Lie ring isomorphisms between nest algebras on Banach spaces

Let ${\mathcal N}$ and ${\mathcal M}$ be nests on Banach spaces $X$ and $Y$ over the (real or complex) field $\mathbb F$ and let $\mbox{\rm Alg}{\mathcal N}$ and $\mbox{\rm Alg}{\mathcal M}$ be the associated nest algebras, respectively. It is shown that a map $\Phi:{\rm Alg}{\mathcal N}\rightarrow{\rm Alg}{\mathcal M}$ is a Lie ring isomorphism (i.e., $\Phi$ is additive, Lie multiplicative and bijective) if and only if $\Phi$ has the form $\Phi(A) = TAT^{-1} + h(A)I$ for all $A\in \mbox{\rm Alg}{\mathcal N}$ or $\Phi(A)=-TA^*T^{-1}+h(A)I$ for all $A\in \mbox{\rm Alg}{\mathcal N}$, where $h$ is an additive functional vanishing on all commutators and $T$ is an invertible bounded linear or conjugate linear operator when $\dim X=\infty$; $T$ is a bijective $\tau$-linear transformation for some field automorphism $\tau$ of $\mathbb F$ when $\dim X<\infty$.