Let N be a nontrivial nest on a Hilbert space H , T (N ) be the associated nest algebra, and φ be a linear map on T (N ). We prove, among other results, that φ preserves the commutant or the double commutant if and only if there is a scalar λ and a linear functional f on T (N ) such that φ(T ) = λT + f (T )I for every T ∈ T (N ). In fact, we show a more general result than this. Also, we show that φ is a local Lie centralizer if and only if there exists a scalar λ and a linear functional f on T (N ) vanishing on each commutator such that φ(T ) = λT + f (T )I for every T ∈ T (N ).
