Jordan maps on triangular algebras

Let T be a triangular algebra and R  ′ be an arbitrary ring. Suppose that M:T→R′M:T→R′ and M∗:R′→TM∗:R′→T are surjective maps such thatM(aM∗(x)+M∗(x)a)=M(a)x+xM(a),M∗(M(a)x+xM(a))=aM∗(x)+M∗(x)afor all a∈T,x∈R′a∈T,x∈R′. In this paper, we give sufficient conditions on T such that both M and M∗ are additive. In particular, if T is a standard subalgebra of a nest algebra, then both M and M∗ are additive.