Maps preserving the $c$-numerical radius of products for operators in $\mathfrak{B}(H)$

Abstract

Let $\mathfrak{𝔅}\left(\mathcal{\mathscr{H}}\right)$ and ${\mathfrak{𝔅}}^s\left(\mathcal{\mathscr{H}}\right)$ be the algebra of all bounded linear operators on a complex Hilbert space $\mathcal{\mathscr{H}}$ and the real Jordan algebra of all self-adjoint operators on $\mathcal{\mathscr{H}}$, respectively. Suppose $\mathcal{𝒲}$ and $\mathcal{𝒱}$ are subsets of $\mathfrak{𝔅}\left(\mathcal{\mathscr{H}}\right)$ or ${\mathfrak{𝔅}}^s\left(\mathcal{\mathscr{H}}\right)$ and $F\left(\cdot \right)$ is the $c$-numerical radius or the diameter of an operator. Under an assumption on $\mathcal{𝒲}$ and $\mathcal{𝒱}$, characterizations are obtained for surjective maps $\mathrm{\Phi }:\mathcal{𝒲}\to \mathcal{𝒱}$ satisfying

$$F\left(AB\right)=F\left(\mathrm{\Phi }\left(A\right)\mathrm{\Phi }\left(B\right)\right)\left(A,B\in \mathcal{𝒲}\right).$$

When $dim\mathcal{\mathscr{H}}\ge 3$, to establish the proofs, some general results are obtained for functions $F:\mathfrak{𝔅}\left(\mathcal{\mathscr{H}}\right)\to \left[0,+\infty \right)$ satisfying $\left(P_1\right)$ $F\left(UA{U}^{\ast }\right)=F\left(A\right)$ for all $A\in \mathfrak{𝔅}\left(\mathcal{\mathscr{H}}\right)$ and unitary $U$ on $\mathcal{\mathscr{H}}$; $\left(P_2\right)$ For $A\in \mathfrak{𝔅}\left(\mathcal{\mathscr{H}}\right)$, $F\left(A\right)=0$ if and only if $A$ is a multiple of the identity; $\left(P_3\right)$ There are nonnegative real numbers $\alpha ,\beta$ with ${\alpha }^2+{\beta }^2\ne 0$ such that $F\left(T\right)=\alpha \parallel T\parallel +\beta |tr\left(T\right)|$ for each rank-one $T\in \mathfrak{𝔅}\left(\mathcal{\mathscr{H}}\right)$.