL (Extract, Transform, Load) processes in Big Data systems.  
CONCLUSION  
This work extends the classical operator-theoretic framework of Fock spaces to computational spaces in  
Computer Technology. By defining n-power stability, robustness, and quasi-properties in algorithmic contexts,  
we establish their role in ensuring convergence, boundedness, and reliability of computational systems. Future  
research may explore automated detection of stable/robust operators in deep learning and quantum circuit design.  
TERMINOLOGY  
The following mathematical words are converted to the computer technology words  
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Mathematical Fock space → Computational/Data spaces  
Posi/paranormal → Stability/Robustness in computing  
Theorems → Propositions with CS relevance  
Added applications in ML, Quantum Computing, DSP  
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INTERNATIONAL JOURNAL OF RESEARCH AND SCIENTIFIC INNOVATION (IJRSI)  
ISSN No. 2321-2705 | DOI: 10.51244/IJRSI |Volume XII Issue XI November 2025  
A block diagram/flowchart showing how “Operator Stability → Applications in ML/Quantum/DSP”  
REFERENCES  
1. Carswell B.J., Mac Cluer B.D., Schuster A., Composition Operator on the Fock Space, Acta Sci. Math.,  
2003.Cowen C., Mac Cluer B., Composition Operators on Spaces of Analytic Functions, CRC Press,  
1995.  
2. Goodfellow I., Bengio Y., Courville A., Deep Learning, MIT Press, 2016.Nielsen M., Chuang  
I., Quantum Computation and Quantum Information, Cambridge Univ. Press, 2010.  
3. Panayappan S., Senthilkumar D., Mohanraj R., Quasi-hyponormal Operators in Weighted Hardy Spaces,  
Int. J. Math. Analysis, 2008.  
4. Ueki S., Hilbert-Schmidt Weighted Operators on Fock Space, Int. J. Math. Analysis, 2007.  
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