Abstract
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.
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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)
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Cui, J.L. Nonlinear preserver problems on B(H). Acta. Math. Sin.-English Ser. 27, 193–202 (2011). https://doi.org/10.1007/s10114-011-9106-y
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DOI: https://doi.org/10.1007/s10114-011-9106-y