We define the notion of -perturbation of a densely defined adjointable mapping and prove that any such mapping f between Hilbert A-modules over a fixed C-algebra A with densely defined corresponding mapping g is A-linear and adjointable in the classical sense with adjoint g. If both f and g are everywhere defined then they are bounded. Our work concerns with the concept of Hyers–Ulam–Rassias stability originated from the Th. M. Rassias’ stability theorem that appeared in his paper [On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72 (1978), 297–300]. We also indicate complementary results in the case where the Hilbert C -modules admit non-adjointable C -linear mappings.

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