Daugavet’s equation and Jordan elementary operators

Abstract

The aim of this paper is to investigate the Daugavet equation for a Jordan elementary operator. More precisely, we study the equation

$$\begin{aligned} \Vert I+U_{\mathfrak {J},A,B} \Vert =1+2 \Vert A \Vert \Vert B \Vert , \end{aligned}$$

where I stands for the identity operator, A and B are two bounded operators acting on a complex Hilbert space \(\mathcal {H}\), \(\mathfrak {J}\) is a norm ideal of operators on \(\mathcal {H}\), and \(U_{\mathfrak {J}, A, B}\) is the restriction of the Jordan operator \(U_{A,B}\) to \(\mathfrak {J}\). In the particular case where \(\mathfrak {J}=\mathfrak {C}_{2}(\mathcal {H})\) is the ideal of Hilbert–Schmidt operators, we give necessary and sufficient conditions under which the above equation holds.