实Nest代数上的广义Jordan~*-左导子

设H是实Hilber空间, (?)是B(H)中含恒等算子I的算子代数,若(?) 是从(?)到B(H)的线性映射,如果(?)满足对任意的T∈(?),有(?)(T2)=T*(?)(T)+ (?)(T)T-T*(?)(I)T,则称(?)是一个广义Jordan*-左导子;如果(?)满足对任意的T∈(?), 有(?)(T)(ker(T))(?)ran(T*),则称(?)是一个左*-核值保持映射.本文主要获得了如下 结果: Nest代数上每个弱算子拓扑连续的左*-核值保持映射是广义Jordan*-左内 导子,即存在A,B∈B(H),使得对任意的T∈(?),有(?)(T)=T*A+BT.特别地,(?) 也是一个广义Jordan*-左导子.

Generalized Jordan*-left Derivations on Real Nest Algebras

Let (?) be a subalgebra with the unit operator I in B(H), where H is a real Hilbert space. We say that a linear mapping (?) from (?) into B(H) is a generalized Jordan*-left derivation if (?)(T2) = T*(?)(T) + (?)(T)T - T*(?)(I)T for any T ∈(?); (?) is called a generalized*-left kernel-into-range preserving mapping if (?)(T)(ker(T)) (?) ran(T*). In this paper, we show the following main result. If a generalized*-left preserving kernel-into-range mapping on real nest algebras is continuous in the weak operator topology, then (?) is a generalized*-left inner derivation, i.e. (?)(T) = T*A + BT for some A, B∈B(H); In particular, (?) is also a generalized Jordan*-left derivation.