Linear expand-contract plasticity of ellipsoids revisited (Q2086080)

Let \(M\) be a subset of a normed space \(X\). Recall that \(M\) is said to be \textit{linearly expand-contract plastic} (LEC-plastic) if every linear operator \(T:X\rightarrow X\) whose restriction to \(M\) is a non-expansive bijection from \(M\) onto \(M\) is an isometry on~$ M$. Ellipsoids in separable Hilbert spaces of the form \[ E=\left\{\sum x_ne_n\in H: \sum\left|\frac{x_n}{a(n)}\right|^2\le1 \right\}, \] where \(a(n)>0\), are known to provide examples of sets which are not LEC-plastic. The authors take this investigation further and study more general ellipsoids of the form \[ E=\left\{x\in H:\langle x,Ax\rangle\le1\right\}, \] where \(A\) is a self-adjoint operator such that \(\inf_{\Vert x\Vert =1}\langle x,Ax\rangle>0\) and \(\sup_{\Vert x\Vert =1}\langle x,Ax\rangle<\infty\). They provide a full characterization of the conditions under which such ellipsoids are LEC-plastic. Namely, the main result of the paper is the following. {Theorem}. Let \(A\) be a bounded self-adjoint operator. Then the ellipsoid generated by \(A\) is LEC-plastic if and only if the following two conditions hold: \begin{itemize} \item[(1)] The continuous part of the spectrum of \(A\) is empty; \item[(2)] every subset of eigenvalues of the operaor $A$ that consists of more than one element either has a maximum of finite multiplicity or has a minimum of finite multiplicity. \end{itemize}