In separable or perfect symmetric Banach ideals C_E of compact operators on Hilbert space (C_E ≠ C_{l_2}), every skew-Hermitian operator on the self-adjoint subspace C_E^h is inner: H(x) = i(xa − ax) for some self-adjoint a ∈ B(H).
and Semenov E.M., Interpolation of Linear Operators, Translations of Mathematical Monographs
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Skew-Hermitian operators in real Banach spaces of self-adjoint compact operators
In separable or perfect symmetric Banach ideals C_E of compact operators on Hilbert space (C_E ≠ C_{l_2}), every skew-Hermitian operator on the self-adjoint subspace C_E^h is inner: H(x) = i(xa − ax) for some self-adjoint a ∈ B(H).