Linear expand-contract plasticity of ellipsoids revisited
This work is aimed to describe linearly expand-contract plastic ellipsoids given via quadratic form of a bounded positively defined self-adjoint operator in terms of its spectrum. Let $Y$ be a metric space and $F\colon Y\to Y$ be a map. $F$ is called non-expansive if it does not increase distance b...
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Ivan Franko National University of Lviv
2022-06-01
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| Series: | Математичні Студії |
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| Online Access: | http://matstud.org.ua/ojs/index.php/matstud/article/view/319 |
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| author | I. Karpenko O. Zavarzina |
| author_facet | I. Karpenko O. Zavarzina |
| author_sort | I. Karpenko |
| collection | DOAJ |
| description | This work is aimed to describe linearly expand-contract plastic ellipsoids given via quadratic form of a bounded positively defined self-adjoint operator in terms of its spectrum.
Let $Y$ be a metric space and $F\colon Y\to Y$ be a map. $F$ is called non-expansive if it does not increase distance between points of the space $Y$. We say that a subset $M$ of a normed space $X$ is linearly expand-contract plastic (briefly an LEC-plastic) if every linear operator $T\colon X \to X$ whose restriction on $M$ is a non-expansive bijection from $M$ onto $M$ is an isometry on $M$.
In the paper, we consider a fixed separable infinite-dimensional Hilbert space $H$. We define an ellipsoid in $H$ as a set of the following form $E =\left\{x \in H\colon \left\langle x, Ax \right\rangle \le 1 \right\}$ where $A$ is a self-adjoint operator for which the following holds: $\inf_{\|x\|=1} \left\langle Ax,x\right\rangle >0$ and $\sup_{\|x\|=1} \left\langle Ax,x\right\rangle < \infty$.
We provide an example which demonstrates that if the spectrum of the generating operator $A$ has a non empty continuous part, then such ellipsoid is not linearly expand-contract plastic.
In this work, we also proof that an ellipsoid is linearly expand-contract plastic if and only if the spectrum of the generating operator $A$ has empty continuous part and every subset of eigenvalues of the operator $A$ that consists of more than one element either has a maximum of finite multiplicity or has a minimum of finite multiplicity. |
| format | Article |
| id | doaj-art-96e14debabc04878887f3dcfbe519f80 |
| institution | Kabale University |
| issn | 1027-4634 2411-0620 |
| language | deu |
| publishDate | 2022-06-01 |
| publisher | Ivan Franko National University of Lviv |
| record_format | Article |
| series | Математичні Студії |
| spelling | doaj-art-96e14debabc04878887f3dcfbe519f802025-08-20T03:28:41ZdeuIvan Franko National University of LvivМатематичні Студії1027-46342411-06202022-06-0157219220110.30970/ms.57.2.192-201319Linear expand-contract plasticity of ellipsoids revisitedI. Karpenko0O. Zavarzina1B. Verkin Institute for Low Temperature Physics and Engineering Kharkiv, Ukraine Universitat Wien, Oskar-Morgenstern-Platz 1 Wien, AustriaDepartment of Mathematics and Informatics V. N. Karazin Kharkiv National University Kharkiv, UkraineThis work is aimed to describe linearly expand-contract plastic ellipsoids given via quadratic form of a bounded positively defined self-adjoint operator in terms of its spectrum. Let $Y$ be a metric space and $F\colon Y\to Y$ be a map. $F$ is called non-expansive if it does not increase distance between points of the space $Y$. We say that a subset $M$ of a normed space $X$ is linearly expand-contract plastic (briefly an LEC-plastic) if every linear operator $T\colon X \to X$ whose restriction on $M$ is a non-expansive bijection from $M$ onto $M$ is an isometry on $M$. In the paper, we consider a fixed separable infinite-dimensional Hilbert space $H$. We define an ellipsoid in $H$ as a set of the following form $E =\left\{x \in H\colon \left\langle x, Ax \right\rangle \le 1 \right\}$ where $A$ is a self-adjoint operator for which the following holds: $\inf_{\|x\|=1} \left\langle Ax,x\right\rangle >0$ and $\sup_{\|x\|=1} \left\langle Ax,x\right\rangle < \infty$. We provide an example which demonstrates that if the spectrum of the generating operator $A$ has a non empty continuous part, then such ellipsoid is not linearly expand-contract plastic. In this work, we also proof that an ellipsoid is linearly expand-contract plastic if and only if the spectrum of the generating operator $A$ has empty continuous part and every subset of eigenvalues of the operator $A$ that consists of more than one element either has a maximum of finite multiplicity or has a minimum of finite multiplicity.http://matstud.org.ua/ojs/index.php/matstud/article/view/319non-expansive map; ellipsoid; linearly expand-contract plastic space |
| spellingShingle | I. Karpenko O. Zavarzina Linear expand-contract plasticity of ellipsoids revisited Математичні Студії non-expansive map; ellipsoid; linearly expand-contract plastic space |
| title | Linear expand-contract plasticity of ellipsoids revisited |
| title_full | Linear expand-contract plasticity of ellipsoids revisited |
| title_fullStr | Linear expand-contract plasticity of ellipsoids revisited |
| title_full_unstemmed | Linear expand-contract plasticity of ellipsoids revisited |
| title_short | Linear expand-contract plasticity of ellipsoids revisited |
| title_sort | linear expand contract plasticity of ellipsoids revisited |
| topic | non-expansive map; ellipsoid; linearly expand-contract plastic space |
| url | http://matstud.org.ua/ojs/index.php/matstud/article/view/319 |
| work_keys_str_mv | AT ikarpenko linearexpandcontractplasticityofellipsoidsrevisited AT ozavarzina linearexpandcontractplasticityofellipsoidsrevisited |