Decomposition of Idempotent Operators on Hilbert C-Modules

This study advances the application of the generalized Halmos’ two projections theorem to idempotent operators on Hilbert C*-modules through a comprehensive study of sums involving adjointable idempotents and their adjoints. We establish fundamental properties including the closedness, orthogonal co...

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Bibliographic Details
Published inMathematics (Basel) Vol. 13; no. 15; p. 2378
Main Author Luo, Wei
Format Journal Article
LanguageEnglish
Published Basel MDPI AG 01.08.2025
Subjects
Adjoints
Algebra
Decomposition
factorization of operators
Hilbert C-modules
idempotents
linear combinations
Modules
Moore–Penrose inverse
Norms
operator decomposition
Operators (mathematics)
Theorems
Web services
Iran
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Summary:This study advances the application of the generalized Halmos’ two projections theorem to idempotent operators on Hilbert C*-modules through a comprehensive study of sums involving adjointable idempotents and their adjoints. We establish fundamental properties including the closedness, orthogonal complementability, Moore–Penrose inverses, and spectral norms of such sums. For arbitrary (not necessarily adjointable) idempotent operators that admit a decomposition into linear combinations or products of two idempotents, we derive explicit representations for all such decompositions. A numerical example is given to show how our main theorem allows for the decomposition of idempotent matrices into linear combinations of two idempotent matrices, and two concrete examples on Hilbert C*-modules validate the theoretical significance of our framework.
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ISSN:2227-7390
2227-7390
DOI:10.3390/math13152378