Bulletin of the
Korean Mathematical Society BKMS

A mapping $f: {\mathcal M} \to {\mathcal N}$ between Hilbert $C^*$-modules approximately preserves the inner product if \[\|\langle f(x), f(y)\rangle - \langle x, y\rangle \| \leq \varphi(x, y)\] for an appropriate control function $\varphi(x,y)$ and all $x, y \in {\mathcal M}$. In this paper, we extend some results concerning the stability of the orthogonality equation to the framework of Hilbert $C^*$-modules on more general restricted domains. In particular, we investigate some asymptotic behavior and the Hyers--Ulam--Rassias stability of the orthogonality equation.

Keywords: Hilbert $C^*$-module, Hyers--Ulam--Rassias stability, superstability, orthogonality equation, asymptotic behavior

MSC numbers: Primary 39B52; Secondary 46L08, 39B82, 46B99, 17A40