Mercer's Theorem for Vector-Valued Reproducing Kernel Hilbert Spaces in Kaplansky-Hilbert Modules over $L_{\infty}(Ω)$
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915629772046336 |
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| author | Arziev, A. Orinbaev, K. Kudaybergenov. P. |
| author_facet | Arziev, A. Orinbaev, K. Kudaybergenov. P. |
| contents | The study presents a vector-valued extension of the classical Mercer theorem within the framework of reproducing kernel Hilbert spaces defined over Kaplansky-Hilbert modules associated with the algebra of essentially bounded measurable functions. The analysis focuses on a partial integral operator with a positive definite kernel depending on a measurable parameter, and establishes the equivalence of three fundamental properties: the completeness of the system of eigenfunctions in the corresponding vector-valued space, the injectivity of the adjoint embedding operator, and the existence of a pointwise spectral decomposition of the kernel in terms of the eigenvalues and eigenfunctions of a parameterized family of operators. The proof relies on constructing an isometric isomorphism between the Kaplansky-Hilbert module and the space of measurable sections of a Hilbert bundle, thereby reducing the problem to the application of the classical Mercer theorem on each fiber of the bundle. Furthermore, a formalism of vector-valued lifting is developed to guarantee the coherence of inner product structures between the original module and its bundle representation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_16981 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Mercer's Theorem for Vector-Valued Reproducing Kernel Hilbert Spaces in Kaplansky-Hilbert Modules over $L_{\infty}(Ω)$ Arziev, A. Orinbaev, K. Kudaybergenov. P. Functional Analysis 46H25, 46E30 The study presents a vector-valued extension of the classical Mercer theorem within the framework of reproducing kernel Hilbert spaces defined over Kaplansky-Hilbert modules associated with the algebra of essentially bounded measurable functions. The analysis focuses on a partial integral operator with a positive definite kernel depending on a measurable parameter, and establishes the equivalence of three fundamental properties: the completeness of the system of eigenfunctions in the corresponding vector-valued space, the injectivity of the adjoint embedding operator, and the existence of a pointwise spectral decomposition of the kernel in terms of the eigenvalues and eigenfunctions of a parameterized family of operators. The proof relies on constructing an isometric isomorphism between the Kaplansky-Hilbert module and the space of measurable sections of a Hilbert bundle, thereby reducing the problem to the application of the classical Mercer theorem on each fiber of the bundle. Furthermore, a formalism of vector-valued lifting is developed to guarantee the coherence of inner product structures between the original module and its bundle representation. |
| title | Mercer's Theorem for Vector-Valued Reproducing Kernel Hilbert Spaces in Kaplansky-Hilbert Modules over $L_{\infty}(Ω)$ |
| topic | Functional Analysis 46H25, 46E30 |
| url | https://arxiv.org/abs/2511.16981 |