# Isometries of perfect norm ideals of compact operators

## TL;DR

This paper characterizes all surjective linear isometries and 2-local isometries on perfect Banach symmetric ideals of compact operators, showing they are essentially implemented by unitary conjugations or transpositions, with implications for operator symmetry.

## Contribution

It provides a complete description of surjective isometries and 2-local isometries on perfect Banach symmetric ideals, extending understanding of their structure beyond previous partial results.

## Key findings

- Surjective linear isometries are implemented by unitary operators or transpositions.
- Any surjective 2-local isometry is a linear isometry.
- Results apply to all perfect Banach symmetric ideals except $\\mathcal{C}_2$.

## Abstract

It is proved that for every surjective linear isometry $V$ on a perfect Banach symmetric ideal $\mathcal C_E\neq \mathcal C_2$ 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 $V(x) = ux^tv$ 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 $\mathcal C_E \neq \mathcal C_2$ is a linear isometry on $\mathcal C_E$.