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

Abstract

Let $\mathcal H$ be a complex infinite-dimensional separable Hilbert space, and let $\mathcal K(\mathcal H)$ be the $C^*$-algebra of compact linear operators in $\mathcal H$. Let $(E,\|\cdot\|_E)$ 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 $\|x\|_{\mathcal C_E}=\|\{\mu(n,x)\}\|_E$, $x\in\mathcal C_E$, be the Banach ideal of compact operators generated by $E$. Let $\mathcal C_E^h=\{x\in\mathcal C_E : x=x^*\}$ be the real Banach subspace of self-adjoint operators in $(\mathcal C_E, \|\cdot\|_{\mathcal C_E})$. We show that in the case when $\mathcal C_E$ is a separable or perfect Banach symmetric ideal, $\mathcal C_E \neq \mathcal C_{l_2}$, for any skew-Hermitian operator $H\colon\mathcal C_E^h \to \mathcal C_E^h$ there exists self-adjoint bounded linear operator $a$ in $\mathcal H$ such that $H(x)=i(xa - ax)$ for all $x\in\mathcal C_E^h$.