Let A be a unital Banach algebra, w ∈ A, and γ : A → A is a continuous linear map. We show that γ satisfies aγ(b) = γ(w) (γ(a)b = γ(w)) whenever a, b ∈ A with ab = w and w is a left (right) separating point in A if and only if γ is a right (left) centralizer. Also, we prove that γ satisfies aγ(b) = γ(a)b = γ(w) whenever a, b ∈ A with ab = w and w is a left or right separating point in A if and only if γ is a centralizer. We also provide some applications of the obtained results for characterization of a continuous linear map γ : A → A on a unital Banach ∗-algebra A satisfying aγ(b)∗ = γ(w∗)∗ (γ(a)∗b = γ(w∗)∗) whenever a, b ∈ A with ab∗ = w (a∗b = w) and w is a left (right) separating point, or γ satisfying aγ(b)∗ = γ(c)∗d = γ(w∗)∗ whenever a, b, c, d ∈ A with ab∗ = c∗d = w and w is a left or right separating point
