Rank-Preserving Maps on NEST Algebras

In the paper, a complete description of all weakly continuous rank-preserving linearmaps on nest algebras is given. As an application, we recapture some results concerning lo-cal automorphisms of nest algebras, which obtained by the seconed author in another paper.The following are our main theorems.Throughout this paper, N will be a nest of a Banach space X with 0≠0or X≠X,AlgN denotes the associated nest algebra with N.Theorem 2.3 Let φ be a weekly continuous rank-1 preserving linear map on AlgN.Then one of the following holds:(1)There exists A and C in B(X) such that φ(T) =ATC.(2)There exists A∈B (X~*,X) and C∈B(X,X~*) such that φ(T) =AT~*C.(3)There exists a weakly-weakly star continuous linear map λ(*) from AlgN into X~*and y∈X such that φ(T) =y(?)λ(T).(4) There exists a weakly-weakly continuous linear map δ(*) from AlgN into X andg∈X~* such that φ(T) =δ(T)(?)g.Theorem 2.5 If φ is a linear map on AlgN, and there exists T. in AlgN such thatrankφ(T)>1, then φ is a wea