中文摘要：

令N和M分别是实或复Banach空间X（dimX〉5）和y中的两个套且AlgN和AlgM分别是与套N和M相关的套代数．符号AlgFA／表示AlgN中所有有限秩算子全体．设Φ：AlgFN→AlgFM是可加映射，且值域包含AlgFM中的所有秩-幂零元．如果Φ双边保秩-幂零性，作者证明了存在一个域自同构τ及τ-线性算子A和C使得要么对所有的秩-幂零元x×f∈AlgFN，Φ（x×f）=Ax×Cf，要么对所有的秩-幂零元x×f∈AlgFN，Φ（x×f）=Af×Cx．特别地，当X和Y是Hilbert空间且Φ是连续映射时，作者得到这类可加映射Φ的完全刻画．

英文摘要：

Let N and M be two nests on real or complex Banach spaces X and Y, respectively, and Φ be an additive map between ideals AlgFN and AlgFM of finite rank operators in nest algebras AlgAl and AlgM, of which the range contains all rank-1 nilpotent operators in AlgM. The authors show that if Φ is rank-1 nilpotcncy preserving in both directions, then Φ has the form either Φ（x × f） = Ax × Cf for every rank-1 nilpotcnt operator x × AlgFN or Φ （x × f） = Af × Cx for every rank-1 nilpotcnt operator x × f ∈ AlgFN, where A and C are certain τ-linear operators with an automorphism τ of the underlying field. And the authors obtain particularly a characterization of such Φ if it is continuous, X and Y are Hilbert spaces with dimX≥6.