Generalized Multiple Operator Integrals for Operators with Finite Dimensions
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913911550246912 |
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| author | Chang, Shih-Yu |
| author_facet | Chang, Shih-Yu |
| contents | Multiple Operator Integrals (MOIs) have played a foundational role in operator theory and functional calculus, particularly for analyzing Hermitian matrices via spectral decomposition. Conventional MOIs rely on the assumption of self-adjointness, making them analytically tractable for computing Frechet derivatives, establishing trace formulas, and deriving commutator estimates. However, many problems in mathematics, science, and engineering involve matrices that are fundamentally non-Hermitian, arising in contexts such as numerical discretization of differential operators, signal processing, control systems, and non-Hermitian physics. These cases necessitate a more general framework for operator integration. In this paper, we propose and develop the theory of Generalized Multiple Operator Integrals (GMOIs), which extends MOI techniques to arbitrary matrices, including non-Hermitian and non-normal cases. We unify conventional MOIs under the perspective of the Spectral Mapping Theorem and present a rigorous construction of Generalized Double Operator Integrals (GDOIs) in finite-dimensional setting via Jordan decomposition of any matrix. We establish key algebraic properties, derive norm estimates, and prove continuity and Lipschitz-type perturbation formulas. Finally, we demonstrate how GMOIs can be used to compute matrix function derivatives in non-Hermitian environments, thus significantly broadening the applicability of MOI-based methods in both theoretical and practical domains. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_19971 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Generalized Multiple Operator Integrals for Operators with Finite Dimensions Chang, Shih-Yu Functional Analysis Operator Algebras Spectral Theory Multiple Operator Integrals (MOIs) have played a foundational role in operator theory and functional calculus, particularly for analyzing Hermitian matrices via spectral decomposition. Conventional MOIs rely on the assumption of self-adjointness, making them analytically tractable for computing Frechet derivatives, establishing trace formulas, and deriving commutator estimates. However, many problems in mathematics, science, and engineering involve matrices that are fundamentally non-Hermitian, arising in contexts such as numerical discretization of differential operators, signal processing, control systems, and non-Hermitian physics. These cases necessitate a more general framework for operator integration. In this paper, we propose and develop the theory of Generalized Multiple Operator Integrals (GMOIs), which extends MOI techniques to arbitrary matrices, including non-Hermitian and non-normal cases. We unify conventional MOIs under the perspective of the Spectral Mapping Theorem and present a rigorous construction of Generalized Double Operator Integrals (GDOIs) in finite-dimensional setting via Jordan decomposition of any matrix. We establish key algebraic properties, derive norm estimates, and prove continuity and Lipschitz-type perturbation formulas. Finally, we demonstrate how GMOIs can be used to compute matrix function derivatives in non-Hermitian environments, thus significantly broadening the applicability of MOI-based methods in both theoretical and practical domains. |
| title | Generalized Multiple Operator Integrals for Operators with Finite Dimensions |
| topic | Functional Analysis Operator Algebras Spectral Theory |
| url | https://arxiv.org/abs/2506.19971 |