Nonlinear preserver problems on B(H)

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 . 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 B s (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.