Maps preserving peripheral spectrum of generalized Jordan products of self-adjoint operators
Summary: Let \(\mathcal{A}_1\) and \(\mathcal{A}_2\) be standard real Jordan algebras of self-adjoint operators on complex Hilbert spaces \(H_1\) and \(H_2\), respectively. For \(k \geq 2\), let \((i_1, \dots, i_m)\) be a fixed sequence with \(i_1, \dots, i_m \in\{1, \dots, k \}\) and assume that at least one of the terms in \((i_1, \dots, i_m)\) appears exactly once. Define the generalized Jordan product \(T_1 \circ T_2 \circ \cdots \circ T_k = T_{i_1} T_{i_2} \cdots T_{i_m} + T_{i_m} \cdots T_{i_2} T_{i_1}\) on elements in \(\mathcal{A}_i\). Let \(\Phi : \mathcal{A}_1 \rightarrow \mathcal{A}_2\) be a map with the range containing all rank-one projections and trace zero-rank two self-adjoint operators. We show that \(\Phi\) satisfies that \(\sigma_\pi(\Phi(A_1) \circ \cdots \circ \Phi(A_k)) = \sigma_\pi(A_1 \circ \cdots \circ A_k)\) for all \(A_1, \ldots, A_k\), where \(\sigma_\pi(A)\) stands for the peripheral spectrum of \(A\), if and only if there exist a scalar \(c \in \{- 1,1 \}\) and a unitary operator \(U : H_1 \rightarrow H_2\) such that \(\Phi(A) = c U A U^*\) for all \(A \in \mathcal{A}_1\), or \(\Phi(A) = c U A^t U^*\) for all \(A \in \mathcal{A}_1\), where \(A^t\) is the transpose of \(A\) for an arbitrarily fixed orthonormal basis of \(H_1\). Moreover, \(c = 1\) whenever \(m\) is odd.
scientific article; zbMATH DE number 7021907
scientific article