Let \(A\) be a unital Banach algebra with norm \(| \cdot| \), \(A_1=\{a\in A \mid | a| =1\}\) and \(_A{\mathcal{H}}\) a left Banach \(A\)-module with norm \(\| \cdot\|\). The authors study the generalized Hyers-Ulam-Rassias stability of the invertible mapping in a Banach module over a unital Banach algebra. The main result is the following: Let \(F:_A{\mathcal{H}}\longrightarrow_A{\mathcal{H}}\) be a mapping for which there exists a function \(\varphi:_A{\mathcal{H}}\times_A{\mathcal{H}}\longrightarrow[0,\infty)\) such that \[ \tilde \varphi (x,y):=\sum_{k=0}^\infty3^{-k}\varphi(3^kx,3^ky)<\infty \] for all \(a\in A_1\) and all \(x,y\in _A{\mathcal{H}}\), and \[ \| 2F\Big(\frac{ax+by}{2}\Big)-aF(x)-aF(y)\| \leq \varphi(x,y) \] for all \(a\in A_1^{+}\cup\{i\}\) and all \(x,y\in_A{\mathcal{H}}\). Assume that \(F(tx)\) is continuous in \(t\in \mathbb{R}\) for each fixed \(x\in_A{\mathcal{H}}\), \(F(3^nx)=3^nF(x)\) for all positive integers \(n\) and all \(x\in_A{\mathcal{H}}\). Then the mapping \(F\) is an \(A\)-linear operator. Also, under supplementary conditions about \(F\) it is proved that \(F\) is a self-adjoint operator, normal operator, unitary operator, or a projection.