Pairs of linear maps on matrix spaces preserving products equal to fixed elements

Abstract

Let ${\mathcal {M}}_{n}$ be the algebra of all $n\times n$ complex matrices, and fix $M,N\in {\mathcal {M}}_{n}.$ In this paper, we characterize linear maps $\varphi ,\psi :{\mathcal {M}}_{n} \rightarrow {\mathcal {M}}_{n}$, with both $\varphi \left( {\mathcal {M}}_{n}\right)$ and $\psi \left( {\mathcal {M}}_{n}\right)$ containing at least one invertible matrix, having the property that $\varphi (A)\psi (B)=M$ whenever $AB=N$.