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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".

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