# Maps preserving the spectrum of generalized Jordan product of operators, and its Addendum - Mathematics > Functional Analysis

Abstract: In the paper -Maps preserving the spectrum of generalized Jordan product ofoperators-, we define a generalized Jordan products on standard operatoralgebras $A 1, A 2$ on complex Banach spaces $X 1, X 2$, respectively. Thisincludes the usual Jordan product $A 1 \circ A 2 = A 1 A 2 + A 2 A 1$, and thetriple $\{A 1,A 2,A 3\} = A 1 A 2 A 3 + A 3 A 2 A 1$. Let a map $\Phi : A 1 \toA 2$ 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 therange 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. Similarresults for maps between self-adjoint operators acting on Hilbert spaces arealso obtained.After our paper -Maps preserving the spectrum of generalized Jordan productof 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 arenot entirely clear and accurate. Here we supply some details in the -Addendum-.

Author: Jinchuan Hou, Chi-Kwong Li, Ngai-Ching Wong

Source: https://arxiv.org/