Jordan product commuting maps with λ-Aluthge transform

F Chabbabi (1) , M Mbekhta (2)

1 Université de Lille, Sciences et Technologies
2 LPP - Laboratoire Paul Painlevé - UMR 8524

Let H and K be two complex Hilbert spaces and B(H) be the algebra of bounded linear operators from H into itself. The main purpose in this paper is to obtain a characterization of bijective maps Φ : B(H) → B(K) satisfying the following condition Δ λ (Φ(A) • Φ(B)) = Φ(Δ λ (A • B)) for all A, B ∈ B(H), where Δ λ (T) stands the λ-Aluthge transform of the operator T ∈ B(H) and A • B = 1 2 (AB + BA) is the Jordan product of A and B. We prove that a bijective map Φ satisfies the above condition, if and only if there exists an unitary operator U : H → K, such that Φ has the form Φ(A) = UAU * for all A ∈ B(H).

Mots clés: normal; quasi-normal operators; polar decomposition; λ-Aluthge transform

Domaines: Analyse fonctionnelle [math.FA]; Algèbres d'opérateurs [math.OA]

F Chabbabi, M Mbekhta. Jordan product commuting maps with λ-Aluthge transform. 2016. ⟨hal-01334047v2⟩