Hill representations for *-linear matrix maps

TL;DR

This paper extends the study of Hill representations for *-linear matrix maps, providing explicit descriptions of the matrices involved, their relations, and implications of *-linearity, building on classical and recent work.

Contribution

It characterizes possible matrices in minimal Hill representations, explicitly determines the Hill matrix, and explores relations between different representations and *-linearity effects.

Findings

Explicit characterization of matrices in Hill representations

Explicit formula for the Hill matrix  H

Analysis of relations between different Hill representations

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 $\mathcal{L}$.