Linear mappings approximately preserving orthogonality (Q1772293)

The author gives several results for linear mappings between real or complex inner product spaces such that for each two orthogonal vectors in the domain, their values are \(\varepsilon\)-orthogonal in the target space. \(\varepsilon\)-orthogonality \((0\leq \varepsilon <1)\) means the relation \[ u\perp^\varepsilon v\Longleftrightarrow | (u| v)| \leq \varepsilon \| u\| \cdot \| v\| . \] The main result of the paper is Theorem 2, where it is shown that if \(X\) and \(Y\) are real or complex inner product spaces and if \(f:X\rightarrow Y\) is a nonzero linear mapping satisfying (AOP) with some \(\varepsilon\in (0,1)\), then \(f\) is injective, continuous and there exists \(\gamma > 0\) such that \[ \left| (\left( f(x)| f(y)\right) -\gamma^2(x| y) \right| \leq \delta \cdot \min \left\{\gamma^2 \| x\| \cdot \| y\| , \| f(x)\| \cdot \| f(y)\| \right\}, \quad x,y\in X, \tag{1} \] with \(\delta =4\varepsilon \left( {1\over 1-\varepsilon} +\sqrt {{1+\varepsilon\over 1-\varepsilon}}\right)\). Conversely, if \(f:X\rightarrow Y\) satisfies \((1)\) with some \(\delta\geq 0\) and \(\gamma > 0\), then \(f\) is a quasi-linear, approximately orthogonality preserving mapping.