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 = wand 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 andw 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.