Maps completely preserving the quadratic operators

TL;DR

This paper characterizes bijections that preserve quadratic operators between standard operator algebras, showing they are essentially isomorphisms or conjugate isomorphisms, thus revealing the structure-preserving nature of such maps.

Contribution

It establishes a complete characterization of bijections preserving quadratic operators as either isomorphisms or conjugate isomorphisms in the complex case.

Findings

Preserving quadratic operators implies the map is an isomorphism or conjugate isomorphism.

The result applies to standard operator algebras on Banach spaces.

The characterization holds in the complex case, highlighting the structure-preserving nature.

Abstract

Let $\mathcal{A}$ and $\mathcal{B}$ be standard operator algebras on Banach spaces $\mathcal{X}$ and $\mathcal{Y}$, respectively. In this paper, we show that every bijection completely preserving quadratic operators from $\mathcal{A}$ onto $\mathcal{B}$ is either an isomorphism or (in the complex case) a conjugate isomorphism.