Linear maps preserving orthogonality (Q2516288)

In this paper, the authors consider linear maps between normed spaces preserving an orthogonality condition. In the introduction, the authors review the notion of B-orthogonality on a normed space, also the \(\rho_+\), \(\rho_{-}\) and \(\rho\) orthogonality conditions on a normed space. If \(X\) is a normed space then \(\rho_{\pm}:X\times X \rightarrow \mathbb{R}\) is defined as follows: \[ \rho_{\pm}(x,y)=\lim_{t \rightarrow 0^{\pm}} \frac{\| x+ty\|^2 -\| x\|^2}{2t}. \] Two elements in \(X\), \(x\) and \(y\) are said \(\rho_+\) (\(\rho_{-}\)) orthogonal if \(\rho_{+}(x,y)=0\) (\(\rho_{-}(x,y)=0\), respectively). Further, \(x\) and \(y\) are \(\rho\) (\(\rho_{*}\)) orthogonal if \((\rho_{+}+\rho_{-})(x,y)=0\) (\(\rho_{+}(x,y)\cdot \rho_{-}(x,y)=0\), respectively). There are two main results proved in this paper. {Theorem 2.2.} Let \(X\), \(Y\) be normed spaces, \(T:X \rightarrow Y\) a non-zero linear map. Then the following conditions are equivalent. {\parindent=6mm \begin{itemize} \item[(1.)] \(T\) preserves \(\rho_{*}\)-orthogonality.\item [(2.)] \(T\) preserves \(\rho_{+}\)-orthogonality.\item [(3.)] \(T\) preserves \(\rho_{-}\)-orthogonality.\item [(4.)] \(T\) preserves B-orthogonality.\item [(5.)] \(\| Tx\|=\| t\|\| x\|\) for all \(x \in X\). \end{itemize}} The last theorem proves an interesting result on linear maps that preserving orthogonality. {Theorem 2.3.} Let \(X\), \(Y\) be normed spaces, \(T:X \rightarrow Y\) a non-zero linear map. Then \(T\) preserves \(\rho\)-orthogonality if and only if \(T\) is a scalar multiple of an isometry.