Linear maps transforming \(H\)-unitary matrices.
Let \(H_1\) be an \(n{\times}n\) invertible Hermitian matrix, and let \(\mathbf{U}(H_1)\) be the group of \(n{\times}n \; H_1\)-unitary matrices, i.e., matrices \(A\) satisfying \(A^{*}H_1A=H_1\). Suppose \(H_2\) is an \(m{\times}m\) invertible Hermitian matrix. The authors show that a linear transformation \(\phi:M_n\to M_m\) satisfies \(\phi(\mathbf{U}(H_1))\subseteq \mathbf{U}(H_2)\) if and only if there exist invertible matrices \(S\in M_m, U,V \in \mathbf{U}(H_2)\) such that \[ S^{*}H_2S=[(I_a\oplus-I_b)\otimes H_1]\oplus[(I_c\oplus-I_d)\otimes(H_1^{-1})^t], \] and \(\phi\) has the form \(A\mapsto US[(I_{a+b}\otimes A)\oplus(I_{c+d}\otimes A^t)]S^{-1}V\), where \(a, b, c\) and \(d\) are nonnegative integers satisfying \((a+b+c+d)n=m\). They also prove that if \(H_1\) has inertia \((p,q)\) and \(H_2\) has inertia \((r,s)\), then there is a linear transformation mapping \(\mathbf{U}(H_1)\) into \(\mathbf{U}(H_2)\) if and only if there are nonnegative integers \(u\) and \(v\) such that \((r,s)=u(p,q)+v(q,p)\). These results generalize those of Marcus, Cheung and Li [cf. \textit{M. Marcus}, Duke Math. J. 26, 156--163 (1959; Zbl 0084.01701); \textit{W. S. Cheung} and \textit{C. K. Li}, Can. Math. Bull. 46, 54--58 (2003; Zbl 1034.15004)].
scientific article; zbMATH DE number 2026890