Nonlinear Lie triple derivations by local actions on triangular algebras

Let [Formula: see text] be a triangular algebra over a commutative ring [Formula: see text]. In this paper, under some mild conditions on [Formula: see text], we prove that if [Formula: see text] is a nonlinear map satisfying [Formula: see text] for any [Formula: see text] with [Formula: see text]. Then [Formula: see text] is almost additive on [Formula: see text], that is, [Formula: see text] Moreover, there exist an additive derivation [Formula: see text] of [Formula: see text] and a nonlinear map [Formula: see text] such that [Formula: see text] for [Formula: see text], where [Formula: see text] for any [Formula: see text] with [Formula: see text].

A Class of Nonlinear Nonglobal Semi-Jordan Triple Derivable Mappings on Triangular Algebras

In this paper, we proved that each nonlinear nonglobal semi-Jordan triple derivable mapping on a 2-torsion free triangular algebra is an additive derivation. As its application, we get the similar conclusion on a nest algebra or a 2-torsion free block upper triangular matrix algebra, respectively.

Generalized Jordan N-Derivations of Unital Algebras with Idempotents

Let A be a unital algebra with idempotent e over a 2-torsionfree unital commutative ring ℛ and S : A ⟶ A be an arbitrary generalized Jordan n-derivation associated with a Jordan n-derivation J . We show that, under mild conditions, every generalized Jordan n-derivation S : A ⟶ A is of the form S x = λ x + J x in the current work. As an application, we give a description of generalized Jordan derivations for the condition n = 2 on classical examples of unital algebras with idempotents: triangular algebras, matrix algebras, nest algebras, and algebras of all bounded linear operators, which generalize some known results.

Jordan {g,h}-derivations on triangular algebras

In this article, we give a sufficient and necessary condition for every Jordan {g,h}-derivation to be a {g,h}-derivation on triangular algebras. As an application, we prove that every Jordan {g,h}-derivation on \tau ({\mathscr{N}}) is a {g,h}-derivation if and only if \dim {0}_{+}\ne 1 or \dim {H}_{-}^{\perp }\ne 1 , where {\mathscr{N}} is a non-trivial nest on a complex separable Hilbert space H and \tau ({\mathscr{N}}) is the associated nest algebra.