On superstability of the Wigner equation (Q2002788)

Let \(M\) and \(N\) be inner product spaces over \(F\in \{\mathbb{R}, \mathbb{C}\}\). The famous Wigner's theorem says that any mapping \(f:M \to N\) which preserves the transition probability, i.e., \[ \| \langle f(x), f(y)\rangle\| = \|\langle x, y\rangle\|, \qquad x,y\in M, \tag{W} \] must be phase-equivalent to a linear or a conjugate-linear isometry. Now suppose that a mapping \(f : M \to N\) satisfies (W) approximately. More precisely, assume that \[ \|\langle(x), f(y)\rangle \| - \|\langle x, y\rangle\| \leq \phi(x,y), \qquad x,y\in M, \] where \(\phi: M \times M \to [0,\infty)\) is an appropriate control function. The authors investigate the superstability of above inequality. In particular, they find that if \(f\) is surjective all solutions of above inequality are in fact solutions of (W).