Additive maps on some operator algebras behaving like (α,β)-derivations or generalized (α,β)-derivations at zero-product elements

Let A be a subalgebra of B(X) containing the identity operator I and an idempotent P. Suppose that α,β:A→A are ring epimorphisms and there exists some nest N on X such that α (P)(X) and β(P)(X) are non-trivial elements of N. Let A contain all rank one operators in AlgN and δ : A→B(X) be an additive mapping. It is shown that, if δ is (α,β)-derivable at zero point, then there exists an additive (α,β)=derivation τ : A→B(X) such that δ(A)=τ(A)+α(A)δ(I) for all A∈ A. It is also shown that if δ is generalized (α,β)-derivable at zero point, then δ is an additive generalized (α,β)-derivation. Moreover, by use of this result, the additive maps (generalized) (α,β)-derivable at zero point on several nest algebras, are also characterized.