Generalized Multiple Operator Integrals and Perturbation Theory for Operators with Continuous Spectra
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915418346618880 |
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| author | Chang, Shih-Yu |
| author_facet | Chang, Shih-Yu |
| contents | Operators with continuous spectra naturally arise in spectral theory, quantum mechanics, automorphic forms, and noncommutative geometry. However, analyzing such operators, particularly in the non-selfadjoint setting, remains challenging due to spectral instability and the lack of an orthonormal basis. This work advances the theory of Multiple Operator Integrals (MOIs) by developing a unified framework for generalized MOIs (GMOIs) associated with general (non-normal, non-selfadjoint) operators possessing continuous spectra. Building on prior work in Generalized Double Operator Integrals (GDOIs) and finite dimensional GMOIs, we extend the theory to include: the formulation of GMOIs in the continuous spectrum setting, their algebraic structure, continuity properties, norm and Lipschitz estimates, and a perturbation formula that generalizes classical results. As a key application, we derive a Krein-type spectral shift formula for GDOIs in the continuous spectrum setting and further extend it to arbitrary-order approximations. These contributions provide a foundation for broader developments in spectral theory, operator algebras, noncommutative geometry, and noncommutative analysis. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_23049 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Generalized Multiple Operator Integrals and Perturbation Theory for Operators with Continuous Spectra Chang, Shih-Yu Functional Analysis Operator Algebras Operators with continuous spectra naturally arise in spectral theory, quantum mechanics, automorphic forms, and noncommutative geometry. However, analyzing such operators, particularly in the non-selfadjoint setting, remains challenging due to spectral instability and the lack of an orthonormal basis. This work advances the theory of Multiple Operator Integrals (MOIs) by developing a unified framework for generalized MOIs (GMOIs) associated with general (non-normal, non-selfadjoint) operators possessing continuous spectra. Building on prior work in Generalized Double Operator Integrals (GDOIs) and finite dimensional GMOIs, we extend the theory to include: the formulation of GMOIs in the continuous spectrum setting, their algebraic structure, continuity properties, norm and Lipschitz estimates, and a perturbation formula that generalizes classical results. As a key application, we derive a Krein-type spectral shift formula for GDOIs in the continuous spectrum setting and further extend it to arbitrary-order approximations. These contributions provide a foundation for broader developments in spectral theory, operator algebras, noncommutative geometry, and noncommutative analysis. |
| title | Generalized Multiple Operator Integrals and Perturbation Theory for Operators with Continuous Spectra |
| topic | Functional Analysis Operator Algebras |
| url | https://arxiv.org/abs/2507.23049 |