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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2503.03188 |
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| _version_ | 1866910059696488448 |
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| author | Zhang, Xia Wei, Leilei Liu, Ming |
| author_facet | Zhang, Xia Wei, Leilei Liu, Ming |
| contents | In this paper, we first introduce the notion of the Laplace transform for an abstract-valued function from $[0, \infty)$ to a $\mathcal{T}_{\varepsilon, λ}$-complete random normed module $S$. Then, combining respective advantages of the $(\varepsilon, λ)$-topology and the locally $L^0$-convex topology on $S$, we prove the differentiability, Post-Widder inversion formula and uniqueness of such a Laplace transform. Second, based on the above work, we establish the Hille-Yosida theorem for an exponentially bounded $C$-semigroup on $S$, considering both the dense and nondense cases of the range of $C$, respectively, which extends and improves several important results. Finally, we also apply such a Laplace transform to abstract Cauchy problems in the random setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_03188 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Laplace transform approach to $C$-semigroups on a $\mathcal{T}_{\varepsilon, λ}$-complete random normed module Zhang, Xia Wei, Leilei Liu, Ming Functional Analysis 46H25, 45R05 In this paper, we first introduce the notion of the Laplace transform for an abstract-valued function from $[0, \infty)$ to a $\mathcal{T}_{\varepsilon, λ}$-complete random normed module $S$. Then, combining respective advantages of the $(\varepsilon, λ)$-topology and the locally $L^0$-convex topology on $S$, we prove the differentiability, Post-Widder inversion formula and uniqueness of such a Laplace transform. Second, based on the above work, we establish the Hille-Yosida theorem for an exponentially bounded $C$-semigroup on $S$, considering both the dense and nondense cases of the range of $C$, respectively, which extends and improves several important results. Finally, we also apply such a Laplace transform to abstract Cauchy problems in the random setting. |
| title | A Laplace transform approach to $C$-semigroups on a $\mathcal{T}_{\varepsilon, λ}$-complete random normed module |
| topic | Functional Analysis 46H25, 45R05 |
| url | https://arxiv.org/abs/2503.03188 |