Isometries of perfect norm ideals of compact operators

Abstract

It is proved that for every surjective linear isometry $V$ on a perfect Banach symmetric ideal ${\mathcal{C}}_E\ne {\mathcal{C}}_2$ of compact operators, acting in a complex separable infnite-dimensional Hilbert space $\mathcal{H}$ there exist unitary operators $u$ and $v$ on $\mathcal{H}$ such that $V(x)=uxv$ or $V(x)=u{x}^{t}v$ for all $x\in {\mathcal{C}}_E$, where $x^t$ is the transpose of an operator $x$ with respect to a fixed orthonormal basis in $\mathcal{H}$. In addition, it is shown that any surjective 2-local isometry on a perfect Banach symmetric ideal ${\mathcal{C}}_E\ne {\mathcal{C}}_2$ is a linear isometry on ${\mathcal{C}}_E$.