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 $\ge 2$, ${\mathcal{B}}_s(H)$ the space of all self-adjoint operators on $H$. We give a complete classification of non-linear surjective maps on ${\mathcal{B}}_s(H)$ preserving respectively numerical radius and numerical range of Lie product.

22 pages