Perturbation of Adjointable Mappings on Hilbert C-modules

Let X and Y be Hilbert C∗-modules over a C∗-algebra, and φ : X ×Y → [0,∞) be a function. A (not necessarily linear) mapping f : X → Y is called a φ-perturbed adjointable mapping if there exists a (not necessarily linear) mapping g : Y → X such that ‖〈f(x), y〉 − 〈x, g(y)〉‖ ≤ φ(x, y) (x ∈ X , y ∈ Y). In this paper, we investigate the generalized Hyers–Ulam–Rassias stability of adjointable mappings on Hilbert C∗-modules. In particular, we show that if φ(x, y) = δ, where δ is a fixed positive number, then every perturbed adjointable mapping is indeed an adjointable mapping.