Numerical radius distance-preserving maps on ℬ(ℋ)

Bai, Zhaofang; Hou, Jinchuan

doi:10.1090/S0002-9939-03-07190-9

Numerical radius distance-preserving maps on $\mathcal {B}(H)$
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by Zhaofang Bai and Jinchuan Hou PDF

Proc. Amer. Math. Soc. 132 (2004), 1453-1461 Request permission

Abstract:

Let $H$ be a complex Hilbert space, $\mathcal {B}(H)$ be the algebra of all bounded linear operators on $H$, $\mathcal {H}(H)$ be the subset of all selfadjoint operators in $\mathcal {B}(H)$ and $\mathcal {V}=\mathcal {B}(H)$ or ${\mathcal H}(H)$. Denote by $w(A)$ the numerical radius of $A\in \mathcal {B}(H)$. We characterize surjective maps $\Phi :{\mathcal V}\rightarrow {\mathcal V}$ that satisfy $w(\Phi (A)-\Phi (B))=w(A-B)$ for all $A,B\in {\mathcal V}$ without the linearity assumption.

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