Skew-Hermitian operators in real Banach spaces of self-adjoint compact operators

Let $\mathcal{H}$ be a complex infinite-dimensional separable Hilbert space, and let $\mathcal{K}(\mathcal{H})$ be the $C^{\ast }$-algebra of compact linear operators in $\mathcal{H}$. Let $(E,\mathrm{\parallel }\cdot {\mathrm{\parallel }}_E)$ be a symmetric sequence space. If $\{\mu (n,x)\}$ are the singular values of $x\in \mathcal{K}(\mathcal{H})$, let ${\mathcal{C}}_E=\{x\in \mathcal{K}(\mathcal{H}):\{\mu (n,x)\}\in E\}$ with $\mathrm{\parallel }x{\mathrm{\parallel }}_{{\mathcal{C}}_E}=\mathrm{\parallel }\{\mu (n,x)\}{\mathrm{\parallel }}_E$, $x\in {\mathcal{C}}_E$, be the Banach ideal of compact operators generated by $E$. Let $\mathcal C_E^h={x\in\mathcal C_E : x=x^*}$betherealBanachsubspaceofself-adjointoperatorsin$(\mathcal C_E, |\cdot|{\mathcal C_E})$\mathrm{.}Weshowthatinthecasewhen$\mathcal C_E$isaseparableorperfectBanachsymmetricideal,$\mathcal C_E \neq \mathcal C{l_2}$,foranyskew-Hermitianoperator$H\colon\mathcal C_E^h \to \mathcal C_E^h$thereexistsself-adjointboundedlinearoperator$a$in$\mathcal H$suchthat$H(x)=i(xa - ax)$forall$x\in\mathcal C_E^h$.