A regularity condition under which integral operators with operator-valued kernels are trace class
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866916350776049664 |
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| author | Zweck, John Latushkin, Yuri Gallo, Erika |
| author_facet | Zweck, John Latushkin, Yuri Gallo, Erika |
| contents | We study integral operators on the space of square-integrable functions from a compact set, $X$, to a separable Hilbert space, $H$. The kernel of such an operator takes values in the ideal of Hilbert-Schmidt operators on $H$. We establish regularity conditions on the kernel under which the associated integral operator is trace class. First, we extend Mercer's theorem to operator-valued kernels by proving that a continuous, nonnegative-definite, Hermitian symmetric kernel defines a trace class integral operator on $L^2(X;H)$ under an additional assumption. Second, we show that a general operator-valued kernel that is defined on a compact set and that is Hölder continuous with Hölder exponent greater than a half is trace class provided that the operator-valued kernel is essentially bounded as a mapping into the space of trace class operators on $H$. Finally, when $\dim H < \infty$, we show that an analogous result also holds for matrix-valued kernels on the real line, provided that an additional exponential decay assumption holds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_04794 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A regularity condition under which integral operators with operator-valued kernels are trace class Zweck, John Latushkin, Yuri Gallo, Erika Functional Analysis We study integral operators on the space of square-integrable functions from a compact set, $X$, to a separable Hilbert space, $H$. The kernel of such an operator takes values in the ideal of Hilbert-Schmidt operators on $H$. We establish regularity conditions on the kernel under which the associated integral operator is trace class. First, we extend Mercer's theorem to operator-valued kernels by proving that a continuous, nonnegative-definite, Hermitian symmetric kernel defines a trace class integral operator on $L^2(X;H)$ under an additional assumption. Second, we show that a general operator-valued kernel that is defined on a compact set and that is Hölder continuous with Hölder exponent greater than a half is trace class provided that the operator-valued kernel is essentially bounded as a mapping into the space of trace class operators on $H$. Finally, when $\dim H < \infty$, we show that an analogous result also holds for matrix-valued kernels on the real line, provided that an additional exponential decay assumption holds. |
| title | A regularity condition under which integral operators with operator-valued kernels are trace class |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2408.04794 |