Maps preserving the fixed points of triple Jordan products of operators (Q298003)

Let \(\mathcal{B(X)}\) be the algebra of all bounded linear operators on a complex Banach space \(\mathcal{X}\). Denote by \(F(A)\) the set of all fixed points of \(A\in \mathcal{B(X)}\).NEWLINENEWLINEIn the paper under review, the authors study surjective maps on \(\mathcal{B(X)} \) which preserve the fixed points of Jordan triple products of operators. The main result states the following.NEWLINENEWLINELet \(\dim \) \(\mathcal{X}\geq 3\) and suppose \(\phi :\mathcal{B(X)}\) \( \rightarrow \mathcal{B(X)}\) is a surjective map which satisfies the following condition: NEWLINE\[NEWLINEF(ABA)=F(A)F(B)F(A)NEWLINE\]NEWLINE for every \(A,\) \(B\in \mathcal{B(X)}\). Then \(\phi (A)=\alpha A\) for every \( A\in \mathcal{B(X)}\), where \(\alpha \in \mathbb{C}\) and \(\alpha ^{3}=1\).NEWLINENEWLINEThe proof is divided into several lemmas. For example, the authors prove that \(\phi (0)=0\) and that \(\phi \) is injective. They also prove that \( \phi \) preserves rank one operators in both directions and that the above result holds on the set of all rank one idempotent operators, i.e., \(\phi (P)=\alpha P\) for every rank one idempotent \(P\in \mathcal{B(X)}\), where \(\alpha \in \mathbb{C}\) with \(\alpha ^{3}=1\). Finally, the authors observe that \(\phi \) is of the above form on the set of all rank one operators and then prove that the assertion holds for all operators from \(\mathcal{B(X)}\).