Isometries of Banach ideals of compact operators

Let $H$ be a complex separable infinite-dimensional Hilbert space and let $J \ne C_2$ be a perfect Banach symmetric ideal of compact linear operators acting in $H$. We show that for any surjective linear isometry $V$ on $J$ there are unitary operators $u$ and $v$ on $H$ such that $V(x) = uxv$ or $V(x) = ux^tv$ for all $x$ in $J$, where $x^t$ is the transpose of an operator $x$ with respect to a fixed orthonormal basis in $H$.