Lie isomorphisms of nest algebras
Let \({\mathcal H}\) be a separable complex Hilbert space, and let \({\mathcal M}\) be a nest on \({\mathcal H}\) (i.e., a totally ordered set of closed subspaces of a given Hilbert space). Let \({\mathcal T}({\mathcal M})\) denote the nest algebra (i.e., the algebra of all bounded linear operators \(T\) on \({\mathcal H}\) such that \(TM\subset M\) for all \(M\in{\mathcal M}\)). Setting \({\mathcal M}^\perp\) as the nest \(\{M^\perp, M\in{\mathcal M}\}\), there exists a conjugate-linear involution \(J\) on \({\mathcal H}\) such that \(J({\mathcal M}^\perp)={\mathcal M}^\perp\) (having some other properties as well). The main result of the paper is the following: Given two nests \({\mathcal M}\), \({\mathcal N}\) on \({\mathcal H}\), a linear map \(\phi:{\mathcal T}({\mathcal M})\to {\mathcal T}({\mathcal N})\) is a Lie isomorphism (i.e., \(\phi\) is onto and \(\phi(AB- BA)= \phi(A)\phi(B)- \phi(B)\phi(A)\)) if and only if either. (a) \({\mathcal M}\) is similar to \({\mathcal N}\) (i.e., there exists a dimension preserving order isomorphism of \({\mathcal M}\) onto \({\mathcal N}\)) and there exists an invertible operator \(Y\) on \({\mathcal H}\) satisfying \(Y({\mathcal N})= {\mathcal M}\), and a linear functional \(\beta\) on \({\mathcal T}({\mathcal M})\) with \(\beta(I)\neq -1\) and \(\beta(C)= 0\) for every commutator \(C= AB- BA\), such that \(\phi(T)= Y^{-1} TY+ \beta(T)I\); or (b) \({\mathcal M}\) is similar to \({\mathcal N}^\perp\) and there exists an invertible operator \(Y\) satisfying \(Y({\mathcal N})= {\mathcal M}^\perp\), and a functional \(\beta\) as above, such that \(\phi(T)= -Y^{-1}JT^* JY- \beta(T)I\).
scientific article; zbMATH DE number 1310884