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Approximately $C^*$-inner product preserving mappings

Jacek Chmielinski·Mohammad Sal Moslehian
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Abstract

A mapping $f: {\mathcal M} \to {\mathcal N}$ between Hilbert $C^*$-modules approximately preserves the inner product if \[\|< f(x), f(y)> - < x, y> \| \leq ϕ(x, y),\] for an appropriate control function $ϕ(x,y)$ 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.

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Jacek Chmielinski, Mohammad Sal Moslehian. 2006-01-12. Approximately $C^*$-inner product preserving mappings. https://arxiv.org/abs/math/0601276

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