Additive maps preserving similarity or asymptotic similarity on

Abstract

Let H be an infinite-dimensional complex separable Hilbert space and denote the algebra of all bounded linear operators acting on H. We show that an additive continuous surjective map Φ on is asymptotic similarity preserving if and only if it is similarity preserving, and in turn, if and only if there exist a scalar c and an invertible bounded linear or conjugate linear operator A on H such that either Φ(T)=cATA ⁻¹ for all T or Φ (T)=cAT*A ⁻¹ for all T.

Keywords:

• Similarity
• Asymptotic similarity
• Asymptotic similarity-preserving additive maps

Keywords:

• AMS Subject Classification: Primary 47B49

Acknowledgement

The authors would like to thank the referee who gave detailed and thoughtful comments to improve this article. Particularly, the arguments of Claim 4 in the proof of Lemma 2.1 by using of Faure's non-surjective version of Fundamental Theorem of Projective Geometry was suggested by the referee, which shortens our original arguments. This work was supported partially by NNSF of China and NSF of Shanxi.