"Maps preserving the spectrum of generalized Jordan product of operators", and its "Addendum"

TL;DR

This paper characterizes spectrum-preserving maps on operator algebras that maintain the spectrum of generalized Jordan products, showing such maps are essentially Jordan isomorphisms under certain conditions.

Contribution

It provides a detailed characterization of spectrum-preserving maps for generalized Jordan products, extending previous results and clarifying earlier proofs with additional details.

Findings

Maps preserving spectra are Jordan isomorphisms multiplied by roots of unity.

Range conditions imply the map is a Jordan isomorphism.

Results extend to self-adjoint operators on Hilbert spaces.

Abstract

In the paper "Maps preserving the spectrum of generalized Jordan product of operators", we define a generalized Jordan products on standard operator algebras $A_1, A_2$ on complex Banach spaces $X_1, X_2$, respectively. This includes the usual Jordan product $A_1 \circ A_2 = A_1 A_2 + A_2 A_1$, and the triple $\{A_1,A_2,A_3\} = A_1 A_2 A_3 + A_3 A_2 A_1$. Let a map $\Phi : A_1 \to A_2$ prserving the spectra of the products $$ \sigma (\Phi (A_1) \circ ... \circ \Phi (A_k)) = \sigma (A_1\circ ... \circ A_k) $$ whenever any one of $A_1, ..., A_k$ has rank at most one. It is shown in this paper that if the range of $\Phi $ contains all operators of rank at most three, then $\Phi $ must be a Jordan isomorphism multiplied by an $m$th root of unity. Similar results for maps between self-adjoint operators acting on Hilbert spaces are also obtained. After our paper "Maps preserving the spectrum of generalized Jordan product of operators" was published in Linear Algebra Appl. 432 (2010), 1049-1069, Jianlian Cui pointed out that some arguments in the proof of Theorem 3.1 are not entirely clear and accurate. Here we supply some details in the "Addendum".