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 $\Phi:{\rm Alg}{\mathcal N}\rightarrow{\rm Alg}{\mathcal M}$isaLieringisomorphism(i\mathrm{.}e\mathrm{.},$\Phi$ is additive, Lie multiplicative and bijective) if and only if $\mathrm{\Phi }$ has the form $\Phi(A) = TAT^{-1} + h(A)I$forall$A\in \mbox{\rm Alg}{\mathcal N}$or$\Phi(A)=-TA^*T^{-1}+h(A)I$forall$A\in \mbox{\rm Alg}{\mathcal N}$,where$h$isanadditivefunctionalvanishingonallcommutatorsand$T$ is an invertible bounded linear or conjugate linear operator when $\dim X=\mathrm{\infty }$; $T$ is a bijective $\tau$-linear transformation for some field automorphism $\tau$ of $\mathbb{F}$ when $\dim X<\mathrm{\infty }$.