Maps preserving peripheral spectrum of Jordan products of operators (Q2885140)

Let \({\mathcal A}\) and \({\mathcal B}\) be (not necessarily unital or closed) standard operator algebras on complex Banach spaces \(X\) and \(Y\), respectively. For a bounded linear operator \(A\) on \(X\), denote, as usual, the spectrum by \(\sigma(A)\) and let \(\sigma_\pi(A)=\{z\in\sigma(A):|z|=\max_{w\in\sigma(A)}|w|\}\) be the peripheral spectrum. For a map \(\Phi:{\mathcal A}\to{\mathcal B}\) for which the range contains all operators of rank at most two, the authors first show that \(\Phi\) satisfies NEWLINE\[NEWLINE\sigma_\pi(\Phi(A)\Phi(B)+\Phi(B)\Phi(A))=\sigma_\pi(AB+BA),\;A,B\in{\mathcal A},NEWLINE\]NEWLINE if and only if either there is a bijective linear operator \(T:X\to Y\) such that \(\Phi(A)=\pm TAT^{-1}\), \(A\in{\mathcal A},\) or \(X\) and \(Y\) are reflexive and there is a bijective linear operator \(T:X^*\to Y\) such that \(\phi(A)=\pm TA^*T^{-1}\), \(A\in{\mathcal A}.\) Next, they show that the same conclusion holds if \({\mathcal A}\) and \({\mathcal B}\) are replaced by standard real Jordan algebras of self-adjoint operators on complex Hilbert spaces. Finally, they show that, if \(X=H\) and \(Y=K\) are complex Hilbert spaces, then \(\Phi\) satisfies NEWLINE\[NEWLINE\sigma_\pi(\Phi(A)\Phi(B)^*+\Phi(B)^*\Phi(A))=\sigma_\pi(AB^*+B^*A),\;A,B\in{\mathcal A},NEWLINE\]NEWLINE if and only if either there is a scalar \(\gamma\) of modulus one and a unitary operator \(U:H\to K\) such that either \(\Phi(A)=\gamma UAU^*\), \(A\in{\mathcal A},\) or \(\phi(A)=\gamma UA^{t}U^*\), \(A\in{\mathcal A}.\)