Stability of Derivations on Hilbert $C^*$-Modules

TL;DR

This paper investigates the stability of derivations on Hilbert $C^*$-modules, analyzing whether approximate derivations can be closely approximated by exact derivations within the Hyers--Ulam--Rassias framework.

Contribution

It extends the stability analysis of derivations to the setting of Hilbert $C^*$-modules, providing new results in this mathematical context.

Findings

Established stability results for derivations on Hilbert $C^*$-modules.

Extended Hyers--Ulam--Rassias stability to this class of modules.

Provided conditions under which approximate derivations are near exact derivations.

Abstract

Consider the functional equation ${\mathcal E}_1(f) = {\mathcal E}_2(f) ({\mathcal E})$ in a certain framework. We say a function $f_0$ is an approximate solution of $({\mathcal E})$ if ${\mathcal E}_1(f_0)$ and ${\mathcal E}_2(f_0)$ are close in some sense. The stability problem is whether or not there is an exact solution of $({\mathcal E})$ near $f_0$. In this paper, the stability of derivations on Hilbert $C^*$-modules is investigated in the spirit of Hyers--Ulam--Rassias.