Lie ring isomorphisms between nest algebras on Banach spaces

Let NN and MM be nests on Banach spaces X and Y   over the (real or complex) field FF and let AlgN and AlgM be the associated nest algebras, respectively. It is shown that a map Φ:AlgN→AlgM is a Lie ring isomorphism (i.e., Φ is additive, Lie multiplicative and bijective) if and only if Φ   has the form Φ(A)=TAT−1+h(A)IΦ(A)=TAT−1+h(A)I for all A∈AlgN or Φ(A)=−TA⁎T−1+h(A)IΦ(A)=−TA⁎T−1+h(A)I for all A∈AlgN, where h is an additive functional vanishing on all commutators and T   is an invertible bounded linear or conjugate linear operator when dimX=∞; T is a bijective τ-linear transformation for some field automorphism τ   of FF when dimX<∞.