套代数上保秩一幂零性的可加映射

${\cal N}$和${\cal M}$分别是实或复Banach空间$X$ ($\dim X >5$)和$Y$中的两个套且Alg${\cal N}$和Alg${\cal M}$分别是与套${\cal N}$和${\cal M}$相关的套代数.符号Alg加映射;秩一幂零算子;套代数Additive map,Rank one nilpotent operator,Nest algebra国家自然科学基金;清华大学校科研和教改项目;教育部高等学校博士点教育基金;国家自然科学基金;山西省自然科学基金2005-02-082007年4月25日${\cal N}$和${\cal M}$分别是实或复Banach空间$X$ ($\dim X >5$)和$Y$中的两个套且Alg${\cal N}$和Alg${\cal M}$分别是与套${\cal N}$和${\cal M}$相关的套代数.符号Alg加映射;秩一幂零算子;套代数Additive map,Rank one nilpotent operator,Nest algebra国家自然科学基金;清华大学校科研和教改项目;教育部高等学校博士点教育基金;国家自然科学基金;山西省自然科学基金2005-02-082007年4月25日令N和M分别是实或复Banach空间X(dim X>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AlgFN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中的所有秩一幂零元．如果Φ-双边保秩一幂零性,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一幂零元x(?)f∈AlgFN,Φ(x(?)f)=Ax(?)Cf,要么对所有的秩一幂零元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．

Additive Maps Preserving Rank-1 Nilpotency on Nest Algebras

Let ${\cal N}$ and ${\cal M}$ be two nests on real or complex Banach spaces X and Y, respectively, and $\Phi$ be an additive map between ideals Alg$_{\cal F}{\cal N}$ and Alg$_{\cal F}{\cal M}$ of finite rank operators in nest algebras Alg${\cal N}$ and Alg${\cal M}$, of which the range contains all rank-1 nilpotent operators in Alg${\cal M}$. The authors show that if $\Phi$ is rank-1 nilpotency preserving in both directions, then $\Phi$ has the form either $\Phi(x\otimes f)=Ax\otimes Cf$ for every rank-1 nilpotent operator $x\otimes {\rm Alg}_{\cal F}{\cal N}$ or $\Phi(x\otimes f)=Af\otimes Cx$ for every rank-1 nilpotent operator $x\otimes f\in {\rm Alg}_{\cal F}{\cal N}$,where $A$ and $C$ are certain $\tau$-linear operators with an automorphism $\tau$ of the underlying field. And the authors obtain particularly a characterization of such $\Phi$ if it is continuous, $X$ and $Y$ are Hilbert spaces with dimX≥ 6.