J-子空间格代数上中心化子和广义导子的刻画

设L是Banach空间X上的J-子空间格,AlgL是相应的(J-子空间格代数.设φ:AlgL→AlgL是可加映射,对每个K∈(J)(L),dimK≥2.该文证明了下列表述等价:(1)φ是中心化子;(2)φ满足AB=0■φ(A)B=Aφ(B)=0;(3)φ满足AB+BA=0■φ(A)B+φ(B)A=Aφ(B)+Bφ(A)=0;(4)φ满足ABC+CBA=0■φ(A)BC+φ(C)BA=ABφ(C)+CBφ(A)=0.作为应用,得到AlgL上在零点广义可导的可加映射的完全刻画.

Characterization of Centralizers and Generalized Derivations onJ-Subspace Lattice Algebras

Let L be a J-subspace lattice on a Banach space X and AlgL the associated J-subspace lattice algebra. Assume that Φ: AlgL→AlgL is an additive map and dimK≥2 for every K∈J(L).  It is shown that the following statements are equivalent: (1) Φ is a centralizer; (2) Φ sataisies Φ(A)B=AΦ(B)=0 whenever A, B∈AlgL with AB=0; (3) Φ satisfies Φ(A)B+Φ(B)A=AΦ(B)+B\Φ(A)=0 whenever A, B∈AlgL with  AB+BA=0; (4) Φ satisfies Φ(A)BC+Φ(C)BA=ABΦ(C)+CBΦ(A)=0 whenever A, B∈AlgL with ABC+CBA=0. As an application, additive maps generalized derivable at zero on AlgL are characterized.