Superstability of adjointable mappings on Hilbert c∗-modules

Abstract

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 every- where 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.