An extension result for $(LB)$-spaces and the surjectivity of tensorized mappings
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| Format: | Preprint |
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2023
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| _version_ | 1866917578777034752 |
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| author | Debrouwere, Andreas Neyt, Lenny |
| author_facet | Debrouwere, Andreas Neyt, Lenny |
| contents | We study an extension problem for continuous linear maps in the setting of $(LB)$-spaces. More precisely, we characterize the pairs $(E,Z)$, where $E$ is a locally complete space with a fundamental sequence of bounded sets and $Z$ is an $(LB)$-space, such that for every exact sequence of $(LB)$-spaces $$ 0 \rightarrow X \xrightarrowι Y \rightarrow Z \rightarrow 0$$ the map $$ L(Y,E) \to L(X, E), ~ T \mapsto T \circ ι$$ is surjective, meaning that each continuous linear map $X \to E$ can be extended to a continuous linear map $Y \to E$ via $ι$, under some mild conditions on $E$ or $Z$ (e.g. one of them is nuclear). We use our extension result to obtain sufficient conditions for the surjectivity of tensorized maps between Fréchet-Schwartz spaces. As an application of the latter, we study vector-valued Eidelheit type problems. Our work is inspired by and extends results of Vogt [24]. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2307_05245 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | An extension result for $(LB)$-spaces and the surjectivity of tensorized mappings Debrouwere, Andreas Neyt, Lenny Functional Analysis 46M18, 46A22, 46A32, 46A13, 46A63 We study an extension problem for continuous linear maps in the setting of $(LB)$-spaces. More precisely, we characterize the pairs $(E,Z)$, where $E$ is a locally complete space with a fundamental sequence of bounded sets and $Z$ is an $(LB)$-space, such that for every exact sequence of $(LB)$-spaces $$ 0 \rightarrow X \xrightarrowι Y \rightarrow Z \rightarrow 0$$ the map $$ L(Y,E) \to L(X, E), ~ T \mapsto T \circ ι$$ is surjective, meaning that each continuous linear map $X \to E$ can be extended to a continuous linear map $Y \to E$ via $ι$, under some mild conditions on $E$ or $Z$ (e.g. one of them is nuclear). We use our extension result to obtain sufficient conditions for the surjectivity of tensorized maps between Fréchet-Schwartz spaces. As an application of the latter, we study vector-valued Eidelheit type problems. Our work is inspired by and extends results of Vogt [24]. |
| title | An extension result for $(LB)$-spaces and the surjectivity of tensorized mappings |
| topic | Functional Analysis 46M18, 46A22, 46A32, 46A13, 46A63 |
| url | https://arxiv.org/abs/2307.05245 |