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

TL;DR

This paper classifies non-linear surjective maps on self-adjoint operators in a complex Hilbert space that preserve the numerical radius and range of the Lie product, providing a comprehensive understanding of their structure.

Contribution

It offers a complete classification of non-linear surjective maps preserving numerical radius and range of the Lie product on self-adjoint operators.

Findings

Characterization of maps preserving numerical radius of Lie product

Characterization of maps preserving numerical range of Lie product

Complete classification of such maps in Hilbert space setting

Abstract

Let $H$ be a complex separable Hilbert space of dimension $\geq 2$, ${\mathcal B}_s(H)$ 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.