Totally bounded sets in the absolute weak topology
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866917692780314624 |
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| author | Ardakani, Halimeh Chen, Jin Xi |
| author_facet | Ardakani, Halimeh Chen, Jin Xi |
| contents | In this paper, almost Dunford-Pettis operators with ranges in $c_0$ are used to identify totally bounded sets in the absolute weak topology. That is, a bounded subset $A$ of a Banach lattice $E$ is $|σ|(E,E^\prime)$-totally bounded if and only if $T(A)\subset c_0$ is relatively compact for every almost Dunford-Pettis operator $T:E\to{c_0}$. As an application, we show that for two Banach lattices $E$ and $F$ every positive operator from $E$ to $F$ dominated by a PL-compact operator is PL-compact if and only if either the norm of $E^{\,\prime}$ is order continuous or every order interval in $F$ is $|σ|(F,F^\prime)$-totally bounded. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_09013 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Totally bounded sets in the absolute weak topology Ardakani, Halimeh Chen, Jin Xi Functional Analysis Primary 46B42, Secondary 46B50, 47B65 In this paper, almost Dunford-Pettis operators with ranges in $c_0$ are used to identify totally bounded sets in the absolute weak topology. That is, a bounded subset $A$ of a Banach lattice $E$ is $|σ|(E,E^\prime)$-totally bounded if and only if $T(A)\subset c_0$ is relatively compact for every almost Dunford-Pettis operator $T:E\to{c_0}$. As an application, we show that for two Banach lattices $E$ and $F$ every positive operator from $E$ to $F$ dominated by a PL-compact operator is PL-compact if and only if either the norm of $E^{\,\prime}$ is order continuous or every order interval in $F$ is $|σ|(F,F^\prime)$-totally bounded. |
| title | Totally bounded sets in the absolute weak topology |
| topic | Functional Analysis Primary 46B42, Secondary 46B50, 47B65 |
| url | https://arxiv.org/abs/2406.09013 |