Isometries of perfect norm ideals of compact operators
Volume 241 / 2018
Studia Mathematica 241 (2018), 87-99
MSC: Primary 46L52; Secondary 46L51, 46B04.
DOI: 10.4064/sm170306-19-4
Published online: 28 September 2017
Abstract
It is proved that for every surjective linear isometry $V$ on a perfect Banach symmetric ideal $\mathcal C_E \not =\mathcal C_{2}$ of compact operators, acting in a complex separable infinite-dimensional Hilbert space $\mathcal H$, there exist unitary operators $u$ and $v$ on $\mathcal H$ such that $V(x) = uxv$ for all $x\in \mathcal C_E$ or $V(x) = ux^tv$ for all $x \in \mathcal C_E$, where $x^t $ is the transpose of $x$ with respect to a fixed orthonormal basis for $\mathcal H$. In addition, it is shown that any surjective 2-local isometry on a perfect Banach symmetric ideal $\mathcal C_E \not =\mathcal C_{2}$ is a linear isometry on $\mathcal C_E$.
