Generalized stability of isometries
Let \(X\), \(Y\) be two Banach spaces. The author introduces the following concept of isometry: the map \(f:X\rightarrow Y\) is an \((e,p)\)-isometry if \[ ||f(x)-f(y)|-|x-y||\leq e|x-y|^p \] holds for all \(x,y \in X\), and studies the \(p\)-stability of a couple of Banach spaces \(X,Y\) with respect to this concept. (The pair of Banach space \((X,Y)\) is \(p\)-stable if for every surjective \((e,p)\)-isometry \(f:X\rightarrow Y\) there exists a surjective isometry \(U:X\rightarrow Y\) satisfiying the inequality \(|f(x)-U(x)|\leq \delta(e)|x|^p\) for all \(x\in X\) and a certain fixed positive function \(\delta(e)\).) It is proved that every pair of Banach spaces \((X,Y)\) is \(p\)-stable for all \(p\in [0,1)\) although \((\mathbb{R}^2,\mathbb{R}^2)\) is a counterexample for \(1-\)stability. A superstability result is also proved for pair of finite dimensional Banach spaces and \(p>1\). The paper is very nice, and is connected to previous work of other people. Another interesting and easy to read paper on the subject is the following: \textit{R. Bhatia} and \textit{P. Šemrl} [Am. Math. Mon. 104, No.~6, 497-504 (1997; Zbl 0901.46016)].
scientific article; zbMATH DE number 1423637