Range strongly exposing operators between Banach spaces
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arXiv
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| Autori principali: | , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866913784294014976 |
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| author | Choi, Geunsu del Río, Helena Fovelle, Audrey Jung, Mingu Martín, Miguel |
| author_facet | Choi, Geunsu del Río, Helena Fovelle, Audrey Jung, Mingu Martín, Miguel |
| contents | We introduce a new class of bounded linear operators, called range strongly exposing (RSE) operators, which form a natural intermediate class: weaker than Bourgain's absolutely strongly exposing operators, yet stronger than both uniquely quasi norm-attaining and classical norm-attaining operators. Several foundational results on norm-attaining operators are extended to the RSE setting. Among our main contributions, we establish that for every infinite-dimensional Banach space $Y$, there exists a Banach space $X$ such that the RSE operators from $X$ to $Y$ are not dense - an RSE analogue of a result by Acosta (1999) which applies only when $Y$ is strictly convex. We also show that the Radon-Nikodým property of $Y$ is sufficient to obtain that RSE operators from $L_1(μ)$ to $Y$ are dense and that this is also necessary if $μ$ is not purely atomic. This extends and sharpens classical results by Uhl (1976). As a consequence, we prove that the set of RSE operators between $L_1(μ)$ and $L_1 (ν)$ is dense if and only if at least one of the measures $μ$ or $ν$ is purely atomic, in contrast with the classical result by Iwanik (1979) which guarantees the denseness of norm-attaining operators for all measures $μ$ and $ν$. We also prove that weakly compact operators from any $C(K)$ space can always be approximated by (weakly compact) RSE operators, thereby strengthening a result of Schachermayer (1983). Additionally, we present several improvements of more recent results concerning finite-rank operators and $Γ$-flat operators which give, in particular, RSE versions of classical results on compact operators by Johnson-Wolfe (1979). Finally, we discuss RSE counterparts of results by Zizler and Lindenstrauss on the denseness of operators whose adjoints attain their norm. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_18581 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Range strongly exposing operators between Banach spaces Choi, Geunsu del Río, Helena Fovelle, Audrey Jung, Mingu Martín, Miguel Functional Analysis 46B04 (Primary) 46B20, 46B22, 46B25, 47B01, 47B07 (Secondary) We introduce a new class of bounded linear operators, called range strongly exposing (RSE) operators, which form a natural intermediate class: weaker than Bourgain's absolutely strongly exposing operators, yet stronger than both uniquely quasi norm-attaining and classical norm-attaining operators. Several foundational results on norm-attaining operators are extended to the RSE setting. Among our main contributions, we establish that for every infinite-dimensional Banach space $Y$, there exists a Banach space $X$ such that the RSE operators from $X$ to $Y$ are not dense - an RSE analogue of a result by Acosta (1999) which applies only when $Y$ is strictly convex. We also show that the Radon-Nikodým property of $Y$ is sufficient to obtain that RSE operators from $L_1(μ)$ to $Y$ are dense and that this is also necessary if $μ$ is not purely atomic. This extends and sharpens classical results by Uhl (1976). As a consequence, we prove that the set of RSE operators between $L_1(μ)$ and $L_1 (ν)$ is dense if and only if at least one of the measures $μ$ or $ν$ is purely atomic, in contrast with the classical result by Iwanik (1979) which guarantees the denseness of norm-attaining operators for all measures $μ$ and $ν$. We also prove that weakly compact operators from any $C(K)$ space can always be approximated by (weakly compact) RSE operators, thereby strengthening a result of Schachermayer (1983). Additionally, we present several improvements of more recent results concerning finite-rank operators and $Γ$-flat operators which give, in particular, RSE versions of classical results on compact operators by Johnson-Wolfe (1979). Finally, we discuss RSE counterparts of results by Zizler and Lindenstrauss on the denseness of operators whose adjoints attain their norm. |
| title | Range strongly exposing operators between Banach spaces |
| topic | Functional Analysis 46B04 (Primary) 46B20, 46B22, 46B25, 47B01, 47B07 (Secondary) |
| url | https://arxiv.org/abs/2503.18581 |