Isometries of real subspaces of self-adjoint operators in banach symmetric ideals

Let $(\mathcal C_E, \|\cdot\|_{\mathcal C_E})$ be a Banach symmetric ideal of compact operators, acting in a complex separable infinite-dimensional Hilbert space $\mathcal H$. 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 $(\mathcal C_E, \|\cdot\|_{\mathcal C_E})$ is a separable or perfect Banach symmetric ideal ($\mathcal C_E \neq \mathcal C_2$) any skew-Hermitian operator $H: \mathcal C_E^h\to \mathcal C_E^h$ has the following form $H(x)=i(xa - ax)$ for same $a^*=a\in \mathcal B(\mathcal H)$ and for all $x\in \mathcal C_E^h$. Using this description of skew-Hermitian operators, we obtain the following general form of surjective linear isometries $V:\mathcal C_E^h \to \mathcal C_E^h$. Let $(\mathcal C_E, \|\cdot\|_{\mathcal C_E})$ be a separable or a perfect Banach symmetric ideal with not uniform norm, that is $\|p\|_{\mathcal C_E}> 1$ for any finite dimensional projection $p \in\mathcal C_E$ with $\dim p(\mathcal H)>1$, let $\mathcal C_E \neq \mathcal C_2$, and let $V: \mathcal C_E^h \to \mathcal C_E^h$ be a surjective linear isometry. Then there exists unitary or anti-unitary operator $u$ on $\mathcal H$ such that $V(x)=uxu^*$ or $V(x)=-uxu^*$ for all $x \in \mathcal C_E^h$.