Shape-preserving projections in low-dimensional settings and the q -monotone case
Let $P: X \rightarrow V$ be a projection from a real Banach space $X$ onto a subspace $V$ and let $S \subset X$. In this setting, one can ask if $S$ is left invariant under $P$, i.e., if $PS \subset S$. If $V$ is finite-dimensional and $S$ is a cone with particular structure, then the occurrence o...
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| Datum: | 2012 |
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| Hauptverfasser: | , , , |
| Format: | Artikel |
| Sprache: | Englisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
2012
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| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/2607 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| Zusammenfassung: | Let $P: X \rightarrow V$ be a projection from a real Banach space $X$ onto a subspace $V$ and let $S \subset X$.
In this setting, one can ask if $S$ is left invariant under $P$, i.e., if $PS \subset S$.
If $V$ is finite-dimensional and $S$ is a cone with particular structure, then the occurrence of the imbedding $PS \subset S$ can be characterized through a geometric description.
This characterization relies heavily on the structure of $S$, or, more specifically, on the structure of the cone $S^{*}$ dual to $S$.
In this paper, шє remove the structural assumptions on $S^{*}$ and characterize the cases where $PS \subset S$.
We note that the (so-called) $q$-monotone shape forms a cone which (lacks structure and thus) serves as an application for our characterization. |
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