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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2512.22497 |
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| _version_ | 1866912791125819392 |
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| author | Monika Dey, Priyadarshi Easley, Zachary |
| author_facet | Monika Dey, Priyadarshi Easley, Zachary |
| contents | This paper investigates the projection operators that lie in the algebra generated by powers of an $n$-potent operator $T$ on a complex Banach space, where $T^n = T$. We give a complete description of all projections in the algebra $\operatorname{comb}(T) = \text{span}\{T, T^2, \dots, T^{n-1}\}$, and prove that each such projection is uniquely determined by, and in bijection with, a subset of the nonzero spectrum of $T$. As a consequence, the family of projections in $\operatorname{comb}(T)$ forms a Boolean algebra isomorphic to the power set of $σ(T)\setminus\{0\}$. We also establish a spectral decomposition for $n$-potent operators in terms of their Riesz projections and derive explicit formulas for the associated Riesz projections using resolvent expansions. We give an illustration of the theory for $5$-potent operators, which highlights the algebraic and spectral structure of finite-order operators on Banach spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_22497 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Projections in the Algebra generated by an n-Potent Operator Monika Dey, Priyadarshi Easley, Zachary Functional Analysis 46 This paper investigates the projection operators that lie in the algebra generated by powers of an $n$-potent operator $T$ on a complex Banach space, where $T^n = T$. We give a complete description of all projections in the algebra $\operatorname{comb}(T) = \text{span}\{T, T^2, \dots, T^{n-1}\}$, and prove that each such projection is uniquely determined by, and in bijection with, a subset of the nonzero spectrum of $T$. As a consequence, the family of projections in $\operatorname{comb}(T)$ forms a Boolean algebra isomorphic to the power set of $σ(T)\setminus\{0\}$. We also establish a spectral decomposition for $n$-potent operators in terms of their Riesz projections and derive explicit formulas for the associated Riesz projections using resolvent expansions. We give an illustration of the theory for $5$-potent operators, which highlights the algebraic and spectral structure of finite-order operators on Banach spaces. |
| title | Projections in the Algebra generated by an n-Potent Operator |
| topic | Functional Analysis 46 |
| url | https://arxiv.org/abs/2512.22497 |