Representation of sequence classes by operator ideals
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914704432037888 |
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| author | Botelho, Geraldo Santiago, Ariel S. |
| author_facet | Botelho, Geraldo Santiago, Ariel S. |
| contents | It is well known that weakly $p$-summable sequences in a Banach space $E$ are associated to bounded operators from $\ell_{p^*}$ to $E$, and unconditionally $p$-summable sequences in $E$ are associated to compact operators from $\ell_{p^*}$ to $E$. Generalizing these results to a quite wide environment, we characterize the classes of Banach spaces-valued sequences that are associated to (or represented by) some Banach operator ideal. Using these characterizations, we decide, among all sequence classes that usually appear in the literature, which are represented by some Banach operator ideal and which are not. Moreover, to each class that is represented by some Banach operator ideal, we show an ideal that represents it. Illustrative examples and additional applications are provided. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_04496 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Representation of sequence classes by operator ideals Botelho, Geraldo Santiago, Ariel S. Functional Analysis 46A45, 46B45, 47B10, 47L20 It is well known that weakly $p$-summable sequences in a Banach space $E$ are associated to bounded operators from $\ell_{p^*}$ to $E$, and unconditionally $p$-summable sequences in $E$ are associated to compact operators from $\ell_{p^*}$ to $E$. Generalizing these results to a quite wide environment, we characterize the classes of Banach spaces-valued sequences that are associated to (or represented by) some Banach operator ideal. Using these characterizations, we decide, among all sequence classes that usually appear in the literature, which are represented by some Banach operator ideal and which are not. Moreover, to each class that is represented by some Banach operator ideal, we show an ideal that represents it. Illustrative examples and additional applications are provided. |
| title | Representation of sequence classes by operator ideals |
| topic | Functional Analysis 46A45, 46B45, 47B10, 47L20 |
| url | https://arxiv.org/abs/2402.04496 |