Hill representations for *-linear matrix maps

Abstract: In a paper from 1973 R.D. Hill studied linear matrix maps $\mathcal{L}:\mathbb{C}^{q \times q}\to\mathbb{C}^{n \times n}$ which map Hermitian matrices to Hermitian matrices, or equivalently, preserve adjoints, i.e., $\mathcal{L}(V^)=\mathcal{L}(V)^$, via representations of the form \begin{equation*} \mathcal{L}(V)=\sum_{k,l=1}^m \mathbb{H}{kl}\, A_l V A_k^*,\quad V\in\mathbb{C}^{q \times q}, \end{equation*} for matrices $A_1,\ldots,A_m \in\mathbb{C}^{n \times q}$ and continued his study of such representations in later work, sometimes with co-authors, to completely positive matrix maps and associated matrix reorderings. In this paper we expand the study of such representations, referred to as Hill representations here, in various directions. In particular, we describe which matrices $A_1,\ldots, A_m$ can appear in Hill representations (provided the number $m$ is minimal) and determine the associated Hill matrix $\mathbb{H}=\left[\mathbb{H}{kl}\right]$ explicitly. Also, we describe how different Hill representations of $\mathcal{L}$ (again with $m$ minimal) are related and investigate further the implication of $*$-linearity on the linear map $\mathcal{L}$.