In this paper some recent results of Petryshyn on linear A-proper mappings are generalized to linear operators A : E? F which form together with a sequence (Ai) of operators AiEi? Fi ? Ao, a so called a-regular pair. The operators Ai have to be understood as approximations to A satisfying the condition of a-regularity, which is a characteristic property in studying the convergence of approximation methods. The theorems proved in this paper are concerned with properties shared by an operator A, if there exists a sequence (A i), such that the pair A,(Ai) is a regular. One of the main results is that, under this condition, A satisfies a generalized Fredholm-Alternative