Mappings approximately preserving orthogonality in normed spaces
A vector \(x\) in a real or complex normed space \(X\) is called orthogonal in the Birkhoff-James sense to a vector \(y \in X\), written as \(x \bot y\), if for every scalar \(\lambda\) we have \(\|x+\lambda y\| \geq \|x\|\). For \(\varepsilon \in [0,1)\), two vectors \(x\) and \(y\) are called \(\varepsilon\)-orthogonal in the sense of Chmieliński, \(x\bot^\varepsilon y\) (in the sense of Dragomir, \(x\bot_n^\varepsilon y\), resp.)\ whenever \(\|x+\lambda y\|\geq \|x\|^2-2\varepsilon\|x\|\,\|\lambda y\|\) for all scalars \(\lambda\) (\(\|x+\lambda y\|\geq (1-\varepsilon)\|x\|\) for all scalars \(\lambda\), resp.). A mapping \(T: X \to Y\) is an \(\varepsilon\)-isometry if \((1-\delta_1(\varepsilon))\|x\| \leq \|Tx\|\leq (1+\delta_2(\varepsilon))\|x\|\), where \(\delta_1(\varepsilon), \delta_2(\varepsilon) \to 0\) as \(\varepsilon\to 0\). A pair \((X, Y)\) of normed spaces, where \(Y\) is uniformly smooth, has the SLI property if there exists a function \(\delta:[0,1) \to (0,\infty)\) satisfying \(\lim_{\varepsilon\to 0}\delta(\varepsilon)=0\) such that, whenever \(T\) is an \(\varepsilon\)-isometry, then there exists an isometry \(U\) such that \(\|T-U\| \leq \delta(\varepsilon)\). The authors show that a pair \((X, Y)\) has the SLI property if and only if the stability of the orthogonality preserving property in sense \(x\bot y\Rightarrow Tx\bot^\varepsilon Ty\) holds or, equivalently, the stability of the orthogonality preserving property in the sense that \(x\bot y\Rightarrow Tx\bot_n^\varepsilon Ty\) holds. A mapping \(T: X \to Y\) is said to be \(\varepsilon\)-approximately orthogonality preserving if \(x\bot y\Rightarrow Tx\bot^\varepsilon Ty\) (or \(x\bot y\Rightarrow Tx\bot^\varepsilon Ty\)). Using some ideas of \textit{S.\,F.\thinspace Bellenot} [Isr.\ J.\ Math.\ 56, 89--96 (1986; Zbl 0619.46012)], the authors present an example showing that approximately orthogonality preserving mappings are in general far from scalar multiples of isometries.
scientific article; zbMATH DE number 5817656
scientific article