On the stability of the orthogonality equation and the orthogonality-preserving property with two unknown functions

Abstract

For two unknown functions $f$, $g$, the equation

$$\langle f\left(x\right)\left|g\right(y)\rangle =\langle x\mid y\rangle$$ and its stability as well as the approximate orthogonality-preserving property $$x\perp y⟹fx{\perp }^{\epsilon }gy$$ are considered.