Weak continuity of skew-Hermitian operators in Banach ideals

Let $\mathcal{H}$ be a separable complex Hilbert space, $\mathcal{B(H)}$ be the $C^{*}$-algebra of all bounded linear operators acting in $\mathcal{H}$, $\mathcal{I}$ be the perfect Banach ideal of compact operators in $\mathcal{B(H)}$, and $\mathcal{I}^h=\{{x\in\mathcal{I}}, \ {x=x^*}\}$. We prove that any skew-Hermitian operator $T:\mathcal{I}^h\to\mathcal{I}^h$ is continuous in the weak topology $\sigma(\mathcal{I},\mathcal{I}^{\times})$, where $\mathcal{I}^{\times}=\{x\in\mathcal{B(H)} \mid xy \in \mathcal{C}_1 \ \forall y \in \mathcal{I}\}$ is the associated Banach ideal for $\mathcal{I}$.