摘要
Let H be a complex Hilbert space of dimension greater than 2, and B（H） denote the Banach algebra of all bounded linear operators on H. For A, B C B（H）, define the binary relation A ≤ B by A＊A = A＊B and AA＊ = AB＊. Then （B（H）, ＂〈.＂） is a partially ordered set and the relation ＂≤＂ is called the star order on B（H）. Denote by Bs（H） the set of all self-adjoint operators in B（H）. In this paper, we first characterize nonlinear continuous bijective maps on Bs（H） which preserve the star order in both directions. We characterize also additive maps （or linear maps） on B（H） （or nest algebras） which are multiplicative at some invertible operator.
Let H be a complex Hilbert space of dimension greater than 2, and B（H） denote the Banach algebra of all bounded linear operators on H. For A, B C B（H）, define the binary relation A ≤ B by A＊A = A＊B and AA＊ = AB＊. Then （B（H）, ＂〈.＂） is a partially ordered set and the relation ＂≤＂ is called the star order on B（H）. Denote by Bs（H） the set of all self-adjoint operators in B（H）. In this paper, we first characterize nonlinear continuous bijective maps on Bs（H） which preserve the star order in both directions. We characterize also additive maps （or linear maps） on B（H） （or nest algebras） which are multiplicative at some invertible operator.
基金
Supported by National Natural Science Foundation of China （Grant Nos. 10871111, 10501029） and the Specialized Research Fund for Doctoral Program of Higher Education （Grant No. 200800030059）
