A fixed point approach to the stability of $\varphi$-morphisms on Hilbert $C^*$-modules

Abstract

Let $E$, $F$ be two Hilbert $C^*$-modules over $C^*$-algebras $A$ and $B$ respectively. In this paper, by the alternative fixed point theorem, we give the Hyers-Ulam-Rassias stability of the equation $$\ip{U(x), U(y)}=\varphi( \ip{x,y})\qquad(x, y\in E),$$ where $U : E\to F$ is a mapping and $\varphi : A\to B$ is an additive map