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

Let (CE,∥⋅∥CE) be a Banach symmetric ideal of compact operators, acting in a complex separable infinite-dimensional Hilbert space H. Let ChE={x∈CE:x=x∗} be the real Banach subspace of self-adjoint operators in (CE,∥⋅∥CE). We show that in the case when (CE,∥⋅∥CE) is a separable or perfect Banach symmetric ideal (CE≠C2) any skew-Hermitian operator H:ChE→ChE has the following form H(x)=i(xa-ax) for same a∗=a∈B(H) and for all x∈ChE. Using this description of skew-Hermitian operators, we obtain the following general form of surjective linear isometries V:ChE→ChE. Let (CE,∥⋅∥CE) be a separable or a perfect Banach symmetric ideal with not uniform norm, that is ∥p∥CE>1 for any finite dimensional projection p∈CE with dimp(H)>1, let CE≠C2, and let V:ChE→ChE be a surjective linear isometry. Then there exists unitary or anti-unitary operator u on H such that V(x)=uxu∗ or V(x)=-uxu∗ for all x∈ChE.