{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,24]],"date-time":"2026-02-24T10:20:09Z","timestamp":1771928409522,"version":"3.50.1"},"reference-count":26,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2026,2,14]],"date-time":"2026-02-14T00:00:00Z","timestamp":1771027200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>We study the Moore\u2013Penrose inverse of idempotent operators on Hilbert C*-modules. First, we extend the computation of the Moore\u2013Penrose inverse of an idempotent operator and its difference from the range projection to this setting. This leads to an explicit formula for the Moore\u2013Penrose inverse of the sum of an idempotent and its adjoint. Furthermore, we establish a decomposition of an idempotent operator into a product of two commuting idempotents and clarify the relationship between their Moore\u2013Penrose inverses and that of the original operator. We also analyze spectral properties and operator norms, obtaining sharp norm bounds.<\/jats:p>","DOI":"10.3390\/axioms15020141","type":"journal-article","created":{"date-parts":[[2026,2,16]],"date-time":"2026-02-16T08:38:39Z","timestamp":1771231119000},"page":"141","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["The Moore\u2013Penrose Inverse and Product Decomposition of Idempotent Operators on Hilbert C*-Modules"],"prefix":"10.3390","volume":"15","author":[{"given":"Wei","family":"Luo","sequence":"first","affiliation":[{"name":"Department of Statistics and Mathematics, Shanghai Lixin University of Accounting and Finance, Shanghai 201209, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2026,2,14]]},"reference":[{"key":"ref_1","first-page":"394","article-title":"On the Reciprocal of the General Algebraic Matrix","volume":"26","author":"Moore","year":"1920","journal-title":"Bull. Am. Math. Soc."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"406","DOI":"10.1017\/S0305004100030401","article-title":"A Generalized Inverse for Matrices","volume":"51","author":"Penrose","year":"1955","journal-title":"Proc. Camb. Philos. Soc."},{"key":"ref_3","first-page":"845","article-title":"On Matrices Whose Moore\u2013Penrose Inverse is Idempotent","volume":"68","author":"Baksalary","year":"2020","journal-title":"Linear Multilinear Algebra"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"681","DOI":"10.1080\/03081080902778222","article-title":"Core Inverse of Matrices","volume":"58","author":"Baksalary","year":"2010","journal-title":"Linear Multilinear Algebra"},{"key":"ref_5","first-page":"691","article-title":"The Weak Core Inverse","volume":"94","author":"Ferreyra","year":"2020","journal-title":"Aequat. Math."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"756","DOI":"10.1080\/03081087.2018.1432546","article-title":"On DMP Inverses and m-EP Elements in Rings","volume":"67","author":"Zhu","year":"2018","journal-title":"Linear Multilinear Algebra"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"310","DOI":"10.1007\/s40314-024-02838-9","article-title":"Similarities Between the Numerical Range of an Operator and of Certain Generalized Inverses","volume":"43","author":"Stankov","year":"2024","journal-title":"Comput. Appl. Math."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Stanojevi\u0107, V., Kazakovtsev, L., Stanimirovi\u0107, P.S., Rezova, N., and Shkaberina, G. (2022). Calculating the Moore\u2013Penrose Generalized Inverse on Massively Parallel Systems. Algorithms, 15.","DOI":"10.3390\/a15100348"},{"key":"ref_9","unstructured":"Xavier, G.M.T., Nava, L.M.F., Casta\u00f1eda, F.G., and Cadenas, J.A.M. (2023, January 25\u201327). FPGA Simulation for Computing Pseudoinverse Matrices. Proceedings of the 2023 20th International Conference on Electrical Engineering, Computing Science and Automatic Control (CCE), Mexico City, Mexico."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"254","DOI":"10.1016\/j.neucom.2022.08.036","article-title":"A Robust Noise Tolerant Zeroing Neural Network for Solving Time-Varying Linear Matrix Equations","volume":"508","author":"Gerontitis","year":"2022","journal-title":"Neurocomputing"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1106","DOI":"10.1049\/sil2.12156","article-title":"An Efficient Second-Order Neural Network Model for Computing the Moore\u2013Penrose Inverse of Matrices","volume":"16","author":"Li","year":"2022","journal-title":"IET Signal Process."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Stanimirovi\u0107, P.S., Wei, Y., Li, S., Gerontitis, D., and Cao, X. (2025). Generalized Matrix Inversion: A Machine Learning Approach, Springer. Preprint\/Book Chapter.","DOI":"10.1007\/978-3-032-01493-1"},{"key":"ref_13","first-page":"124957","article-title":"Computing the Moore-Penrose Inverse Using Its Error Bounds","volume":"371","author":"Roy","year":"2020","journal-title":"Appl. Math. Comput."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1007\/s40314-019-0893-6","article-title":"Perturbation Theory for Moore\u2013Penrose Inverse of Tensor via Einstein Product","volume":"38","author":"Ma","year":"2019","journal-title":"Comput. Appl. Math."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"4163","DOI":"10.1007\/s12190-023-01920-5","article-title":"Perturbations of Moore-Penrose Inverse and Dual Moore-Penrose Generalized Inverse","volume":"69","author":"Cui","year":"2023","journal-title":"J. Appl. Math. Comput."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1023\/A:1009729204027","article-title":"Geometrical Aspects of Hilbert C*-Modules","volume":"3","author":"Frank","year":"1999","journal-title":"Positivity"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Manuilov, V.M., and Troitsky, E.V. (2005). Hilbert C*-Modules, American Mathematical Society. Translations of Mathematical Monographs.","DOI":"10.1090\/mmono\/226"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"7","DOI":"10.1007\/s11117-022-00960-8","article-title":"Extensions of the Hilbert-Multi-Norm in Hilbert C*-Modules","volume":"27","author":"Abedi","year":"2023","journal-title":"Positivity"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"381","DOI":"10.1090\/S0002-9947-1969-0251519-5","article-title":"Two Subspaces","volume":"144","author":"Halmos","year":"1969","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_20","first-page":"1099","article-title":"Common Complements of Two Subspaces and an Answer to Gro\u00df\u2019s Question","volume":"49","author":"Deng","year":"2006","journal-title":"Acta Math. Sin."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Lance, E.C. (1995). Hilbert C*-Modules: A Toolkit for Operator Algebraists, Cambridge University Press.","DOI":"10.1017\/CBO9780511526206"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"992","DOI":"10.1016\/j.laa.2007.08.035","article-title":"Positive Semi-Definite Matrices of Adjointable Operators on Hilbert C*-Modules","volume":"428","author":"Xu","year":"2008","journal-title":"Linear Algebra Appl."},{"key":"ref_23","first-page":"91","article-title":"Range Projections of Idempotents in C*-Algebras","volume":"34","author":"Koliha","year":"2001","journal-title":"Demonstr. Math."},{"key":"ref_24","first-page":"201","article-title":"Closed Range and Nonclosed Range Adjointable Operators on Hilbert C*-Modules","volume":"14","author":"Vosough","year":"2020","journal-title":"Oper. Matrices"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Wang, G., Wei, Y., and Qiao, S. (2018). Generalized Inverses: Theory and Computations, Springer. [2nd ed.].","DOI":"10.1007\/978-981-13-0146-9"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Luo, W. (2025). Decomposition of Idempotent Operators on Hilbert C*-Modules. Mathematics, 13.","DOI":"10.3390\/math13152378"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/15\/2\/141\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,2,24]],"date-time":"2026-02-24T09:28:54Z","timestamp":1771925334000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/15\/2\/141"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,2,14]]},"references-count":26,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2026,2]]}},"alternative-id":["axioms15020141"],"URL":"https:\/\/doi.org\/10.3390\/axioms15020141","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,2,14]]}}}