A Study of orthogonal projectors and their fundamental properties in Hilbert spaces

Hillary Amonyela Isabu; Kevin Alwanyi Shilaviga; Violet Agala Khaveri

doi:10.51867/scimundi.6.2.21

Auteurs

Hillary Amonyela Isabu Masinde Muliro University of Science and Technology, Kakamega, Kenya https://orcid.org/0000-0001-9305-8233
Kevin Alwanyi Shilaviga Masinde Muliro University of Science and Technology, Kakamega, Kenya https://orcid.org/0009-0002-7702-9285
Violet Agala Khaveri Masinde Muliro University of Science and Technology, Kakamega, Kenya

DOI :

https://doi.org/10.51867/scimundi.6.2.21

Mots-clés :

Best approximation, closed convex set, direct sum decomposition, Hilbert space, orthogonal complement, orthogonal projector, projection theorem

Résumé

This paper presents a systematic study of orthogonal projectors in Hilbert spaces and examines their fundamental properties within the framework of functional analysis. Basic concepts related to projectors, orthogonal complements, convex subsets, and direct sums are reviewed and developed. The existence and uniqueness of best approximations from closed convex subsets of a Hilbert space are established using the parallelogram law. The projection theorem is then proved, showing that every element of a Hilbert space H can be uniquely decomposed into the sum of an element of a closed subspace M and an element of its orthogonal complement M⊥. As a consequence, the identity M⊥⊥ = M is established for every closed linear subspace M. Additional characterizations of dense subspaces through orthogonality conditions are presented. The results provide a theoretical foundation for applications in numerical analysis, linear algebra, approximation theory, Gram–Schmidt orthogonalization, QR decomposition, and orthogonal polynomial theory.

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Références

Baksalary, J. K., Baksalary, O. M., & Szulc, T. (2002). A property of orthogonal projectors. Linear Algebra and Its Applications, 354(1-3), 35-39. https://doi.org/10.1016/S0024-3795(02)00337-3 DOI: https://doi.org/10.1016/S0024-3795(02)00337-3

Berberian, S. K. (1999). Introduction to Hilbert space. American Mathematical Society.

Damle, A., Lin, L., & Ying, L. (2015). Compressed representation of Kohn-Sham orbitals via selected columns of the density matrix. Journal of Chemical Theory and Computation, 11(4), 1463-1469. https://doi.org/10.1021/ct500985f DOI: https://doi.org/10.1021/ct500985f

Friedrichs, K. O. (2012). Spectral theory of operators in Hilbert space. Springer. https://doi.org/10.1007/978-1-4612-6396-8 DOI: https://doi.org/10.1007/978-1-4612-6396-8

Graybill, F. A. (1976). Theory and application of the linear model. Duxbury Press.

Groß, J. (1999). On the product of orthogonal projectors. Linear Algebra and Its Applications, 289(1-3), 141-150. https://doi.org/10.1016/S0024-3795(98)10064-2 DOI: https://doi.org/10.1016/S0024-3795(98)10064-2

Sengupta, A. (2002). Orthogonal projections. Journal of Mathematical Analysis and Applications.

Simiyu, A. N., Shilaviga, A. K., & Olege, F. (2022). Characterization of orthogonal projectors. Journal of Advances in Mathematics and Computer Science, 37(3), 33-42. https://doi.org/10.9734/JAMCS/2022/v37i330440 DOI: https://doi.org/10.9734/jamcs/2022/v37i330440

Stahlecker, P., & Trenkler, G. (1993). Some further results on the use of proxy variables in prediction. The Review of Economics and Statistics, 75, 707-711. https://doi.org/10.2307/2110026 DOI: https://doi.org/10.2307/2110026

Vasilevski, N. L. (1998). $C^*$-algebras generated by orthogonal projections and their applications. Integral Equations and Operator Theory, 31(1), 113-132. https://doi.org/10.1007/BF01203459 DOI: https://doi.org/10.1007/BF01203459

Yanai, H., Takeuchi, K., & Takane, Y. (2011). Projection matrices, generalized inverse matrices, and singular value decomposition. Springer.

https://doi.org/10.1007/978-1-4419-9887-3 DOI: https://doi.org/10.1007/978-1-4419-9887-3