Isometries of perfect symmetrically normed ideals of compact operators

It is well known that for every surjective linear isometry $V$ on minimal symmetrically norm ideal $C_E \ne C_2$ of compact operators acting in the Hilbert space $H$ there exist unitary operators $u$ and $v$ on $H$ such that $V(x) = uxv$ (or $V(x) = ux^tv$) for all $x \in C_E$, where $x^t$ is the transpose of the operator $x$ with respect to a fixed orthonormal basis of $H$. We show that a similar description of all surjective linear isometries holds for perfect symmetrically normed ideals of compact operators.