Pairs of linear maps on matrix spaces preserving products equal to fixed elements
Let \(M_n(\mathbb C)\) denote the algebra of \(n \times n\) matrices over \(\mathbb C\). Given a linear map \(\phi: M_n(\mathbb C) \to M_n(\mathbb C)\) satisfying the property \(\phi(A)\phi(B) = M\) whenever \(AB = N\), where \(M\) and \(N\) are fixed matrices, some recent papers (such as [\textit{L. Catalano} and \textit{H. Julius}, J. Algebra 575, 220--232 (2021; Zbl 1475.15036); \textit{C. Costara}, Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat., RACSAM 116, No. 4, Paper No. 176, 10 p. (2022; Zbl 07600374)]) established that \(\phi\) is fairly close to an automorphism or an anti-automorphism of \(M_n(\mathbb C)\), usually under surjectivity conditions. Heuristically speaking, this means that automorphisms of the complex matrix algebra are mostly characterized by products equal to a single fixed matrix (instead of the abstract requirement of being multiplicative at every point). The present author varies the above problem by allowing for a pair of linear maps \(\phi,\psi: M_n(\mathbb C) \to M_n(\mathbb C)\) to satisfy the property \(\phi(A)\psi(B) = M\) whenever \(AB = N\), where \(M\) and \(N\) are certain fixed matrices. Despite the relatively weak hypotheses, much can be said; the main theorem (Theorem 1) on the description of such maps \(\phi\) and \(\psi\) may be summarized as follows. Theorem. If \(M \neq 0\) is not invertible and \(\phi\) and \(\psi\) both have invertible elements in their range, then there exist invertible matrices \(U, V, \) and \(W\) satisfying the equation \(UNW = M\) such that \[\phi(T) = UTV, \qquad \psi(T) = V^{-1}TW\] for all \(T \in M_n(\mathbb C)\). The same conclusion holds if \(M = 0\). Now, if \(M\) is invertible, then \(N\) is invertible as well and the previous description of \(\phi\) and \(\psi\) may hold but we also have the following second possible description: there exist invertible matrices \(U,V,\) and \(W\) satisfying \(UW = M\) such that \[\phi(T) = UT^tV, \qquad \psi(T) = V^{-1}(N^t)^{-1} T^tW\] for all \(T \in M_n(\mathbb C)\), where \(T^t\) denotes the transpose of \(T\). The proof of the theorem is an interesting blend of matrix theory and topological properties of \(M_n(\mathbb C)\) and its invertible elements.
scientific article; zbMATH DE number 7600374