Jordan {g,h}-derivations on triangular algebras

Liang Kong,Jianhua Zhang

doi:10.1515/math-2020-0044

Abstract

Abstract 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 τ ( N ) \tau ({\mathscr{N}}) is a {g,h}-derivation if and only if dim 0 + ≠ 1 \dim {0}_{+}\ne 1 or dim H − ⊥ ≠ 1 \dim {H}_{-}^{\perp }\ne 1 , where N {\mathscr{N}} is a non-trivial nest on a complex separable Hilbert space H and τ ( N ) \tau ({\mathscr{N}}) is the associated nest algebra.

Highlights

In this article, we assume that all algebras and rings are associative and 2-torsion free
Herstein [2] showed that every Jordan derivation from a prime ring into itself is a derivation
Zhang and Yu [4] proved that every Jordan derivation on triangular algebras is a derivation

Summary

Introduction

We assume that all algebras and rings are associative and 2-torsion free. Zhang and Yu [4] proved that every Jordan derivation on triangular algebras is a derivation. The triples (δ, δ1, δ2), where δ is a generalized derivation and δ1, δ2 are R-linear maps associated with it, are called ternary derivations. F is said to be a Jordan generalized derivation with an associated map h if. Li and Benkovič [18] proved that every Jordan generalized derivation is a generalized derivation on triangular algebras. Brešar [19] proved that every Jordan {g,h}-derivation of a semiprime unital algebra A is a {g,h}-derivation. Derivation is a {g,h}-derivation on triangular algebras It is not true (see the proof of Theorem 3.1). The purpose of this article is to give a sufficient and necessary condition for every Jordan {g,h}-derivation to be a {g,h}-derivation on triangular algebras. We prove that every Jordan {g,h}-derivation on algebras is

Main result

An application

2: If dim