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

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".

1. 29 pages, the "orginal paper". 2. 5 pages, the "Addendum". 3. Replace the latex file of the "original paper" to avoid the conflict of using an old version of 'natbib' at April 23, 2010. The newer version simply does not use `natbib' at all, and nothing else is changed.

References in corpus (1)

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

Cited by in corpus (1)

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