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Decomposition of Idempotent Operators on Hilbert C *-Modules

Wei Luo ()
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Wei Luo: Department of Statistics and Mathematics, Shanghai Lixin University of Accounting and Finance, Shanghai 201209, China

Mathematics, 2025, vol. 13, issue 15, 1-32

Abstract: 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.

Keywords: idempotents; linear combinations; factorization of operators; Hilbert C *-modules; operator decomposition; Moore–Penrose inverse; spectral norm (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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