Hill representations for [formula omitted]-linear matrix maps

Abstract

In the paper (Hill, 1973) from 1973 R.D. Hill studied linear matrix maps L:ℂq×q→ℂn×n which map Hermitian matrices to Hermitian matrices, or equivalently, preserve adjoints, i.e., L(V∗)=L(V)∗, via representations of the form L(V)=∑k,l=1mHklAlVAk∗,V∈ℂq×q,for matrices A1,…,Am∈ℂn×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 A1,…,Am can appear in Hill representations (provided the number m is minimal) and determine the associated Hill matrix H=Hkl explicitly. Also, we describe how different Hill representations of L (again with m minimal) are related and investigate further the implication of ∗-linearity on the linear map L.