A Study of a Basic Sufficient Condition for the Compactness of Linear Operators on Banach Spaces
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Banach Spaces., Compact Operators., Total Boundedness., Finite Dimensionality., Completeness.
Abstract
This work examines conditions for the compactness of linear operators in Banach spaces, a key question in functional analysis with broad applications. Compactness ensures that bounded sets are mapped to relatively compact sets, making it a fundamental tool in the study of operators on infinite-dimensional spaces. This paper provides a detailed investigation of three conditions ensuring compactness: total boundedness, finite dimensionality, and completeness. It addresses a significant gap in the literature and provides a sound theoretical framework.
This paper aims to (1) explain the connection between finite dimensionality and total boundedness as conditions for compactness, (2) present unified sufficient conditions for the compactness of linear operators with proofs, and (3) offer new insights into operator theory for broader mathematical applications. This study employs advanced functional analytic techniques to deduce and validate these well-founded conditions.
This work addresses gaps in compact operator theory, with implications for quantum physics, differential equations, and numerical analysis. By enhancing the understanding of Banach spaces and operator theory, this study may inspire further exploration of their properties.
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References
. Tao, J., Yang, D., Yuan, W., & Zhang, Y. (2021). Compactness characterizations of commutators on ball Banach function spaces. Potential Analysis, 1-35.
. Kim, J. M., Lee, K. Y., & Zheng, B. (2020). Banach compactness and Banach nuclear operators. Results in Mathematics, 75, 1-28.
. Nowakowski, E. (2021). Some compact operators on the Hahn space. Scientific Research Communications, 1(1), 18-33.
. Ledzewicz, U., & Nowakowski, A. (2020). Necessary and sufficient conditions for optimality for nonlinear control problems in Banach spaces. Optimal Control of Differential Equations (pp. 195-216). CRC Press.
. Jung, M., Martín, M., & Zoca, A. R. (2023). Residuality in the set of norm-attaining operators between Banach spaces. Journal of Functional Analysis, 284(2), 109746.
. Fourie, J. H., & Zegosek, E. D. (2023). Summability of sequences and extensions of operator ideals. Quaestiones Mathematicae, 46(3), 599-619.
. Koshino, K. (2024). Compactness of averaging operators on Banach function spaces. Banach Journal of Mathematical Analysis, 18(4), 73-85.
. Ishikawa, I. (2023). Bounded composition operators on functional quasi-Banach spaces and stability of dynamical systems. Advances in Mathematics, 424, 109048.
. van Velthoven, J. T., & Voigtlaender, F. (2022). Coorbit spaces associated to quasi-Banach function spaces and their molecular decomposition. arXiv preprint arXiv:2203.07959.
. Mastyło, M., & Silva, E. B. (2020). Interpolation of compact bilinear operators. Bulletin of Mathematical Sciences, 10(02), 2050002.
. Bachir, M. (2023). A strong Bishop-Phelps property and a new class of Banach spaces with the property $(A) $ of Lindenstrauss. arXiv preprint arXiv:2304.12611.
. García, D., Maestre, M., Martín, M., & Roldán, Ó. (2021). On the compact operators case of the Bishop–Phelps–Bollobás property for numerical radius. Results in Mathematics, 76(3), 122.
. Mebarki, L., Messirdi, B., & Benharrat, M. (2021). Quasi-compactness of linear operators on Banach spaces: New properties and application to Markov chains. Asian-European Journal of Mathematics, 14(04), 2150060.
. Fukushima, H., Yajima, H., Sugimura, K., Hosokawa, T., Omukai, K., & Matsumoto, T. (2020). Star cluster formation and cloud dispersal by radiative feedback: dependence on metallicity and compactness. Monthly Notices of the Royal Astronomical Society, 497(3), 3830-3845.
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