Equivalent characterization of right (left) centralizers or centralizers on Banach algebras

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 γsatisfying 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.