Maps preserving numerical radius and cross norms of operator products

Abstract

Let H be a complex Hilbert space with dim H ≥ 3 and 𝒲, 𝒱 ⊆ ℬ(H) be subsets that contain all rank-one idempotents. Let F(·) stand for numerical radius w(·) or a cross operator norm N(·). All the general surjective maps Φ: 𝒲 → 𝒱 satisfying F(AB) = F(Φ(A)Φ(B)) (resp., F(ABA) = F(Φ(A)Φ(B)Φ(A))) for all A, B ∈ 𝒲 are characterized.

Keywords:

• non-linear map
• numerical radius
• products of operators

AMS Subject Classifications:

• 47H20
• 47B49
• 47A12

Acknowledgements

This work is partially supported by National Natural Science Foundation of China (10771157, 1087111), Provincial Natural Science Foundation (2007011016) and Research Fund of Shanxi for Returned Scholars (2007-38). The authors thank the referee for pointing out a gap in the proof of Theorem 2.1 in the original version and for many helpful comments.