Rank-1 nilpotency linear preservers on nest algebras

Let and be two nests on Banach spaces X and Y, respectively, and let Φ be a linear map between nest algebras and of which the range contains all rank-1 nilpotent operators in . We give some structure theorems for Φ to be rank-1 nilpotency preserving in both directions, and obtain particularly a characterization of such Φ if it is bounded and X and Y are Hilbert spaces with dimX 6.