Linear expand-contract plasticity of ellipsoids revisited

  • I. Karpenko B. Verkin Institute for Low Temperature Physics and Engineering Kharkiv, Ukraine Universitat Wien, Oskar-Morgenstern-Platz 1 Wien, Austria
  • O. Zavarzina Department of Mathematics and Informatics V. N. Karazin Kharkiv National University Kharkiv, Ukraine

Анотація

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.

Біографії авторів

I. Karpenko, B. Verkin Institute for Low Temperature Physics and Engineering Kharkiv, Ukraine Universitat Wien, Oskar-Morgenstern-Platz 1 Wien, Austria

B. Verkin Institute for Low Temperature Physics and Engineering
Kharkiv, Ukraine
Universitat Wien, Oskar-Morgenstern-Platz 1 Wien, Austria

O. Zavarzina, Department of Mathematics and Informatics V. N. Karazin Kharkiv National University Kharkiv, Ukraine

Department of Mathematics and Informatics
V. N. Karazin Kharkiv National University
Kharkiv, Ukraine

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Опубліковано
2022-06-27
Як цитувати
Karpenko, I., & Zavarzina, O. (2022). Linear expand-contract plasticity of ellipsoids revisited. Математичні студії, 57(2), 192-201. https://doi.org/10.30970/ms.57.2.192-201
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