Non-linear maps on self-adjoint operators preserving numerical radius and numerical range of Lie product

Abstract

Let H be a complex separable Hilbert space of dimension ≥ 2, Bs(H) the space of all self-adjoint operators on H. We give a complete classification of non-linear surjective maps on Bs(H) preserving respectively numerical radius and numerical range of Lie product. F (A • B) = F (Φ(A) • Φ(B)) for all A, B ∈ A. Assume that Φ : A → A satisfy Eq.(1.1). For the case F = W and Φ is surjective, it was shown in [15] that if A • B = AB and A = B(H), then there exists a unitary operator U such that Φ(A) = ǫU AU * for all A ∈ A, where ǫ ∈ {−1, 1}; if A • B = ABA and A = B(H), then Φ is the multiple of a C * -isomorphism (by a cubic root of unity); if A • B = AB and A = B s (H), then there exists a unitary operator U such that Φ(A) = ǫU AU * for all A ∈ A, where ǫ ∈ {−1, 1}. For the case F = w, A • B = AB, A = B(H) and Φ is surjective, it was proved in [6] that there exist a unitary or anti-unitary operator U and a unit-modular functional f : A → C such that Φ(A) = f (A)U AU * for all A ∈ A. The case of F = w, A • B = ABA 2002 Mathematical Subject Classification. 47H20, 47B49, 47A12.

Maps Preserving the Spectrum of Skew Lie Product of Operators

“…Molnár in [19] studied maps preserving the spectrum of operator and matrix products. His results have been extended in several directions [1,2,7,8,[13][14][15]17] and [18]. In [1], the problem of characterizing maps between matrix algebras preserving the spectrum of polynomial products of matrices is considered.…”