Lie σ-centralizers on algebras when σ is an automorphism

Let [Formula: see text] be an algebra over a unital commutative ring [Formula: see text], and let [Formula: see text] be an algebraic homomorphism. In this paper, we consider a linear map [Formula: see text] that satisfies one of the following conditions: [Formula: see text] [Formula: see text] or [Formula: see text] where [Formula: see text] is the Lie product in [Formula: see text]. We characterize linear maps [Formula: see text] under automorphisms [Formula: see text] by reducing to the case [Formula: see text] (the identity map on [Formula: see text]). Using these equivalences, we further characterize the linear maps [Formula: see text] on generalized matrix algebras; triangular algebras; von Neumann algebras; standard operator algebras and nest algebras for every automorphism [Formula: see text] on these algebras. We also obtain similar results for Lie [Formula: see text]-centralizers. Some of our results generalize some of the previous results.