TY - JOUR
AU - Karpenko, I.
AU - Zavarzina, O.
PY - 2022/06/27
Y2 - 2022/12/02
TI - Linear expand-contract plasticity of ellipsoids revisited
JF - Matematychni Studii
JA - Mat. Stud.
VL - 57
IS - 2
SE - Articles
DO - 10.30970/ms.57.2.192-201
UR - http://www.matstud.org.ua/ojs/index.php/matstud/article/view/319
SP - 192-201
AB - 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.
ER -