Residual-Weighted Decomposition of Positive Operators
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866915644576890880 |
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| author | Tian, James |
| author_facet | Tian, James |
| contents | This paper investigates an iterative rank-one decomposition scheme for positive operators on a Hilbert space based on a residual-weighted congruence update. At each step the operator is compressed along a chosen unit vector while remaining inside the positive cone, and the resulting map defines a monotone dynamical system on the cone of positive operators. We prove that the associated residuals admit a canonical telescoping decomposition into rank-one terms and a limiting positive operator, and we identify this limit together with an exact energy identity expressing the defect between the initial and limiting operators as a convergent series of rank-one contributions. In the case where the iteration exhausts the operator, the residual directions form a Parseval frame for the natural range space, yielding a constructive procedure that produces Parseval frames without spectral calculus. We further solve the inverse problem by characterizing those decreasing chains with rank-one steps that arise from such dynamics via an intrinsic normalization condition involving the Moore-Penrose inverse. For trace-class operators we obtain a scalar energy identity and show that mild greedy or density conditions on the chosen directions guarantee exhaustion. An application to reproducing kernel Hilbert spaces illustrates the abstract results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_00167 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Residual-Weighted Decomposition of Positive Operators Tian, James Functional Analysis Primary: 47B65, Secondary: 42C15, 47A62 This paper investigates an iterative rank-one decomposition scheme for positive operators on a Hilbert space based on a residual-weighted congruence update. At each step the operator is compressed along a chosen unit vector while remaining inside the positive cone, and the resulting map defines a monotone dynamical system on the cone of positive operators. We prove that the associated residuals admit a canonical telescoping decomposition into rank-one terms and a limiting positive operator, and we identify this limit together with an exact energy identity expressing the defect between the initial and limiting operators as a convergent series of rank-one contributions. In the case where the iteration exhausts the operator, the residual directions form a Parseval frame for the natural range space, yielding a constructive procedure that produces Parseval frames without spectral calculus. We further solve the inverse problem by characterizing those decreasing chains with rank-one steps that arise from such dynamics via an intrinsic normalization condition involving the Moore-Penrose inverse. For trace-class operators we obtain a scalar energy identity and show that mild greedy or density conditions on the chosen directions guarantee exhaustion. An application to reproducing kernel Hilbert spaces illustrates the abstract results. |
| title | Residual-Weighted Decomposition of Positive Operators |
| topic | Functional Analysis Primary: 47B65, Secondary: 42C15, 47A62 |
| url | https://arxiv.org/abs/2512.00167 |