Properties and preservers of numerical radius on skew Lie products of operators

Abstract Let H be a complex separable Hilbert space with dim ⁡ H ≥ 3 and B ( H ) the Banach algebra of all bounded linear operators on H. Denote by w ( A ) the numerical radius of a bounded linear operator A ∈ B ( H ) and A B − B A ⁎ the skew Lie product of two operators A , B ∈ B ( H ) . In this paper, it is shown that, if a surjective map Φ : B ( H ) → B ( H ) satisfies w ( A B − B A ⁎ ) = w ( Φ ( A ) Φ ( B ) − Φ ( B ) Φ ( A ) ⁎ ) for all A , B ∈ B ( H ) , then there exist a unitary operator U ∈ B ( H ) , a functional h : B ( H ) → { − 1 , 1 } and a subset S ⊆ B ( H ) consisting of some normal operators such that Φ ( A ) = h ( A ) U A U ⁎ if A ∈ B ( H ) ∖ S and Φ ( A ) = h ( A ) U A ⁎ U ⁎ if A ∈ S . Particularly, if dim ⁡ H ∞ , a complete characterization of S can be obtained.